Extremal Norms for Fiber Bunched Cocycles

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Ma\~n\'e Lemma, in its various forms. In this paper, we prove a non-commutative Ma\~n\'e Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditions are essentially irreducibility and sufficiently strong fiber bunching. Therefore we extend the classic concept of Barabanov norm, which is used in the study of the joint spectral radius. We obtain several consequences, including sufficient conditions for the existence of Lyapunov maximizing sets.


Extremal norms
Let E be a d-dimensional real vector bundle over a compact metric space X, with projection map π. Let T : X Ñ X be a homeomorphism. We say that Φ is an automorphism of E covering T if the diagram E E X X Φ π π T commutes and moreover the restriction of Φ to each fiber E x :" π´1pxq is a linear isomorphism Φ x onto the fiber E T x . The set of such automorphisms is denoted AutpE, T q. The simplest situation is when the vector bundle is trivial, say E " XˆR d . Then Φ takes the form Φpx, uq " pT pxq, F pxquq , (1.1) for some continuous map F : X Ñ GLpd, Rq. The pair pT, F q is called a (linear) cocycle.
A Finsler norm 1 on E is a continuous map }¨} : E Ñ R whose restriction to each fiber E x is a norm. If L is a linear map from a fiber E x to another fiber E y , then we define the operator norm: When no confusion is likely to arise we denote this simply by }L}.
Fix an automorphism Φ covering T and a Finsler norm }¨}. Given x P X, the limit if it exists, is called the (first) Lyapunov exponent of Φ at the point x. The Lyapunov exponent is obviously independent of the choice of the Finsler norm. If µ is a T -invariant Borel probability measure for T , then the Lyapunov exponent χ 1 pΦ, xq exists for µ-almost every x P X; this is a well-known consequence of Kingman's subadditive ergodic theorem; see e.g. [Kr]. Let us denote χ 1 pΦ, µq :" ş χ 1 pΦ,¨q dµ. If the measure µ is ergodic then χ 1 pΦ, xq " χ 1 pΦ, µq for µ-almost every x P X.
In this paper we are interested in the maximal Lyapunov exponent, defined as: βpΦq :" sup µPMT χ 1 pΦ, µq , where M T denotes the set of all T -invariant Borel probability measures. The supremum is always attained by an ergodic measure -this follows from upper semicontinuity of χ 1 pΦ,¨q with respect to the weak-star topology, and the fact that M T is a compact convex set whose extreme points are exactly the ergodic measures. Let us mention that the maximal Lyapunov exponent can also be characterized in more elementary terms as follows: (1.4) (We use "linf" to denote a limit that is also an infimum.) These equalities follow from general results on "subadditive ergodic optimization": see [Mo3,Appendix A].
A trivial upper bound for the maximal Lyapunov exponent, which depends on the chosen Finsler norm, is given by: (1.5) If equality holds then }¨} is called an extremal norm for Φ. More precisely, the norm is so "tight" that there is no vector u ‰ 0 in E whose expansion factor }Φpuq}{}u} exceeds the maximum asymptotic expansion rate e βpΦq . In particular, if βpΦq ď 0 then the extremal norm is a (non-strict) Lyapunov function for Φ. Extremal norms first appeared in the 1960 paper [RS] by Rota and Strang, who considered the particular setting of one-step cocycles (details are given below), but apparently were not considered in our level of generality before.
The existence of an extremal norm is far from automatic 2 , and has strong consequences. In this paper we construct extremal norms for a large and natural class of vector bundle automorphisms.

Previous results
Consider the case of an 1-dimensional vector bundle E, with an arbitrary Finsler norm }¨}. Given Φ P AutpE, T q, there exists a unique continuous function f : X Ñ R such that u P E x ñ }Φpuq} T x " e f pxq }u} x . (1.6) Note that in this case the maximal Lyapunov exponent βpΦq equals: βpf q :" sup µPMT ż f dµ . (1.7) Any other Finsler norm |||¨||| is of the form: |||u||| x " e hpxq }u} x , for some continuous function h : X Ñ R. Then |||¨||| is a extremal norm if and only if h satisfies the "cohomological inequality": Such a function h is called a subaction for pT, f q. Existence of subactions can fail dramatically: see e.g. [BJ,§3] and [Gar, Appendix]. However, if the dynamics T is in some sense hyperbolic (e.g., a shift) and the function f is regular enough (e.g., Hölder) then subactions h do exist. Results of this type are sometimes called Mañé lemmas; see [CG,CLT,Bou1,Bou2]. (See also [Boc2,Prop. 2.1] for a negative result.) Important applications include [BMa, Con]. The study of invariant measures that attain that supremum in (1.7) is called ergodic optimization; we refer the reader to [Je1,Je2,Gar] for much more information.
For a discussion of ergodic optimization in a more general context, including optimization of Lyapunov exponents, see [Boc2].
When dim E ą 1, commutativity is lost and much less is known. The most studied situation is the following one. Let T : X Ñ X be the full shift on N symbols, defined on the space X :" t0, 1, . . . , N´1u Z . Given a N -tuple pA 0 , . . . , A N´1 q of invertible dˆd matrices, let F : X Ñ GLpd, Rq be given by F pxq " A x0 . We say that pT, F q is a one-step cocycle. Let Φ the associated automorphism (1.1). In that case, the quantity e βpΦq is known as the joint spectral radius of the set tA 0 , . . . , A N´1 u. 3 It was introduced by Rota and Strang [RS].
If, for example, N " 1 and A 0 " p 1 1 0 1 q, then no extremal norm exists. However, if the set tA 0 , . . . , A N´1 u is irreducible, in the sense that there is no common invariant non-trivial subspace, then extremal norms |||¨||| do exist, and can be taken so that |||u||| x is independent of x P X. Actually, Barabanov [Bar] proved that there exists a norm |||¨||| on R d with the following stronger property: |||A i u||| " e βpΦq |||u||| . (1.8) For more information on the joint spectral radius and Barabanov norms, see [Wir, Jun]. Further applications of extremal norms were obtained by Morris [Mo2,Mo3]. Still in the setting of one-step cocycles, a modification of the concept of Barabanov norm was used in [BR, BMo] to study Lyapunov-maximizing and also Lyapunov-minimizing measures.
Extremal norms for certain locally constant cocycles over sofic shifts are studied in the paper [PEDJ].
The main purpose of this paper is to establish existence of extremal norms in a far more general setting.

The main result
We now describe the hypotheses on the automorphism Φ and the underlying dynamics T from which we will prove the existence of extremal norms. We first describe then informally, leaving the precise definitions for later sections.
First, we assume that T : X Ñ X is a transitive hyperbolic homeomorphism of a compact metric space X. Hyperbolicity means that T has local stable and unstable sets with uniform exponential bounds, which satisfy a local product property. Examples include subshifts of finite type and Anosov diffeomorphisms.
Second, we assume that the vector bundle E has a Hölder structure, and that the automorphism Φ respects this structure. In the case of trivial vector bundles, this means that the matrix function F in formula (1.1) is Hölder continuous.
Third, we assume that the automorphism Φ is fiber bunched. In crude terms, this means that the non-conformality of the linear maps Φ x is small when compared to the hyperbolicity rates of T . The precise condition involves the Hölder exponent of the automorphism, so that more regular automorphisms are allowed to be less conformal. In the case that T and Φ are differentiable, fiber bunching means that the projectivization of Φ is a partially hyperbolic diffeomorphism.
Actually, for d ě 3 we need to assume a stronger form of fiber bunching. Our last assumption is irreducibility, meaning that Φ admits no nontrivial regular subbundle, where regular means as regular as the automorphism itself. We remark that this condition is satisfied for typical fiber bunched automorphisms: it holds on an open and dense subset of infinite codimension.
The main result of this paper is that under the conditions above, extremal norms exist. See Corollary 5.1 for a precise statement.
In the case where the base dynamics T is a subshift of finite type, we are able to improve our main result and obtain an extremal norm with a further property akin to the Barabanov property: see Subsection 5.3. Classical Barabanov norms are usually non-Riemannian (that is, they do not come from inner products), and it is easy to produce examples 4 . On the other hand, in our setting, there is more flexibility as the norm is allowed to depend on the basepoint. So one could wonder if the Finsler extremal norms in our main result could be taken Riemannian. Unfortunately, that is not the case: we construct an explicit example in Appendix B.2.

Consequences
As a consequence of our result on the existence of extremal norms, we can show that the maximal Lyapunov exponent is a locally Lipschitz function on the space of strongly bunched irreducible automorphisms (see Proposition 5.4 for a more precise statement), thus extending a result of Wirth [Wir].
We are also able to obtain several general properties of strongly bunched automorphisms Φ (not necessarily irreducible): • Their growth obeys certain uniform bounds: see Theorem 4.6.
• They obey the subordination principle: if µ and ν are invariant probability measures such that ν is Lyapunov maximizing in the sense that χ 1 pΦ, νq " βpΦq, and supp µ Ď supp ν, then µ is Lyapunov maximizing as well: see Theorem 7.1. This property is far from being tautological, even in the commutative setting; in fact it was introduced in this setting by Bousch [Bou1].
• The maximal Lyapunov exponent βpΦq can be approximated by Lyapunov exponents of measures supported on periodic orbits, and moreover the 4 The pair of matrices (B.1) is one such example.
quality of this approximation is superpolynomial with respect to the period: see Theorem 7.2. This extends a result of Morris [Mo2], who gave a quantitative version of the celebrated theorem of Berger-Wang [BeW].
We also introduce Mather sets in our context; these sets are the habitat of Lyapunov maximizing measures. We prove an important structural result on the existence of dominated splittings on the Mather sets, namely Theorem 6.5, which is an essential ingredient in the proof of the aforementioned Theorem 7.2.

Organization of the paper
In Section 2 we introduce the setting for our results, providing the definitions and properties of fiber bunched automorphisms and related concepts.
In Section 3 we study irreducibility and related concepts.
In Section 4 we provide sufficient conditions for relative product boundedness, an intermediate property which is required for the existence of extremal norms.
The construction of extremal norms is given in Section 5, together with the construction of Barabanov-like norms for shifts and an application to the regularity of βp¨q.
In Section 6 we introduce Mather sets in a very general setting and, under the assumption of existence of an extremal norm, establish fine properties about them.
In Section 7 we collect several applications of our results. Appendix A contains the proofs of several subsidiary results, therefore making the paper self-contained.
In Appendix B we exhibit some "pathological" examples, including an example that fits in the setting of our main results, but where no Riemannian extremal norm exists.

The fiber bunched setting
In this section, we fix the basic setting for our theorems. Namely, we define and state the basic properties of Hölder vector bundles, intrinsically hyperbolic homeomorphisms, fiber bunching, holonomies, and irreducibility. Our approach is influenced by [BGV, Via, KS], and we tried to make it as general as possible. We also obtain some new regularity results that are essential for the main theorems of the paper. However, to make the presentation more fluid, we postpone most proofs to Appendix A.

The Hölder exponent
From now on, assume that pX, dq is a compact metric space. We also fix θ ą 0 such that the algebra of θ-Hölder functions on X is normal, that is, given any two disjoint compact subsets of X, there exists a θ-Hölder function that takes values in the interval r0, 1s and equals 0 on one set and 1 on the other. This assumption is automatically satisfied if θ ď 1. If X is a Cantor set, then the assumption holds for any θ ą 0. Normality implies the existence of θ-Hölder partitions of unity: see e.g. [Katz,p. 221].

Hölder vector bundles
Let E be a d-dimensional vector bundle over X. We recall the definition and fix the terminology. E is a topological space endowed with a continuous map π : E Ñ X (called the projection), a cover of X by open sets U i (called coordinate neighborhoods), and a family of homeomorphisms (called coordinate maps) which is required to have the following compatibility property: whenever x P U i X U j , the map is linear. Therefore we obtain a family of continuous maps: which are called coordinate transformations. Moreover, each fiber E x :" π´1pxq has a unique structure of d-dimensional vector space such that the maps become isomorphisms. Since X is assumed to be compact, we will from now on assume that the cover tU i u is finite.
We say that E is a θ-Hölder vector bundle if the coordinate transformations (2.1) are locally θ-Hölder. By compactness, we can reduce the coordinate neighborhoods so that the coordinate transformations become (uniformly) θ-Hölder.
As mentioned in Subsection 1.1, a Finsler norm is a continuous function }¨} on E that restricts to a norm }¨} x on each fiber E x . A Finsler norm }¨} is called Riemannian if each }¨} x is induced by an inner product x¨,¨y x . A Finsler norm }¨} is called θ-Hölder if for every u P R d and every coordinate neighborhood, the function x P U i Þ Ñ }h i pxqu} is θ-Hölder. Every θ-Hölder vector bundle E admits a θ-Hölder Riemannian norm; the proof is straightforward using a θ-Hölder partition of unity.
We will also need a way of "transporting" vectors from one fiber to another: Proposition 2.1 (Transport maps). Let E be a θ-Hölder vector bundle. There exists a family of linear maps I yÐx : E x Ñ E y with the following properties: (a) For every point x P X, the linear map I xÐx equals the identity.
(b) For every pair of indices i, j, the matrix-valued map px, yq P U iˆUj Þ Ñ rh j pyqs´1˝I yÐx˝hi pxq is θ-Hölder.
See Appendix A.1 for the proof of Proposition 2.1. The next propositions, also proved in Appendix A.1, give additional quantitative properties of the transport maps that will be useful in subsequent calculations. Recall that we agree to denote a norm and its induced operator norm by the same symbol, as in (1.2).
Proposition 2.2. Let E be a θ-Hölder vector bundle, endowed with a Finsler norm. Let tI yÐx u be the family of transport maps provided by Proposition 2.1. Then there is C ą 0 such that for all points x, y, z P X, }I yÐz˝IzÐx´IyÐx } ď C maxtdpx, zq θ , dpy, zq θ u , Proposition 2.3. Let E be a θ-Hölder vector bundle, endowed with a Finsler norm }¨}. Let tI yÐx u be the family of transport maps provided by Proposition 2.1. Then the Finsler norm }¨} is θ-Hölder if and only if there exists C ą 0 such that for all points x, y P X,ˇ} I yÐx }´1ˇˇď Cdpx, yq θ .

θ-Hölder bundle automorphisms
Assume that E is a θ-Hölder vector bundle over the compact metric space X. Fix a θ-Hölder Riemannian norm on E.
A map Φ : E Ñ E is called an endomorphism of E if there exists a continuous map T : X Ñ X such that π˝Φ " T˝π (we say that Φ covers T ) and for each x P X, the restriction of Φ to the fiber E x is a linear map Φ x to the fiber E T x . If T is a homeomorphism and each Φ x is a isomorphism then we say that Φ is an automorphism.
We say that the endomorphism Φ covering T is θ-Hölder if T is Lipschitz and the maps x P U i X T´1pU j q Þ Ñ rh j pT xqs´1˝Φ x˝hi pxq P GLpd, Rq are θ-Hölder. 5 As an immediate consequence, the function x P X Þ Ñ }Φ x } is θ-Hölder.
We can characterize θ-Hölder automorphisms in terms of the transport maps from Proposition 2.1: Proposition 2.4. An endomorphism Φ : E Ñ E covering a Lipschitz map T is θ-Hölder if and only if there exists K ą 0 such that for all x, y P X, we have Next, we want to topologize the set of θ-Hölder automorphisms. Let End θ pE, T q denote the vector space of θ-Hölder endomorphisms Φ : E Ñ E covering T . Define the C 0 norm: (2. 3) The stronger θ-Hölder norm makes End θ pE, T q a Banach space: The set Aut θ pE, T q of θ-Hölder automorphisms is a C 0 -open subset of End θ pE, T q. Given K ě 1, let:

Hyperbolic homeomorphisms
The concept of hyperbolicity in differentiable dynamical systems was introduced by Anosov [Ano] and Smale [Sma]. Even without recourse to a differentiable structure, it is possible to define hyperbolicity (and to prove interesting theorems); this has been done in various ways: [Bow, Rue, AY, Akin, AH]. In this paper, we will use essentially the same definition of hyperbolic homeomorphism given by Sakai [Sak].
Recall that X is a compact metric space. Let T : X Ñ X be a homeomorphism. Given x P X and ε ą 0, we define the following sets: • local unstable set W u ε pxq :" y P X ; dpT´ny, T´nxq ď ε for all n ě 0 ( ; • local stable set W s ε pxq :" y P X ; dpT n y, T n xq ď ε for all n ě 0 ( . We say that T is a hyperbolic homeomorphism if the following axioms hold: (a) T is bi-Lipschitz, i.e., both T and T´1 are Lipschitz; (b) there exist a constant ε 0 ą 0 and a pair of continuous positive functions λ u , λ s (called the hyperbolicity exponents) such that: x P X, x 1 , x 2 P W u ε0 pxq ñ dpT´1x 1 , T´1x 2 q ď e´λ u pxq dpx 1 , x 2 q , (2.6) y P X, y 1 , y 2 P W s ε0 pyq ñ dpT y 1 , T y 2 q ď e´λ s pyq dpy 1 , y 2 q ; (2.7) (c) there exists a constant ε 1 P p0, ε 0 q such that for any pair of points x, y P X with dpx, yq ď 2ε 1 , the intersection W u ε0 pxq X W s ε0 pyq contains exactly one point, denoted by rx, ys and called bracket, which depends continuously on x and y; (d) there exists a constant C ą 0 such that: x, y P X, dpx, yq ď 2ε 1 ñ max dprx, ys, xq, dprx, ys, yq ( ď Cdpx, yq . (2.8) Remark 2.5. Sakai [Sak] uses the terminology L-hyperbolic homeomorphism, while Ruelle [Rue] uses Smale spaces. Modulo a change of metric, both definitions are equivalent to ours, and also to expansivity plus the shadowing property: see [Sak] and references cited there.
Let us also define other sets associated with T : • unstable set W u pxq :" ty P X ; dpT´ny, T´nxq Ñ 0 as n Ñ`8u; • stable set W s pxq :" ty P X ; dpT n y, T n xq Ñ 0 as n Ñ`8u; If T is a hyperbolic homeomorphism then, as an immediate consequence of part (b) of the definition, for every ε P p0, ε 0 s we have the following set relations: T´npW s ε pT n xqq . (2.9) The transverse regularity of the unstable and stable sets is a classical subject, and fine results about hyperbolicity rely on it: see [KH,Chapter 19]. Nevertheless, we could not find a reference for the following property for hyperbolic homeomorphisms: Proposition 2.6. Let T be a hyperbolic homeomorphism. There exist constants 0 ă κ s ď 1 and C ą 0 such that if x, x 1 , y, y 1 P X satisfy (see Fig. 1): The proof, which includes an estimate for the constant κ s , is given in Appendix A. If T is the restriction of a C 2 diffeomorphism to a basic hyperbolic basic set then a better estimate for κ s is given in [SS]. Even better regularity estimates can be obtained under various types of extra assumptions: see [PR] and references therein.

Fiber bunched automorphisms and their holonomies
We now discuss the notion of fiber bunching. It was introduced in a setting very similar to ours by Bonatti, Gómez-Mont, and Viana [BGV], though related concepts can be traced back to Brin and Pesin [BP] and Hirsch, Pugh, and Shub [HPS]. The paper [BGV] uses a different terminology ("dominated cocycles"), but subsequently the term "fiber bunched cocycles" prevailed: [AV1,AV2,KS].
If L is a linear isomorphism between inner product spaces, we define its bolicity 6 as bolpLq :" }L} }L´1} , (2.11) which measures the lack of conformality of L (see Proposition 3.1). Let E be a θ-Hölder d-dimensional vector bundle over X. Assume that T is a hyperbolic homeomorphism, and that Φ is a θ-Hölder automorphism of E covering T . We say that E is fiber bunched if there exists a Riemannian norm (sometimes called an adapted norm) such that for all x P X, log bolpΦ x q ă min tθλ u pxq, θλ s pxqu , (2.12) where λ u , λ s are the hyperbolicity rates of T . By perturbing the adapted norm if necessary, we can assume it is also θ-Hölder, if necessary. Consider the subset of fiber bunched automorphisms in the space Aut θ pE, T q of θ-Hölder automorphisms; then this set is open with respect to the C 0 norm (2.3), and therefore also open with respect to the stronger θ-Hölder norm (2.4).
Sometimes we need stronger bunching: we say that it is pη u , η s q-bunched for certain constants η u , η s P p0, θs if, for some adapted norm, and all x P X, log bolpΦ x q ă min tη u λ u pxq, η s λ s pxqu . (2.13) Remark 2.7. We have used the pointwise definition of fiber bunching; the more stringent notion of absolute fiber bunching requires the same condition with constant hyperbolicity exponents λu, λs. Furthermore, our definition of fiber bunching is immediate in the sense that it manifests itself in a single iterate; one can also define a notion of eventual fiber bunching.
The most basic and fruitful consequence of fiber bunching is the existence of certain unstable and stable holonomy maps. Like the transport maps from Proposition 2.1, unstable and stable holonomy maps provide a way of linearly transporting vectors from a fiber E x to another fiber E y (as long as the points x, y belong to the same unstable or stable set), but with several extra properties: Proposition 2.8 (Holonomy maps). Let Φ P Aut θ K pE, T q be a fiber bunched automorphism. For each ‹ P tu, su, there exist a unique family of linear maps H ‹ yÐx : E x Ñ E y , defined whenever y P W ‹ pxq, such that the following properties hold: There exist a constant C ą 0 such that: (2.14) (e) The following map is continuous: Furthermore, the constant C in (2.14) works for all automorphisms in a C 0neighborhood of Φ in Aut θ K pE, T q, and the the right-hand side in (2.15) depends continuously on the automorphism in that neighborhood.
The maps H u and H s are called unstable and stable holonomies, respectively. Properties (a) and (b) are called groupoid properties, and property (c) is called equivariance.
The stable holonomies are actually defined by the following formula: and unstable holonomies are defined likewise, taking n Ñ´8 instead. The proof of Proposition 2.8 consists essentially in proving uniform convergence in this formulas, and it turns out that fiber bunching is the precise condition for this to work. Except for minor adjustments, the argument is the same as in [BGV,§1.4], [KS,§4.1], but for completeness and convenience of the reader we present the proof in Appendix A.2.
Remark 2.9. Fiber bunched automorphisms satisfy a non-commutative version of Walters' condition [Bou1], namely: Indeed, consider z :" rx, ys and note the following identity: Then, using the continuity of the bracket and the regularity of the holonomies, it is straightforward to obtain the non-commutative Walters' condition.
We use the holonomies to define certain subsets of E. For ε ą 0, u P E, and x " πpuq, let: Analogously we define W s ε puq and W s puq. The sets W u (resp. W s ) form a Φinvariant partition of E and project by π onto the sets W u (resp. W s ).
Part (d) of Proposition 2.8 basically says that the "leaves" W u , W s are Hölder-continuous. We will need the transverse regularity of the holonomies: Proposition 2.10. Let Φ P Aut θ K pE, T q be a fiber bunched automorphism. There exist θ s P p0, θκ s s and C ą 0 such that if x, x 1 , y, y 1 P X satisfy conditions (2.10) as in Fig. 1 then: Furthermore, the same constants θ s and C work for every automorphism in a C 0 -neighborhood of Φ in Aut θ K pE, T q. We were not able to find such a statement in the literature, so we provide a proof in Appendix A.3 3 Invariant subbundles

Subbundles and rigidity
Let E be an inner product space of dimension d, and let p P v1, dw. We denote by G p pEq the p-th Grassmannian of E, i.e., the set of the all p-dimensional subspaces of E. There are many metrics on this set that are "natural" in the sense that they are preserved by the action of orthogonal linear maps: see [QZL]. As shown in Appendix A.4, we can find one such metric d with the useful properties stated in the following propositions: Proposition 3.1. If L : E Ñ F is a linear isomorphism between d-dimensional inner product spaces and p ă d, then the induced map G p pEq Ñ G p pF q is Lipschitz with a constant equal to the bolicity of L (2.11).
Proposition 3.2. If L : E Ñ E is a linear isomorphism of a inner product space such that }L´id} ď δ then the induced map on G p pEq is Opδq-close to the identity, provided δ is sufficiently small. Proposition 3.3. The map that associates to a p-tuple of linearly independent vectors its span is locally Lipschitz.
In particular, the metric d induces the usual topology on the Grassmannian. Now consider a θ-Hölder d-dimensional vector bundle E over X. For each p P v1, d´1w, let G p pEq denote the fiber bundle whose fiber over x P X is G p pE x q. As just explained, the fixed Riemannian norm on E induces a distance on each fiber of this bundle.
Let F be a continuous p-dimensional subbundle of E. We say that F is η-Hölder, for some η P p0, θs, if there exists C ą 0 such that for all sufficiently close points x, y P X we have: where the d in the left hand side is the distance in G p pE y q. (Note that I yÐx is an isomorphism when x and y are close enough.) Let Φ be a fiber bunched automorphism of E. We say that a subbundle F Ď E is Φ-invariant if for all x P X, we have We say that F is H u -invariant (or W u -saturated ) if for all x P X and all y P W u pxq, we have H u yÐx pF x q " F y . We say that F is η-Hölder along unstable sets, for some η P p0, θs, if there exists C ą 0 and ε ą 0 such that the estimate (3.1) holds whenever y P W u ε pxq. Equivalently, there exists C ą 0 for all x P X and all y P W u ε0 pxq, we have: To see the equivalence, use θ-Hölderness of the holonomy (2.14) and Proposition 3.2. H s -invariance and η-Hölderness along stable sets are defined analogously.
Proposition 3.4 (Rigidity). Let η P p0, θs. Suppose that Φ is pη, θq-bunched. Let F Ď E be a continuous Φ-invariant subbundle. If F is η-Hölder along unstable sets then F is H u -invariant, and in particular F is actually θ-Hölder along unstable sets.
Proof. Since Φ is pη, θq-bunched, there is a constant r P p0, 1q such that bolpΦ x q ă re ηλupxq for every x P X. Now fix x P X and y P W u ε0 pxq. For each n ě 0, let x n :" T´nx and y n :" T´ny. Then: d`F y , H u yÐx pF x q˘" d`Φ n yn pF yn q, Φ n yn pH u ynÐxn pF xn q˘(by Φ-invariance of F) ď bolpΦ n yn q d`F yn , H u ynÐxn pF xn q˘(by Proposition 3.1) ď C bolpΦ n yn q dpx n , y n q η (by η-Hölderness of F).
Combining these estimates, we have: where B ě 1 is the maximal bolicity. As n Ñ 8, the right hand side tends to zero. So F y " H u yÐx pF x q, proving that the subbundle F is H u -invariant.
Corollary 3.5. Let Φ be a fiber bunched automorphism of E. Let F Ď E be a Φ-invariant continuous subbundle. Then the following conditions are equivalent: (a) F is a θ-Hölder subbundle; Proof. If condition (a) holds then F is θ-Hölder along unstable sets, and so Proposition 3.4 (with η " θ) guarantees that F is H u -invariant. By symmetry, F is also H s -invariant. That is, condition (b) holds. Conversely, assume that condition (b) holds, and consider a pair x, y of nearby points. Then the bracket z :" rx, ys is well-defined, and by property (2.8), it is Opdpx, yqq-close to either x or y. By hypothesis, F y " H s yÐz˝H u zÐx pF x q. Using Proposition 2.2 and θ-Hölderness of the holonomies (2.14), we see that }I yÐx´H s yÐz˝H u zÐx } " Opdpx, yq θ q. It follows from Proposition 3.2 that dpF y , I yÐx pF x qq " Opdpx, yq θ q, i.e., condition (a) holds.

Irreducibility
The trivial subbundles of E are the zero section and E itself. A fiber bunched automorphism Φ is called reducible if it has a nontrivial Φ-invariant subbundle F satisfying any of the equivalent conditions of Corollary 3.5, and irreducible otherwise.
While the existence of continuous Φ-invariant subbundles is common, the existence of θ-Hölder ones is not. For example, if the automorphism admits a dominated splitting 7 , then the two subbundles are Φ-invariant, continuous, and actually Hölder, but usually with smaller Hölder exponent. Actually, the dominating bundle is H u -invariant and so θ-Hölder along unstable sets, but usually not so well behaved along stable sets.
A precise formulation of the fact that reducibility is uncommon among fiber bunched automorphisms is provided by Proposition A.6.
Remark 3.6. Irreducibility is weaker than the twisting condition used in [AV1].

The strong bunching hypothesis
If d " 2 then ordinary fiber bunching suffices for our main results, while if d ě 3 we need Φ to be not only fiber bunched, but pη 0 , θq-bunched, where η 0 is given by the following: Lemma 3.7. There exists η 0 P p0, θs that depends only on the hyperbolic homeomorphism T (or, more precisely, on its hyperbolicity exponents) and on the Hölder exponent θ such that if Φ is a pη 0 , θq-bunched automorphism then the associated regularity exponent θ s from Proposition 2.10 satisfies: Let us say that a θ-Hölder automorphism Φ : E Ñ E covering T is strongly bunched if: • the vector bundle E has fibers of dimension d ď 2, and Φ is fiber bunched; or The precise point of our proofs where we need strong bunching is for the validity of Theorem 3.8, explained in the next subsection.

Spannability
The following concept of spannability will play an important role in this paper; it is vaguely similar to the concept of accessibility in partially hyperbolic dynamics (see e.g. [Pes,§8.1]).
Let us say that a fiber bunched automorphism Φ is spannable if for all x, y P X, and all nonzero u P E x , there exist: • points x 1 , . . . , x d P W u pxq; • integers n 1 , . . . , n d ě 0 such that the points y i :" T ni x i all belong to W s pyq; with the property that the vectors v 1 , . . . , v d P E y defined by form a basis for E y . It is clear that every spannable automorphism is irreducible. The following important result provides a converse under extra assumptions: Theorem 3.8 (Sufficient conditions for spannability). Let T be a transitive hyperbolic homeomorphism. Let Φ be a strongly bunched irreducible automorphism covering T . Then Φ is spannable.
In particular (see Proposition A.6), typical strongly bunched automorphisms are spannable (provided T is transitive).
It would be interesting to know whether or not strong bunching is really necessary for the validity of Theorem 3.8.
In order to prove the theorem, we need the following easy property of the unstable and stable sets for the base dynamics: Lemma 3.9. For every x P X, the sets Ť ně0 W u pT n xq and Ť ně0 W s pT´nxq are dense in X.
Proof. Let D be the set of points whose forward orbits are dense. Since T is transitive, D is itself dense. Moreover, D is W s -saturated (i.e., it is a union of stable sets). By definition of hyperbolic homeomorphism, local stable and unstable sets whose basepoints are sufficiently close always intersect. It follows that D intersects all unstable sets. This implies that for every x P X, the set Ť ně0 W u pT n xq is dense. Applying this to T´1 we obtain that Ť ně0 W s pT´nxq is also dense.
Proof of Theorem 3.8. Fix a point x P X and a nonzero vector u P E x . Let Λ :" Ť ně0 W u pT n xq, which by Lemma 3.9 is a dense subset of X. Define the following subsets of the vector bundle E: where the latter equation means that for each y P X, the fiber F y :" E y X F is the vector space spanned by S y :" E y X S. In order to prove the theorem, we need to show that F " E. Clearly, • U projects onto Λ, and is both forward-Φ-invariant (i.e., ΦpUq Ď U) and W u -saturated (i.e., it is a union of W u sets); • S projects onto X, and is both forward-Φ-invariant and W s -saturated; therefore F has the same properties.
We claim that the function y P X Þ Ñ dim F y has the following properties: (a) it is non-decreasing along orbits of T (i.e., dim F T y ě dim F y ); (b) it is constant along W s sets; (c) it is lower semicontinuous.
Indeed, properties (a) and (b) follow from the facts that F is forward-invariant and W s -saturated, respectively. In order to check property (c), fix an arbitrary point y P X and let p :" dim F y . Then there exist points x 1 , . . . , x p P W u pxq and integers n 1 , . . . , n p ě 0 such that the points y i :" T ni x i all belong to W s pyq, and the vectors v i given by formula (3.2) span F y . If y 1 is sufficiently close to y, then for each i we can find y 1 i P W u py i q X W s py 1 q such that the holonomies H u yiÐy pv i q is close to v i , and so the span of tv 1 1 , . . . , v 1 p u has dimension p. Since each v 1 i belongs to F y 1 , we conclude that dim F y 1 ě p, therefore proving the semicontinuity property (c).
Let C be the set of points y P X where dim F y attains its minimum. By the properties (a), (b), and (c) that we have just proved, the set C is nonempty, backwards-invariant (i.e., T´1pCq Ď C), W s -saturated, and closed. It follows from Lemma 3.9 that C " X. In other words, F has constant dimension, say p.
Let θ s be given by Proposition 2.10. We claim that F is θ s -Hölder along unstable sets, in the sense defined in Subsection 3.1. By compactness, it suffices to prove this claim on a neighborhood of each point y P X. Take points x 1 , . . . , x p P W u pxq and integers n 1 , . . . , n p ě 0 such that the points y i :" T ni x i all belong to W s pyq, and the vectors v i given by formula (3.2) span F y . Take k ě 0 large enough so that the points T k y i all belong to W s ε1 pT k yq, where ε 1 is constant from condition (c) in the definition of hyperbolic homeomorphism. If we prove that F is θ s -Hölder along unstable sets on a neighborhood of T k y then, by invariance, it will follow that F is θ s -Hölder along unstable sets on a neighborhood of y. So let us assume that k " 0, for simplicity of notation. Let y 1 P W u ε1 pyq be close to y. Then the brackets y 1 i :" ry i , y 1 s are well-defined; see Fig. 2. We need to compare the following two subspaces of E y 1 : By Proposition 2.10, } 1´2 } " Opdpy, y 1 q θs q; moreover }H s by Proposition 3.3, we conclude that d`F y 1 , H u y 1 Ðy pF y q˘" Opdpy, y 1 q θs q. This concludes the proof that F is θ s -Hölder along unstable sets. A fortiori, F is continuous (since it is invariant under stable holonomies). Figure 2: Proof of Theorem 3.8 The proof ends differently according to the dimension of E. If d " 1 then F " E and we are done.
Next, consider the case d " 2. Assume by contradiction that F ‰ E, i.e., that F has 1-dimensional fibers. The set U defined at the beginning has 1-dimensional fibers U y for every y P Λ, and since U Ď F, these fibers must coincide with F y . But Λ is dense in X and F is continuous. Therefore F is the closure of U. In particular, F is W u -saturated. Recalling that F is also W s -saturated, we contradict irreducibility. This concludes the proof in the case d " 2. Now consider the case d ě 3. Then, by definition of strong bunching, Φ is pη 0 , θq-bunched. Recall from Lemma 3.7 that η 0 ď θ s . So Proposition 3.4 (rigidity) applies and the regularity of the subbundle is upgraded: it is actually θ-Hölder along unstable sets. Irreducibility implies that F " E, thus concluding the proof.
We will use an apparently stronger, but equivalent form of spannability. Recall that ε 1 ą 0 is one of the constants that appear in the definition of hyperbolic homeomorphism (Subsection 2.4).
Proposition 3.10 (Uniform spannability). Suppose Φ is a spannable automorphism. Then there exist constantsn ě 0 and C 0 ą 0 with the following properties: For all points x, y P X, and all unit vectors u P E x , there exist: • points x 1 , . . . , x d P W u ε1 pxq; • integers n 1 , . . . , n d P v0,nw such that the points y i :" T ni x i all belong to W s ε1 pyq; with the property that the vectors v 1 , . . . , v d P E y defined by (3.2) form a basis for E y ; moreover, if L : E y Ñ E y is a linear map that sends this basis to an orthonormal basis then }L} ă C 0 .
Proof. If u P E is a nonzero vector, let rus denote its class in the projective bundleÊ :" G 1 pEq. Let Φ be a spannable automorphism. Given prus, yq PÊˆX, consider x " πpuq and let x i , n i , y i , and v i , where i P v1, dw, be as in the definition of spannability. Note that if prũs,ỹq belongs to a sufficiently small neighborhood of prus, yq then we can find the corresponding data px i ,ñ i ,ỹ i , rṽ i sq close to px i , n i , y i , rv i sq (soñ i " n i ) and actually depending continuously on prũs,ỹq. Since the spaceÊˆX is compact, we can cover it by finitely many such neighborhoods U j . We can also assume that the sets U j are compact. Fix any set U j and an element prus, yq P U j . Let x " πpuq and let px i , n i , y i , rv i sq, i P v1, dw be the corresponding spannability data. For each k ě 0, the pair prΦ´kpuqs, T k yq PÊˆX has pT´kx i , n i`2 k, T k y i , rΦ k pv i qsq, i P v1, dw as valid spannability data. By (2.9), if k is large enough then By continuity of the spannability data on the compact set U j , this conclusion holds provided k is bigger than some k j . There are finitely many indices j to consider, so let us fix a definitive k bigger than all k j 's. The compact sets V j :" tprΦ´kpuqs, T k yq ; prus, yq P U j u also cover the spaceÊˆX. They provide the spannability data with the required uniformity properties. This proves the proposition.
Corollary 3.11. Given a spannable automorphism Φ P Aut θ K pE, T q, we can choosen ě 0 and C 0 ą 0 such that the the statement of Proposition 3.10 holds for all automorphisms in a C 0 -neighborhood of Φ in the space Aut θ K pE, T q. In particular, spannable automorphisms form a C 0 -open subset of Aut θ K pE, T q. Proof. By part (e) of Proposition 2.8, holonomies depend continuously on the fiber-bunched automorphism Φ, with respect to the C 0 -norm. So, in the situation of Proposition 3.10, if we make a C 0 -perturbation of Φ (among θ-Hölder automorphisms) then the vectors v 1 , . . . , v d change little and therefore stay linearly independent.
Of course, if Φ has an extremal norm then it is relatively product bounded. The converse is true in the 1-step case, as noted by Rota and Strang [RS]. But the converse is not true in general 8 ; in fact it may fail even in dimension 1, as shown by Morris [Mo1, Proposition 2]. In Morris' example, the dynamics is uniformly hyperbolic (actually a full shift), but the function is not Hölder.
In this paper, we need to prove relative product boundedness as an essential preliminary step in the construction of extremal norms. We will show the following: Proposition 4.1. Every spannable automorphism is relatively product bounded.
The proof, which will occupy Subsections 4.2 and 4.3, is roughly as follows: first, we find pieces of W u sets of uniform size that stay relatively product bounded for a long time (Lemma 4.3), then we use a compactness argument to find small pieces of W u sets that stay relatively product bounded forever (Lemma 4.4), and finally we use spannability to spread this property to the whole bundle (Lemma 4.5).
Let us close this subsection with some remarks about product boundedness and relative product boundedness.
It was shown by Blondel and Tsitsiklis [BT] that the product boundedness of a pair of rational matrices is algorithmically undecidable.
A result of Coronel, Navas, and Ponce [CNP] states if T is a minimal homeomorphism (i.e. all its orbits are dense) and Φ and Φ´1 are both product bounded then there exists an invariant Riemannian norm.
It is easy to give examples of regular (e.g. Hölder) automorphisms that are not relatively product bounded: any cocycle constant equal to p 1 1 0 1 q will do. Here is a more interesting example: Example 4.2. Let the base dynamics T be an irrational rotation of the circle R{Z, and consider the matrix-valued map F pxq :"`2 0 0 1{2˘R2πx , where R θ denotes the rotation matrix by angle θ. As shown by Herman [Her,, the SLp2, Rq-cocycle pT, F q has a positive Lyapunov exponent, but it is not uniformly hyperbolic. Therefore it cannot be relatively product bounded, because otherwise it would contradict a result of Morris [Mo2, Theorem 2.1].

Existence of local unstable sets with relatively bounded orbits
Let Φ be a fiber bunched automorphism in the set Aut θ K pE, T q. By the definition (2.5) of this set, (4.1) By equicontinuity of local holonomies, there exists a constant C 1 ą 1 such that: Moreover, by Proposition 2.8, it is possible to choose a constant C 1 that works for all automorphisms in a C 0 -neighborhood of Φ in Aut θ K pE, T q. Let Eˆdenote the complement of the zero section in E. Recall that ε 0 comes from the definition of hyperbolic homeomorphism. Lemma 4.3. Let Φ be a fiber bunched automorphism. Then there exists ε 2 P p0, ε 0 q, depending only on T , such that for every integer m ą 0 there exists u P Eˆwith the following property: Proof. Multiplying Φ by a nonzero constant, we can assume that βpΦq " 0. Let λ u be the hyperbolicity exponent of T along unstable sets, and let a :" sup xPX e´λ upxq ă 1 . Hyperbolicity implies: Let ε 2 :" p1´aqε 0 . In order to show that the conclusion of the lemma holds for this ε 2 , let us assume for a contradiction that there exists an integer m ą 0 such that: (4.5) We recursively define sequences pu k q, pv k q in Eˆand pn k q in v1, mw as follows: We choose u 0 P Eˆarbitrarily. Assuming u k was already defined, we let n k :" npu k q and v k :" vpu k q be given by (4.5), and let u k`1 :" Φ n k pv k q. Note that for each k ě 0 we have }u k`1 } ą 2}u k } and so }u k } ě 2 k }u 0 }. Now let ℓ k :" n 0`n1`¨¨¨`nk´1 (so ℓ 0 :" 0), and w k :" Φ´ℓ k pv k q. We claim that each w k belongs to W u proving the claim. In particular, by (4.2) we obtain }w k } ď C 1 }u 0 }. Since v k P W u ε2 pu k q, using (4.2) again we have }u k } ď C 1 }v k }. Therefore: Since ℓ k ď mk, using (1.4) we obtain This contradiction concludes the proof.
The next lemma supersedes the previous one: Lemma 4.4. Let Φ P Aut θ K pE, T q be a fiber bunched automorphism. Then there exist a constant C 2 ą 1 and a vector u˚P Eˆsuch that Moreover, the same constant C 2 works for all automorphisms in a C 0 -neighborhood of Φ in Aut θ K pE, T q. Proof. Again, multiplying Φ by a nonzero constant (and increasing K if necessary), we can assume that βpΦq " 0.
For each integer m ě 1, Lemma 4.4 provides u m P Eˆ, say with }u m } " 1, such that for every n P v1, mw and every v P W u ε2 pu m q we have }Φ n pvq} ď 2. Passing to a subsequence if necessary, we assume that pu m q converges to somē u, which has }ū} " 1. Let x m :" πpu m q andx :" πpūq.
We claim that sup Indeed, givenv P W u ε3 pūq and n ě 0, considerȳ :" πpvq. Since x m Ñx, for every sufficiently large m ě n we have dpx m ,ȳq ă 2ε 3 , and in particular y m :" rx m ,ȳs is well-defined and belongs to W u ε2 px m q. Let v m :" H u ymÐxm pu m q and w m :" H s yÐym pv m q (see Fig. 3). Then: As m Ñ 8 (recall that n is fixed), we have y m Ñ rx,ȳs "ȳ and so: by continuity of holonomies. It follows that }Φ n pvq} ď 2C 1 , completing the proof of the claim (4.6).
by (4.6). On the other hand, recalling (4.1), we have }u˚} ě K´ℓ}u} " K´ℓ. So the vector u˚has the desired property with C 2 :" 2K ℓ C 1 , completing the proof of the lemma.

Proof of relative product boundedness
The next lemma uses spannability to spread local product boundedness from a local unstable set to the whole space: Lemma 4.5. Let Φ P Aut θ K pE, T q be a spannable automorphism. There exists C 3 ą 1 with the following properties. Suppose u P E is a nonzero vector such the following quantity is finite: Furthermore, the same constant C 3 works for all automorphisms in a C 0 -neighborhood of Φ in Aut θ K pE, T q. Note that the hypothesis of Lemma 4.5 is non-void by Lemma 4.4, and that its conclusion implies that Φ is relatively product bounded. So Lemma 4.5 implies Proposition 4.1. However, the more technical statement of Lemma 4.5 is necessary for the construction of an extremal norm in the next section.
Proof of Lemma 4.5. It is sufficient to consider βpΦq " 0. Letn and C 0 be the uniform spannability constants provided by Proposition 3.10. Fix a nonzero vector u for which the associated quantity r is finite, and without loss of generality, let us assume that }u} " 1. Let r 1 ą r be arbitrary. Then there exists ns uch that sup něn˚s up vPW u ε 1 puq }Φ n pvq} ď r 1 .
Consider arbitrary y P X and w P E y . Apply Proposition 3.10 to the points x :" πpuq and y and the vector u, obtaining points x 1 , . . . , x d P W u ε1 pxq and times n 1 , . . . , n d P v0,nw such that each point y i :" T ni x i belongs to W s ε1 pyq and the vectors v i defined by (3.2) form a basis for E y . Moreover, if we express w as a linear combination a 1 v 1`¨¨¨`ad v d , then the proposition also yields that We have } 1 } ď C 1 by (4.2), and } 2 } ď r 1 by definition. Combining these estimates, we obtain: }Φ n pwq} ď dC 0 C 1 r 1 }w} , that is, }Φ n y } ď C 3 r 1 , where C 3 :" dC 0 C 1 . So lim sup nÑ8 sup yPX }Φ n y } is bounded by C 3 r 1 , and actually by C 3 r, since r 1 ą r is arbitrary. This proves the desired inequality.
Now consider a C 0 -perturbation of Φ in the set Aut θ K pE, T q. By Corollary 3.11, this perturbation is also spannable, and we can use the same constantsn and C 0 . So the argument above applies verbatim for the perturbed automorphism.

Application: polynomial bounds
Let us give an application of what we have proved so far, namely that under the hypothesis of strong fiber bunching, relative product boundedness fails at most by a polynomial factor. The reader anxious to see extremal norms may skip this subsection.
Theorem 4.6. Let T be a transitive hyperbolic homeomorphism. Let Φ : E Ñ E be strongly bunched automorphism covering T . Then there exists an integer d 1 P v0, d´1w and C ą 0 such that }Φ n x } ď Cn d 1 e nβpΦq for all x P X and n ě 0.
Before proving this theorem, let us fix some terminology. Suppose E is a θ-Hölder vector bundle over X, with a fixed θ-Hölder Riemannian norm, and that F Ď E is a θ-Hölder subbundle. Let F K Ď E be the orthogonal complement subbundle, which is also θ-Hölder. Then the orthogonal projections P : E Ñ F and Q : E Ñ F K (4.7) are θ-Hölder endomorphisms covering id X . Now suppose Φ : E Ñ E is θ-Hölder automorphism covering T and that F is Φ-invariant. Then there are two induced θ-Hölder automorphisms, both covering T , namely the obvious restricted automorphism Φ| F : F Ñ F, and the quotient automorphism Φ {F : F K Ñ F K defined by Φ {F :" pQ˝Φq| F K . If the automorphism Φ is fiber bunched (or strongly bunched) then so are Φ| F and Φ {F.
Proof of Theorem 4.6. Let Φ : E Ñ E be a strongly bunched automorphism. If Φ is irreducible then by Theorem 3.8 it is spannable, and by Proposition 4.1 it is relatively product bounded, hence our claim holds with d 1 " 0. In particular, the theorem holds when d " 1. Now suppose Φ is reducible, that is, there exists a θ-Hölder Φ-invariant nontrivial subbundle F Ă E. By induction on dimension, we can assume that the theorem holds for the restricted automorphism Φ| F and the quotient automorphism Φ {F, that is, there are nonnegative integers d 1 ă dim F and d 2 ă d´dim F such that: Note that, by the definitions of the automorphisms Φ| F and Φ {F, for all x P X and n ě 0, and therefore (4.9) Letting P and Q be the orthogonal projections (4.7), note the identity: More generally, for every n ě 1, we have: (4.10) which can be checked by induction. Using the bounds (4.8), it follows that: Noting that d 1`d2`1 ă d and recalling (4.9), we obtain the desired polynomial bound.
Incidentally, note that (4.10) implies that (4.9) is an equality, that is: Actually, a more general fact holds: for any T -invariant ergodic probability measure µ, χ 1 pΦ, µq " max χ 1 pΦ| F , µq, χ 1 p Φ {F, µq ( . (4.12) We will use this fact in Subsection 7.1. We were not able to find a precise reference for it, but it follows easily from the identity (4.10) together with an estimate such as [Boc1,Lemma 12].
5 Construction of extremal norms

Extremal norms for spannable automorphisms
In this subsection we state and prove the central result of this paper, Theorem 5.2 below. Let us present a simple consequence first: Corollary 5.1. Let T be a transitive hyperbolic homeomorphism. Let Φ be a strongly bunched irreducible automorphism covering T . Then Φ admits an extremal norm.
Here is the full statement of our result on extremal norms. Let θ u be the exponent provided by applying Proposition 2.10 to Φ´1.
(5.2) (c) |||¨||| isθ-Hölder along unstable sets withθ :" maxtθ, 1u, that is, there is a constant C 6 ą 0 such that for all x P X and x 1 P W u ε0 pxq,ˇ| Furthermore, for every sufficiently C 0 -small perturbation of the automorphism Φ in the set Aut θ K pE, T q, we can find an extremal norm that satisfies the properties above with the same constants θ u , C 4 , C 5 , C 6 .
Combining the theorem above with Theorem 3.8 we immediately obtain Corollary 5.1.
Note that part (b) of the statement of Theorem 5.2 is compatible with the characterization of Hölderness of a norm given by Proposition 2.3. In summary, our extremal norm is Hölder, but perhaps with a smaller Hölder exponent than the original Φ. 9 Nevertheless, part (c) says that the norm is more regular along unstable sets: there is no loss of exponent, and if θ ă 1 there is a gain.
Concerning the final part of the statement of Theorem 5.2, recall from Corollary 3.11 that the set of spannable automorphisms is a C 0 -open subset of the set Aut θ K pE, T q. So the theorem also says that our extremal norms vary in a bounded way if the automorphism is perturbed; this is useful to certain applications (see Subsection 5.2).
If θ ą 1 then the existence of ζ is an immediate consequence of the fact that the algebra of θ-Hölder functions on XˆX is normal (Lemma A.1).
Take arbitrary nonzero u P E. Since ζpx, yq " 1 whenever y P W u aε1 pxq, The left-hand side is at least 1; indeed by (1.4), for every n ą 0 there exists y P X such that }Φ n y } ě e nβpΦq " 1. Therefore: In particular, the seminorm |||¨||| is actually a norm. Since 0 ď ζ ď 1 and ζpx, yq " 0 whenever y P W u ε1 pxq, inequality (5.5) implies: |||Φpuq||| ď |||u||| , for all u P E , that is, |||¨||| is an extremal norm. Now consider the vector u˚P Eˆgiven by Lemma 4.4. It satisfies |||u˚||| ď C 2 }u˚}, where C 2 ą 1 is a constant depending only on Φ. Applying (5.6) to this vector we obtain: Therefore, for all u P E, where C 1 ą 1 is the constant from (4.2). So, letting C 4 -KC 1 C 2 C 3 and recalling the lower bound (5.7), we obtain (5.1): the extremal norm is uniformly comparable to the original norm by a factor C 4 that works not only for Φ but also for its C 0 perturbations in Aut θ K pE, T q. Before proving regularity properties of the extremal norm, let us establish a few auxiliary facts. For all u P E, v P W u ε1 puq, and n ě 0, using (5.1), extremality, and (4.2), we obtain: Fix a constant b ă 1 sufficiently close to 1 so that: dpx, yq ě bε 1 ñ ζpx, yq ă 1 2 C´1 1 C´3 4 . Then: v P W u ε1 puq W u bε1 puq ñ ζpπpuq, πpvqq}Φ n pvq} ď 1 2 |||u||| . So vectors v outside W u bε1 puq do not contribute in formula (5.4), which therefore can be rewritten as: (5.9) We will prove property (c) first, and use it later in the proof of property (b). In order to simplify writing, let us use the O notation to denote constants that depend only on Φ and can be taken uniform on a C 0 -neighborhood of Φ in Aut θ K pE, T q. In order to prove property (c), we need to show: x P X, x 1 P W u ε0 pxq, u P E x , }u} " 1 ñˇˇ|||H u x 1 Ðx puq|||´|||u|||ˇˇ" O`dpx, x 1 qθ˘.
Noting that W u bε1 pu 1 q Ď W u ε1 puq, we have: Using (5.9) and (5.4), we obtain: On the other hand, using W u bε1 puq Ď W u ε1 pu 1 q, a similar argument shows that: This completes the proof of property (c).

Application: Lipschitz continuity of the maximal Lyapunov exponent
As a simple application of Theorem 5.2, let us establish a local regularity result for the maximal Lyapunov exponent. A similar property for the joint spectral radius (under the assumption of irreducibility) was established by Wirth [Wir,Corol. 4.2], also using extremal norms; see also [Koz] for a more precise result.
Let S K denote the set of spannable automorphisms in Aut θ K pE, T q, which by Corollary 3.11 is relatively C 0 -open.
Proposition 5.4. The maximal Lyapunov exponent βp¨q is a locally Lipschitz function on the set S K , with respect to the C 0 -norm (2.3).
Proof. Let Φ P S K . Let U Ă S K be a C 0 -neighborhood of Φ where Theorem 5.2 applies with uniform constants. Take any two automorphisms Φ 1 and Φ 2 in U, and let |||¨||| 1 and |||¨||| 2 be the corresponding extremal norms provided by Theorem 5.2. Then, using the bound (5.1), we obtain: where }¨} 0 is the C 0 -norm (2.3). By symmetry, we obtain |e βpΦ1q´eβpΦ1q | ď C 4 }Φ 1´Φ2 } 0 . This shows that the function e βp¨q is Lipschitz on the neighborhood U, with respect to the C 0 -norm. Since the function βp¨q is uniformly bounded on U (and in the whole set Aut θ K pE, T q, in fact), it is Lipschitz as well.
Remark 5.5. For reducible automorphisms, it is clear that β is not locally Lipschitz: see e.g. [Wir,p. 27]. Nevertheless, β is continuous on the whole space of θ-Hölder automorphisms: indeed, upper semicontinuity is automatic from (1.4), while lower semicontinuity follows by a theorem of Kalinin [Kal,Theorem 1.4] that allows to approximate β by the Lyapunov exponents of periodic orbits. Let us also remark that if T is no longer hyperbolic, then β becomes discontinuous with respect to the C 0 topology. For example, the cocycle from Example 4.2 can be C 0 -perturbed so that β drops to 0, as it follows e.g. from the the result of [AB].

Barabanov-like norms for linear cocycles over shifts
Let us consider subshifts of finite type, that is, X is the set of two-sided sequences px n q nPZ in an alphabet t0, 1, . . . , N´1u whose neighboring pairs are those allowed by a fixed 0-1 matrix, and T : X Ñ X is the (left) shift map. As usual, we consider on X the (ultra)metric dpx, yq :" e´λ k where k " mint|n| ; x n ‰ y n u, (5.12) and λ ą 0 is a fixed parameter. Then T is a hyperbolic homeomorphism. Indeed letting ε 0 :" e´λ, the corresponding local unstable and stable sets at x " px n q P X are: W u loc pxq :" W u ε0 pxq " py n q P X ; y n " x n for all n ď 0 ( , W s loc pxq :" W s ε0 pxq " py n q P X ; y n " x n for all n ě 0 ( . and so hyperbolicity property (b) holds with λ u " λ s " λ, property (c) holds with 2ε 1 " ε 0 , and property (d) holds with C " 1. Also note that Proposition 2.6 holds with κ s " 1 " C. We will consider θ-Hölder automorphisms covering the subshift T : X Ñ X. Since X is a Cantor set, every θ-Hölder vector bundle is trivial, i.e., θ-Hölder isomorphic to the product bundle. So we are actually dealing with θ-Hölder linear cocycles; nevertheless, we will keep using the vector bundle terminology.
Example 5.6. As mentioned in Subsection 1.2, the one-step cocycle determined by a Ntuple of matrices pA0, . . . , AN´1q P GLpd, Rq N is the pair pT, F q where T is the full shift on N symbols and F : X Ñ GLpd, Rq is given by F pxq :" Ax 0 . Let Φ the associated automorphism (1.1). Then e βpΦq is joint spectral radius of the set tA0, . . . , AN´1u. Since F is locally constant, it is θ-Hölder for any θ P p0,`8q. Choosing θ large enough, the automorphism Φ becomes fiber bunched. (Alternatively, we can take θ " 1, say, and then take the parameter λ large enough.) The holonomies are locally trivial: ‹ P tu, su, y P W ‹ loc pxq ñ H ‹ yÐx " id . (5.13) A useful generalization of one-step cocycles are the sofic cocycles from [BPS,§ 5.1]; the same concept appears in [PEDJ] under the terminology constrained switching systems.
Let us present an improved version of Theorem 5.2 for subshifts of finite type. We obtain an extremal norm with an additional Barabanov-like property: given any vector u P E, there always exists a vector in its local unstable set W u loc puq :" W u ε0 puq whose expansion factor in a single iterate equals the maximum asymptotic expansion rate e βpΦq . Furthermore, the norm is invariant under local unstable holonomies. Therefore, for the case of one-step cocycles, we reobtain the Barabanov property (1.8).
Let us comment on the hypotheses. Given θ ą 0, Lemma 3.7 holds with the value η 0 " θ{3; indeed, this follows from formula (A.24), recalling that λ u " λ s " λ, and noting that we can also take Λ u " λ in (A.7). Therefore θ-Hölder automorphism Φ : E Ñ E covering T is strongly bunched if Φ is fiber bunched and the fibers of E have dimension d ď 2, or Φ is a pθ{3, θq-bunched. In that case, it follows from Theorem 3.8 that Φ is spannable, provided it is irreducible and T is transitive.
Proof of Theorem 5.7. Since X is a Cantor set, we can simplify the construction in Theorem 5.2 and dispense with the bump function ζ from Lemma 5.3. Ultimately, we can replace the definition (5.4) of the extremal norm by the simpler formula (5.14). Note that in the latter formula we maximize over W u loc puq :" W u ε0 puq instead of W u ε1 puq " W u ε0{2 puq; this is possible because the metric on X is an ultrametric. It is straightforward to check that the proof in Theorem 5.2 applies, with simplifications. It is immediate from its definition that the norm satisfies local H u -invariance, that is property (1), which of course subsumes property (c) from Theorem 5.2.
We only left to check the calibration property (2). Given u P E, by definition, there exist sequences n i Õ 8 and v i P W u loc puq such that Denote y i " πpv i q. By compactness, we may suppose that y i Ñ y P W u loc pxq and v i Ñ v P W u loc puq. For i large enough, y 1 i :" T py i q P W u loc pT pyqq. We can assume that this property is true for all i. Let v 1 i :" Φpv i q. Thus, we have: By extremality, the inequality is actually an equality. This proves calibration.

Mather sets
In traditional ergodic optimization, that is, the optimization of Birkhoff averages (see [Je1,Je2,Gar]), a maximizing set is a closed subset such that an invariant probability is maximizing if and only if its support lies on this subset. The existence of such sets is guaranteed in any context where a Mañé Lemma holds. The Mather set is the smallest maximizing set: it is defined as the union of the supports of all maximizing measures. The nomenclature is borrowed from Lagrangian dynamics, where the concept of minimizing measures proved to be a useful generalization of the notion of action minimizing orbits: see [Mat]. There are other canonical maximizing sets, such as the Aubry set: see e.g. [Gar]. Some of these concepts have been already considered in the optimization of the top Lyapunov exponent: see [Mo3,GG].
In this section, we are going to study some notions of Mather sets for continuous vector bundle automorphisms, not necessarily with any Hölder or hyperbolicity structures. Our approach was profoundly influenced by some works of Morris [Mo2,Mo3].
Throughout the section, we assume that X is a compact metric space, T : X Ñ X is a homeomorphism, E is a d-dimensional vector bundle over X, and Φ is an automorphism covering T .

The first Mather set
By a Lyapunov maximizing measure we mean any T -invariant probability µ whose upper Lyapunov exponent χ 1 pΦ, µq equals βpΦq. Following Morris [Mo3], we define the (first) Mather set M pΦq Ď X as the union of the supports of all Lyapunov maximizing measures.
Proposition 6.1. The Mather set M pΦq is the support of some Lyapunov maximizing measure and, a fortiori, it is a nonempty, compact, and T -invariant set.
Proof. The argument is quite standard, but we add it for completeness. For simplicity, write M " M pΦq. As explained at the introduction, at least one Lyapunov maximizing measure exists, so M ‰ ∅. Given a countable basis tB j u jPN for the topology of X, consider the subset of indices J :" tj P N ; B j X M ‰ ∅u. For each j P J, we assign a Lyapunov maximizing measure µ j such that B j X supp µ j ‰ ∅, which means µ j pB j q ą 0. Define then µ :" ř jPJ α j µ j , where α j ą 0 and ř jPJ α j " 1. As a convex combination of Lyapunov maximizing measures, µ is also Lyapunov maximizing. Now consider an arbitrary k P N such that B k X supp µ " ∅. Then for all j P J we have B k X supp µ j " ∅ and therefore µ j pB k q " 0. This implies that k R J, and so B k X M " ∅. We have shown that X supp µ Ď X M , which yields M pΦq " supp µ.
On the other hand, if µ is the Lyapunov maximizing measure with supp µ " M pΦq constructed in Proposition 6.1 then, by Kingman's subadditive theorem, inf n 1 n ş f n dµ " βpΦq. It follows that 1 n ş f n dµ " βpΦq for each n ě 1. Since the functions f n are continuous, they must be identically equal to βpΦq over supp µ " M pΦq, as we wanted to show.

Mather sets of higher index
Up to here we have only considered the first Lyapunov exponent χ 1 , but not we will need to consider the full Lyapunov spectrum. Let us recall the definitions and main properties, referring to [Arn] for details.
If µ is a T -invariant probability measure then the Lyapunov exponents of the automorphism Φ with respect to µ are the numbers χ 1 pΦ, µq ě χ 2 pΦ, µq ě¨¨¨ě χ d pΦ, µq (6.1) uniquely defined by the following equations: for every p P v1, dw, where the automorphism Λ p Φ : Λ p E Ñ Λ p E is the p-fold exterior power of the automorphism Φ : E Ñ E. Suppose µ is ergodic, and that λ is a Lyapunov exponent with respect to µ of multiplicity k, in the sense that it appears k times in the list (6.1). Then Oseledets' theorem says that for µ-a.e. x P X, there exists a k-dimensional subspace O x pλq of the fiber E x , called a Oseledets space, such that: Moreover, Oseledets spaces form a splitting of E x , depend measurably on the point x, and are Φ-equivariant.
We now consider other Mather sets that take multiplicity into account. Define a chain of sets as follows: the p-th Mather set M p pΦq is the union of the supports of all Tinvariant probabilities µ whose p first Lyapunov exponents are all maximal, that is, χ 1 pΦ, µq " χ 2 pΦ, µq "¨¨¨" χ p pΦ, µq " βpΦq .
Repeating the argument of the proof of Proposition 6.1, we see that if the set M p pΦq is nonempty then there exists a measure µ with p maximal Lyapunov exponents and whose support is exactly M p pΦq; in particular, M p pΦq is compact and T -invariant. The following properties follow immediately from the definition of Mather sets and relations (6.1) and (6.2): Proposition 6.3. For any p P v1, dw, we have: βpΛ p Φq ď pβpΦq and M p pΦq Ď M 1 pΛ p Φq .

Furthermore, these two relations become equalities if and only if M p pΦq ‰ ∅.
Let us extend the chain of sets (6.3). Let M d`1 pΦq :" ∅ and let M 0 pΦq be defined as the union of the supports of all T -invariant probability measures (which is also the support of one of them). So M 0 pΦq only depends on T , and is in fact the classical minimal center of attraction of T : see [NS,p. 363], [Akin,p. 164].

Dominated splittings over the Mather sets
We want to provide more information about the action of Φ on the fibers above the Mather sets, assuming the existence of a extremal norm. We will use the notion of dominated splitting, which is very useful in Differentiable Dynamics (see [BDV]). It appears in the celebrated book [HPS] as relative pseudo hyperbolicity. It also appears in ODE and Control Theory under the terminology exponentially separated splitting (see [CK]), and is intimately related to the concept of Anosov representations in Geometric Group Theory (see [BPS]).
Let Y Ď X be a nonempty T -invariant compact set, and let E Y :" π´1pY q be the restricted vector bundle. (Recall that π : E Ñ X denotes the bundle projection.) Let }¨} be a Finsler norm on E. Suppose that the bundle E Y splits as a direct sum F ' G of two (continuous) subbundles whose fibers F x , G x have constant dimensions, and are equivariant in the sense that Φ x pF x q " F T x , Φ x pG x q " G T x . We say that F ' G is a dominated splitting with dominating bundle F and dominated bundle G if there are positive constants c and τ such that for each point x P Y , if u P F x , v P G x are unit vectors then }Φ n x pvq} ď ce´τ n }Φ n x puq} for all n ě 0. (6.4) An equivalent definition is to say that there exists an adapted norm }¨} for which relation (6.4) holds with c " 1 (and therefore only needs to be checked for n " 1): see [Gou]. Dominated splittings are unique given the dimensions: see [CP,Prop. 2.2]. Continuity of the subbundles actually follows from the uniform estimates (6.4) and therefore could be removed from the definition: see [CP,Prop. 2.5]. Actually, if Φ is Hölder then the bundles of a dominated splitting are always Hölder (with a smaller exponent): see [CP,Thrm. 4.11]. Domination can be characterized in terms of existence of invariant cone fields: see [CP,Thrm. 2.6]. This implies strong robustness properties: see [CP,Corol. 2.8].
We will use another criterion for the existence of dominated splittings, expressed in terms of singular values. Recall that if L : E Ñ F is a linear map between d-dimensional inner product spaces, then the singular values σ 1 pLq ě¨¨ě σ d pLq are the eigenvalues of the symmetric operator pL˚Lq 1{2 . So σ 1 pLq coincides with the euclidian operator norm }L}. Endowing the exterior power spaces with the induced inner products, the exterior powers of L have norm: }Λ p L} " σ 1 pLqσ 2 pLq¨¨¨σ p pLq ; (6.5) see e.g. [Arn,p. 120]. Another useful characterization of the singular values is: (6.6) see e.g. [Ste,p. 68].
A theorem from [BG] says that the domination is equivalent to a uniform exponential gap between singular values of the powers of Φ (computed with respect to a Riemannian norm fixed a priori). More precisely: Theorem 6.4 (Bochi-Gourmelon). The bundle E Y admits a dominated splitting with a dominating bundle of dimension p if and only if there exist positive constants c and τ such that σ p`1 pΦ n x q ď ce´τ n σ p pΦ n x q for all x P Y and n ě 0.
We now come back to the Mather sets: Theorem 6.5. Suppose Φ admits an extremal norm |||¨|||. Let p P v1, dw. Suppose Y is a nonempty compact T -invariant set contained in M p pΦq M p`1 pΦq. Then the restricted bundle E Y admits a dominated splitting F'G where the dominating bundle F has fibers of dimension p and is calibrated in the sense that |||Φpuq||| " e βpΦq |||u||| for every u P F.
In particular, if exactly one of the sets M p pΦq M p`1 pΦq is nonempty then we obtain a dominated splitting over the whole Mather set M pΦq.
Related results were previously obtained by Morris: [Mo2, Theorem 2.1] produces a dominated splitting under the weaker assumption of relative product boundedness, but with the strong hypothesis that the set Y is minimal (i.e., all orbits in Y are dense). Assuming existence of an extremal norm, Morris also proves the calibration property of the dominating bundle in his Theorem 2.2.
For a complement to Theorem 6.5, see Proposition B.4 in Appendix B.2.
This is a closed, Φ-invariant subset of E. Denote its fibers by K x :" E x X K.
Proposition 6.6. There exists a T -invariant Borel set R Ď M pΦq such that: • µpM pΦq Rq " 0 for every T -invariant probability measure µ; • for all x P R, the Oseledets space corresponding to the Lyapunov exponent βpΦq exists and coincides with K x .
Proof. Without loss of generality, assume that βpΦq " 0. Let R 0 Ď X be the Borel set of points that satisfy the conclusions of Oseledets theorem. For each x P R 0 X M pΦq, the Oseledets space O x " O x p0q Ď E x is well-defined and has positive dimension, say ppxq. Calibrated vectors have zero Lyapunov exponent, so K x Ď O x . Consider the unit balls on these Oseledets spaces, i.e., B x :" tu P O x ; |||u||| ď 1u. Then Φ x pB x q Ď B T x , and therefore the following function is non-positive: ψpxq :" log volpΦ x pB x qq volpB T x q .
Here vol means ppxq-dimensional volume with respect to a fixed Riemannian norm on the bundle E; of course, the choice of this metric does not affect the function ψ. Then ψ is cohomologous to the functionψpxq :" log det Φpxq| Ox , where det denotes the signless determinant induced by the Riemannian metric (see [Arn,p. 213]); indeed ψ "ψ`ϕ´ϕ˝T where ϕpxq :" log volpB x q. All these functions are Borel measurable and bounded. Let µ be any T -invariant probability measure supported on M pΦq, that is, any Lyapunov maximizing measure. As a consequence of Oseledets theorem, we have şψ dµ " 0 (see [Arn,p. 214]). Since ψ is cohomologous toψ, its integral is zero as well. But ψ ď 0, so ψ " 0 µ-a.e. Let R 1 :" tx P R 0 X M pΦq ; ψpxq " 0u and R :" Ş nPZ T´npR 1 q; then µpRq " 1. Noting that ψpxq " 0 if and only if Φpxq| Ox preserves |||¨|||, we see that if x P R then O x Ď K x . As remarked before, the reverse inclusion is automatic, so O x " K x for every x P R. Since set R has full measure with respect to any Lyapunov maximizing measure, the set M pΦq R has zero measure with respect to any T -invariant probability measure, as we wanted to show.
Corollary 6.7. Suppose Φ admits an extremal norm |||¨|||. For each p P v1, dw and x P M p pΦq, the set K x contains a vector space of dimension p.
Proof. Let µ be a measure whose p first Lyapunov exponents equal βpΦq and whose support equals M p pΦq. Given x P M p pΦq, take a sequence of neighborhoods U i converging to x. Since µpU i q ą 0, by Proposition 6.6 we can find x i P U i such that the Oseledets space corresponding to the Lyapunov exponent βpΦq exists and coincides with K xi . Moreover, these spaces have dimensions at least p. Passing to subsequences, we can assume that these dimensions are constant equal to some q ě p, and that K xi converges to some q-dimensional space V . As K is a closed subset of E, we conclude that V Ď K x , completing the proof.
Remark 6.8. It is not necessarily the case that Kx is a subspace: see Example B.1 in Appendix B.1. On the other hand, if Kx is a subspace, then by Corollary 6.7 its dimension is at least the number p such that x P MppΦq Mp`1pΦq. However, it is not necessarily true that dim Kx " p: see Example B.2 in Appendix B.1.
Proof of Theorem 6.5. As usual, it is sufficient to consider βpΦq " 0.
In the case p " d, we have Y Ď M d pΦq and so by Corollary 6.7 the extremal norm is preserved along the bundle E Y . So the trivial splitting E Y ' 0 has the required properties. So let us suppose that p ă d.
Fix a Riemannian norm }¨} on E. For each x P Y , by Corollary 6.7 the fiber E x contains a p-dimensional subspace formed by vectors u such that for every n ě 0, we have |||Φ n puq||| " |||u||| and therefore c´1 1 }u} ď }Φ n puq} ď c 1 }u}, for some constant c 1 ě 1. Recalling the maxmin characterization of singular values (6.6), we conclude that: c´1 1 ď σ p pΦ n x q ď¨¨¨ď σ 1 pΦ n x q ď c 1 for all x P Y and n ě 0. (6.8) On the other hand, note that the set M p`1 pΦ| EY q is contained in Y X M p`1 pΦq and therefore is empty. So Proposition 6.3 yields β`Λ p`1 pΦ| EY q˘ă pp`1qβpΦ| EY q " 0 .
Recalling that βp¨q can also be characterized by (1.4), we conclude that there exist positive constants c 2 and τ such that for all x P Y and n ě 0, So, using (6.5) and (6.8), we have: x } ď c p 1 c 2 e´τ n and σ p pΦ n x q ě c´1 1 . So we have a uniform exponential gap between the p-th and p`1-th singular values. By Theorem 6.4, the bundle E Y admits a dominated splitting F ' G with a dominating bundle F of dimension p.
To conclude, we need to check that Φ preserves the extremal norm along the bundle F. We will actually show that, in terms of notation (6.7), K x " F x for every x P Y . Since Φ is product bounded, domination implies that vectors in G are uniformly contracted in the future, and therefore uniformly expanded in the past. Furthermore, any vector in u P E x F x is uniformly expanded in the past, since we can write u " v`w with v P F x , w P G x t0u and then |||Φ´npuq||| ě |||Φ´npwq|||ˆ1´| ||Φ´npvq||| |||Φ´npwq|||˙Ñ 8 as n Ñ`8 .
In particular, vectors in E x F x cannot be calibrated; that is, K x Ď F x . This inclusion cannot be strict, thanks to Corollary 6.7. So K x " F x , as claimed.
7 Further applications of extremal norms and Mather sets

Subordination
By definition, the Mather set M pΦq contains the support of every Lyapunov maximizing measure. Let us see that the converse holds under the hypothesis of strong fiber bunching, regardless of reducibility: Theorem 7.1. Let T be a transitive hyperbolic homeomorphism. Let Φ be strongly bunched automorphism covering T . Then every T -invariant probability measure whose support is contained in the Mather set M pΦq is Lyapunov maximizing.
Proof. Let Φ : E Ñ E be a strongly bunched automorphism and let ν be a Tinvariant probability measure whose support is contained in M pΦq; we want to prove that χ 1 pΦ, νq " βpΦq. By ergodic decomposition, it is sufficient to consider the case of ergodic ν.
If Φ is irreducible (which is certainly the case if d " 1) then Φ is spannable by Theorem 3.8, and so Φ admits an extremal norm by Theorem 5.2. Then Proposition 6.2 yields the desired conclusion.
From now on, assume that Φ is reducible, that is, there exists a θ-Hölder Φ-invariant nontrivial subbundle F Ă E. By induction on dimension, we can assume that the theorem holds for the restricted automorphism Φ| F and the quotient automorphism Φ {F. Recall from (4.11) that βpΦq " maxtβpΦ| F q, βp Φ {Fqu.
As a first case, suppose that βpΦq " βpΦ| F q ą βp Φ {Fq. Then it follows from (4.12) that an ergodic measure µ is Lyapunov maximizing for Φ if and only if it is Lyapunov maximizing for Φ| F . Therefore the Mather sets coincide: M pΦq " M pΦ| F q. The measure ν fixed at the beginning is supported on this set; so, by the induction hypothesis, it is Lyapunov maximizing for Φ| F , that is, it is Lyapunov maximizing for Φ, as we wanted to show.
In the last case, we have βpΦq " βpΦ| F q " βp Φ {Fq. Then it follows from (4.12) that an ergodic measure µ is Lyapunov maximizing for Φ if and only if it is Lyapunov maximizing for Φ| F or for Φ {F. Therefore M pΦq " M pΦ| F qYM p Φ {Fq. So the measure ν fixed at the beginning has a support contained in the union of the two closed T -invariant sets M pΦ| F q and M p Φ {Fq. By ergodicity, this support must be contained in one of the two sets. By the induction hypothesis, ν is Lyapunov maximizing for Φ| F or for Φ {F. In either case, it is Lyapunov maximizing for Φ, as we wanted to show.

Lyapunov almost-maximizing periodic orbits of low period
Let Φ be a θ-Hölder automorphism covering a hyperbolic homeomorphism. For each integer n ě 1, let β n pΦq :" max χ 1 pΦ, µq ; µ is supported on a periodic orbit of period ď n ( . This is a bounded non-decreasing sequence, and so it is convergent. Actually, the limit is: lim nÑ8 β n pΦq " βpΦq . (7.1) Indeed, this follows from a much more general result of Kalinin [Kal,Theorem 1.4] on the approximation of Lyapunov exponents using measures supported on periodic orbits. In the case of one-step cocycles, formula (7.1) is known as the Berger-Wang theorem, and it was first proved in [BeW]. For other extensions of Berger-Wang theorem, see [O-R, BF]. It is quite possible that the limit (7.1) is attained for some finite n (and indeed this is expected to be the typical situation). On the other hand, in the worst-case scenario, what can we say about the speed of the approximation in formula (7.1)? A result of Morris [Mo2] says that for one-step cocycles, this speed is always superpolynomial. Here we show that the same is true for strongly bunched automorphisms: Theorem 7.2. If Φ is a strongly bunched automorphism then for every τ ą 0, βpΦq´β n pΦq " Opn´τ q .
The first result of superpolynomial approximation was actually obtained in the context of ergodic optimization of Birkhoff averages by Bressaud and Quas [BQ], who also showed that this type of bound is essentially sharp. The key ingredient is a quantitative version of Anosov Closing Lemma, also due to Bressaud and Quas [BQ], which we state as follows: Theorem 7.3 (Bressaud-Quas Closing Lemma). Let T : X Ñ X be a hyperbolic homeomorphism. Let Y Ď X be a nonempty compact T -invariant set. Then for every τ ą 0 there exists C ą 0 such that for every sufficiently large n, there exists a periodic orbit of period at most n supported on the Cn´τ -neighborhood of Y .
This result is proved in [BQ] for the one-sided full shift; as remarked in that paper, one can use standard techniques to reduce to that case. Alternatively, one can prove Theorem 7.3 directly, and we do so in Appendix A.6.
Proof of Theorem 7.2. Recall from (4.11) that if Φ is reducible then we can replace it by either a restricted or a quotient automorphism with the same maximal Lyapunov exponent. Repeating this procedure a finite number of times, we eventually find a irreducible automorphism with the same maximal Lyapunov exponent; this induced automorphism will also be strongly bunched. So, without loss of generality, we assume that Φ is irreducible. By Theorem 3.8, Φ is spannable, and by Theorem 5.2, Φ admits a Hölder extremal norm |||¨|||. Let p P v1, dw be maximal such that the p-th Mather set M p pΦq -Y is nonempty. By Theorem 6.5, the restricted bundle E Y admits a dominated splitting F ' G where the dominating bundle F has fibers of dimension p and is calibrated in the sense that for every x P Y and u P F x , we have |||Φpuq||| " e βpΦq |||u|||.
By robustness of dominated splittings [CP,Corol. 2.8], there exists a closed neighborhood U of Y such that if Z :" Ş kPZ T´kpU q Ě Y is the maximal invariant set in this neighborhood, then the restricted bundle E Z over the compact invariant set admits a dominated splitting F' G, extending the previously found dominated splitting on E Y . Recall that the bundles of a dominated splitting are Hölder-continuous [CP,Thrm. 4.11]. Furthermore, the extremal norm is also Hölder-continuous. It follows that there exist ρ ą 0 and C 0 ą 0 such that for every x P Z and every u P F x , e βpΦq´C0dpx,Y q ρ |||u||| ď |||Φpuq||| ď e βpΦq |||u||| .
Fix τ ą 0. Let C be given by Theorem 7.3; so for all sufficiently large n, there exists a periodic orbit of period at most n supported on the Cn´τneighborhood of Y , and so contained in Z. Let ν n be the invariant probability measure supported on that orbit. The bound obtained before implies: So βpΦq´β n pΦq " Opn´ρ τ q. Since τ ą 0 is arbitrary, the theorem is proved.
A Appendix: Proof of some technical results

A.1 Basic constructions on θ-Hölder bundles
Recall our assumption from Subsection 2.1 that the algebra of θ-Hölder functions on X is normal. Let us metrize the product XˆX by d`px, yq, px 1 , y 1 q˘max dpx, yq, dpx 1 , y 1 q ( .
Lemma A.1. The algebra of θ-Hölder functions on XˆX is normal.
Proof. Let K 0 , K 1 Ă XˆX be two disjoint nonempty compact sets. Let ε ą 0 be a lower bound for the distance between a point in K 0 and a point in K 1 . Let tB i u be a finite cover of X by open sets of diameter less than ε. Let tρ i u be a partition of unity subordinated to this cover and formed by θ-Hölder functions.
Define a function f : XˆX Ñ R by: Then f is θ-Hölder, takes values in the interval r0, 1s, equals 0 on K 0 , and equals 1 on K 1 . This proves normality.
Proof of Proposition 2.1 (existence of transport maps I yÐx ). 10 Consider the finite cover of XˆX formed by the following open sets: where ∆ Ď XˆX is the diagonal. Consider a partition of unity subordinate to this cover, composed of θ-Hölder functions ρ k,ℓ ; its existence is a consequence of Lemma A.1. Given any pair of points x, y P X, define a linear map from E x to E y by: where the sum is taken over the indices pk, ℓq such that V k,ℓ Q px, yq. If px, yq P U iˆUj then the matrix: rh j pyqs´1˝I yÐx˝hi pxq " ÿ pk,ℓq ρ k,ℓ px, yq g jÐℓ pyq˝g kÐi pxq is θ-Hölder continuous as a function of px, yq, and equals the identity when x " y.
For the following proofs, it is convenient to fix another open cover tV i u of X such that V i Ă U i for each k. Note that for any Finsler norm }¨} on E, we have: where these operators norms are relative to the norm }¨} x on E x and the euclidian norm }¨} eucl on R d .
Proof of Proposition 2.2 (composition of transport maps). In order to prove the assertion, it is sufficient to consider triples of points x, y, z that are close enough so that they belong to a common coordinate neighborhood V i . Consider the matrixĨ yÐx :" rh i pyqs´1˝I yÐx˝hi pxq , (A.2) which by Proposition 2.1 is Opdpx, yq θ q-close to the identity matrix. Using a similar notation for the other points, we have: }Ĩ zÐx´I d} eucl " Opdpx, zq θ q and }Ĩ yÐz´I d} eucl " Opdpy, zq θ q . Therefore: }Ĩ yÐz˝ĨzÐx´ĨyÐx } eucl " O`maxtdpx, zq θ , dpy, zq θ u˘.
Proof of Proposition 2.3 (characterization of θ-Hölder norms). Let }¨} be a θ-Hölder Finsler norm. In order to prove the desired estimate, it is sufficient to consider pairs of points x, y that are close enough so that they belong to a same set V i . By definition, for every u P R d , the map x P V i Þ Ñ }h i pxqu} is θ-Hölder, and so there is a constant C ą 0 such that, for all x, y P V i ,ˇ} Using the boundedness property (A.1) and compactness of the unit sphere, we can find a uniform C so that the estimate above holds for every u P R d .
The proof of the converse is entirely analogous.
Proof of Proposition 2.4 (characterization of θ-Hölder endomorphisms). Suppose Φ is θ-Hölder. In order to prove the desired estimate, it is sufficient to consider pairs of points x, y that are close enough so that they belong to a same set V i X T´1pV j q. LetΦ x :" rh j pT xqs´1˝Φ x˝hi pxq and similarly definẽ Φ y . LetĨ yÐx be defined by (A.2), and similarly defineĨ T xÐT y ; by Proposition 2.1 these matrix-valued maps are θ-Hölder as functions of px, yq. So the map px, yq Þ ÑĨ T yÐT x˝Φx´Φy˝ĨyÐx is also θ-Hölder, and since it vanishes on px, xq we conclude that: Using the boundedness property (A.1) we obtain: as desired. Conversely, assume that such an estimate holds; then (A.4) follows from (A.1). By Proposition 2.1, the matricesĨ yÐx andĨ T yÐT x are Opdpx, yq θ q-close to the identity. It follows that the matricesΦ x andΦ y are Opdpx, yq θ q-close. This means that Φ is θ-Hölder.

A.2 Existence of holonomies
We begin with a straightforward estimate: For every x, y P X and n ě 0 we have: where K 1 depends only on K.
Proof. By submultiplicativity of norms and the definition of bolicity, we have: and so the claimed inequality holds with K 1 being the θ-Hölder constant of log }Φ}. This constant can be estimated in terms of K, using (2.3).
Proof of Proposition 2.8 (existence of holonomies). By symmetry, it is sufficient to consider ‹ " s. The stable holonomy is defined as: where x and y are in a same stable set. Let us establish convergence. Assume first that y P W s ε0 pxq. We have: H n`1´Hn " pΦ n`1 y q´1˝`I T n`1 yÐT n`1 x˝ΦT n x´ΦT n y˝IT n yÐT n xl ooooooooooooooooooooooooooooomooooooooooooooooooooooooooooon ∆n˝Φ n x , and so, using the definition (2.5) of the set Aut θ K pE, T q and Lemma A.2, By property (2.7) in the definition of hyperbolicity, for every j ě 0 we have dpT j x, T j yq ď e´λ pjq s pyq dpx, yq , where λ pjq s pyq :" Since λ s is strictly positive, the series ř 8 j"0 dpT j x, T j yq θ is convergent. Therefore where K 2 ą 0 is another constant. Take a small constant η ą 0 such that the fiber bunching condition (2.12) still holds if the right hand side is multiplied by 1´η. In particular, bolpΦ T j y q ă e p1´ηqθλspT j yq and so }H n`1´Hn } ď K 2 e´η θλ pnq s pyq dpx, yq θ , (A.6) This establishes uniform exponential convergence in formula (A.5) when y P W s ε0 pxq. Using (2.9) we see that convergence holds whenever y P W s pxq. The groupoid properties (a) and (b) are the equivariance property (c) are automatic from the definition. The Hölderness property (d) follows by summing (A.6) for n " 0 to 8, and noting that H 0 " I yÐx . The joint continuity property (e) also follows from the uniformity of our estimates. Finally, if we consider a small C 0 perturbation of Φ in the set Aut θ K pE, T q, then we can use the same constants K 2 and η in (A.6), and so the remaining assertions of the proposition follow.

A.3 Regularity estimates
In this section, we prove Propositions 2.6 and 2.10. Before going into the proofs, let us state our estimates for Hölder exponents.
Since T is Lipschitz, we can find ε 1 P p0, ε 0 q and a continuous strictly positive function Λ u such that for all x, x 1 , x 2 P X, x 1 , x 2 P W u ε1 pxq ñ " T x 1 , T x 2 P W u ε0 pT xq , dpT x 1 , T x 2 q ď e Λupxq dpx 1 , x 2 q . (A.7) We will show that the conclusion of Proposition 2.6 holds for any κ s in the range: Let ϕpxq :" log bolpΦ x q; (A.9) We will show that the conclusion of Proposition 2.10 holds for any κ s in the range: (A.10) (Note that the numerator is positive by fiber bunching.) The idea of the proof of Proposition 2.6 is roughly as follows. Without loss of generality, we can assume that the "quadrilateral" in Fig. 1 has a "base" dpx, x 1 q much smaller than the two "legs" dpx, x 1 q, dpx, yq. We take N ě 0 as big as possible for which we can guarantee that the T N -image of that quadrilateral has a base smaller than the legs. We estimate the "summit" dpT N y, T N y 1 q using the triangle inequality, and finally we iterate backwards to obtain the desired estimate for dpy, y 1 q. The proof of Proposition 2.10 uses the same "there and back again" idea. Formal proofs follow.
Let us denote the Birkhoff sums of a function f : X Ñ R as: Lemma A.3. For any strictly positive continuous function f on X and any a P p0, 1q, there exists bpf, aq ą 0 such that for any z P X and any n ě 0, z 1 P W s ε0 pzq Y T´npW u ε0 pT n zqq ñ f pnq pz 1 q ě af pnq pzq´bpf, aq .
Proof. It is sufficient to consider the case z 1 P W s ε0 pzq, since the case z 1 P T´npW u ε0 pT n zqq follows by reversing the time. By uniform continuity of f and uniform contraction on local stable sets, we can find an integer k " kpf, aq ě 0 such that if z 1 P W s ε0 pzq then f pT j z 1 q ě af pT j zq for every j ě k. Letting bpf, aq :" ak sup X f , we obtain the desired conclusion.
We start the proofs of Propositions 2.6 and 2.10 with some estimates that are common to them. Fix κ s and θ s satisfying (A.8) and (A.10), respectively. Fix a number a P p0, 1q sufficiently close to 1 such that: Fix four points x, x 1 , y, y 1 satisfying (2.10). Let δ :" dpx, x 1 q. Note that to prove Propositions 2.6 and 2.10, it is sufficient to consider δ smaller than a fixed positive constant, say ε 1 from (A.7). Let N be the largest nonnegative integer such that: aλ pN q s pxq`Λ pN q u pxq ă logpε 1 {δq ; (A.13) Then: aλ pN q s pxq`Λ pN q u pxq ě logpε 1 {δq´c , (A.14) for some constant c, namely c :" sup X paλ s`Λu q. In particular, assuming that δ is small enough, N will be large and so the following inequality will hold: where in the last inequality we used Lemma A.3. Therefore the "summit" is: dpT n y, T n y 1 q ď dpT n x, T n yq`dpT n x, T n x 1 q`dpT n x 1 , T n y 1 q (A.19) " O`δ κs˘( by (A.14)) .
We proceed to the proof of Proposition 2.10. In what follows, the constants implicit in O can be taken uniform on a a C 0 -neighborhood of Φ in Aut θ K pE, T q. For n ě 0, define Γ n :" H u T n y 1 ÐT n y˝H s T n yÐT n x´H s T n y 1 ÐT n x 1˝H u T n x 1 ÐT n x .
Let us estimate the norm of these linear maps. First, We estimate the first term: Using (2.14) and Proposition 2.2, we conclude that 1 " O`max dpx, yq θ , dpy, y 1 q θ (˘.
Any of these four distances, say dpy, y 1 q, is less than the sum of the other three; so: Now if n P v0, N w, the corresponding quadrilateral has sides are bounded by ε 0 (estimates (A.16)-(A.21)), and the exact same argument yields: }Γ n } " O`max dpT n x, T n x 1 q θ , dpT n x, T n yq θ , dpT n x 1 , T n y 1 q θ (˘. As a consequence of estimates (A.16) and (A.18), if j P v0, N w then dpT j y 1 , T j xq " O`e´a λ pjq s pxq˘. So it follows from Lemma A.2 that }pΦ n y 1 q´1} }Φ n x } " O`e ϕ pnq py 1 q˘, for n P v0, N w . (A.23) Now we want to iterate backwards to obtain a finer estimate for }Γ 0 }. By the groupoid properties of holonomies, Γ 0 " pΦ n y 1 q´1˝Γ n˝Φ n x . Therefore: " O`δ θs˘( by (A.14)) .
Proposition 2.10 is proved.
Proof of Lemma 3.7 (the strong bunching constant). Fix any positive Suppose Φ is a pη 0 , θq-bunched automorphism. This means that the function ϕ defined by (A.9) is less than η 0 λ u . Therefore we have pointwise inequalities: So there exists θ s ě η 0 that satisfies (A.10).

A.4 The metric on the Grassmannian
Let E be an inner product space of dimension d. If V 1 , V 2 Ď E are subspaces of the same dimension p ą 0, we define: where each F i runs over all linear isomorphisms F i : R p Ñ V i such that }F´1 i } ď 1. (We consider R p endowed with the canonical inner product, and }¨} always denotes the operator norm.) Proposition A.4. d is a metric on the Grassmannian G p pEq.
Proof of Proposition 3.1 (linearly induced maps are Lispchitz). Consider a linear isomorphism L : E Ñ F between d-dimensional inner product spaces. For each i P t1, 2u, let V i P G p pEq, and let W i :" Then G i is a linear isomorphism and }G´1 i } ď 1. So Taking infimum over the F i 's, we obtain dpW 1 , W 2 q ď bolpLqdpV 1 , V 2 q. This proves that the map induced by L has Lipschitz constant bolpLq.
Proof of Proposition 3.3 (span is locally Lipschitz). This is an easy consequence of the definition (A.25), and details are left to the reader.

A.5 Typical fiber bunched automorphisms are irreducible
Recall from Subsection 2.3 that End θ pE, T q denotes the vector space of θ-Hölder endomorphisms, which becomes a Banach space with the θ-Hölder norm (2.4). The set Aut θ pE, T q of θ-Hölder automorphisms and the subset B Ă Aut θ pE, T q of fiber bunched automorphisms are both open subsets of End θ pE, T q (actually they are C 0 -open).
A subset of a Banach space is said to be of infinite codimension if it is locally contained in the union of finitely many closed submanifolds of arbitrarily large codimension.
Proposition A.6. Suppose X is infinite, T : X Ñ X is a transitive hyperbolic homeomorphism, and E is a θ-Hölder vector bundle over X. Then there exists an open and dense subset I of the set B Ă Aut θ pE, T q of fiber bunched automorphisms such that every Φ P I is irreducible. Furthermore, the set B I has infinite codimension.
The proof is an obvious adaptation of arguments from [BGV, Via], so we will make it concise.
Proof. As a consequence of shadowing and expansivity, the hyperbolic homeomorphism T has infinitely many periodic points (see e.g. [Akin,p. 228]). Select one of these, say a point p of period k, and a homoclinic point q associated to p. If Φ is reducible then it admits a non-trivial θ-Hölder Φ-invariant subbundle F which by Corollary 3.5 is both H u -and H s -invariant. Then the subspace F p Ď E p is invariant under two different linear maps, namely Φ k p and H u pÐq˝H u qÐp . On the other hand, we claim that the property that these two maps admit a common nontrivial invariant subspace is atypical in the space Aut θ pE, T q; more precisely, it has positive codimension and, a fortiori, empty interior.
First note that the property that an element of GLpd, Rq admits infinitely many invariant subspaces is atypical (because it implies the existence of a complex eigenvalue of geometric multiplicity bigger than 1). So for typical Φ P Aut θ pE, T q, the collection of Φ k p -invariant subspaces is finite.
On the other hand, choose a closed neighborhood U of q that is disjoint from the future and past iterates of q. If we perturb the automorphism in this neighborhood (or rather in π´1pU q) then the maps Φ k p and H u qÐp are unaffected, but H s pÐq changes, and actually any small perturbation of H s pÐq can be realized with a perturbation of Φ supported in U . In particular, with a well-chosen perturbation, the composition of holonomies sends each of the (finitely many) Φ k p -invariant nontrivial subspaces of E p into something transverse to it. Such an automorphism Φ cannot be reducible. This shows that irreducibility has dense interior in Aut θ pE, T q.
The argument actually shows that if Φ is reducible then it must satisfy infinitely many independent conditions of positive codimension, at least one for each homoclinic orbit. Therefore reducibility has infinite codimension. See [Via,§4] for full details.

A.6 Bressaud-Quas Closing Lemma
Here we will prove Theorem 7.3. Though our formulation is different, the key ideas come from [BQ].
Let f : Y Ñ Y be any homeomorphism of a compact metric space pY, dq. For ε ą 0, an pε, d, f q-pseudoorbit is a string of points px 0 , x 1 , . . . , x n´1 q such that dpf px i q, x i`1 q ă ε for every i P v0, n´2w. If additionally dpf px n´1 q, x 0 q ă ε then we say that the pseudo-orbit is periodic, with period n; in that case indices can be taken as integers mod n instead. Let Rpε, d, f q denote the minimal period of a periodic pε, d, f q-pseudoorbit. Note that: Rpε, d, f q ď nRpε, d, f n q for every n ě 1. (A.26) A set E Ď Y is called pε, dq-separated if dpx, yq ě ε for every pair of distinct points x, y P E. Let Spε, dq be the minimal cardinality of a pε, dq-separated set.
Define a sequence of metrics by: d n,f px, yq :" max iPv0,n´1w dpf i pxq, f i pyqq .
Lemma A.7. Let ε ą 0. Suppose that Rpε, d, f q ą m ą 0. Then: log m ď log Sp ε 2 , dq´1 m log Spε, d m,f q`1 . Proof. Let E be a pd, ε 2 q-separated set of maximal cardinality. Note that the d-balls of radius ε 2 and centers at the points of E cover Y , because otherwise we could enlarge E by adding any point not covered.
Let F be a pd m,f , εq-separated set of maximal cardinality. For each y P F , choose a m-tuple px 0 , . . . , x m´1 q of points in E such that dpx j , f j pyqq ă ε 2 for each j P v0, m´1w. First, we claim that these x j 's are all distinct. Indeed, if x j " x k with j ă k, then dpf j pyq, f k pyqq ă ε, so`f i pyq˘i Pvj,k´1w is a periodic pε, dq-pseudoorbit of period k´j ď m´1 ă Rpε, dq, contradiction.
Second, we claim that if y ‰ y 1 P F then the corresponding m-tuples px 0 , . . . , x m´1 q and px 1 0 , . . . , x 1 m´1 q are distinct. Indeed, if the two m-tuples coincide then for each j P v0, m´1w we have dpf j pyq, f j py 1 qq ă ε. This means that d m,f py, y 1 q ă ε. Since the set F is pd m,f , εq-separated, we conclude that y " y 1 . Third, we claim that if y ‰ y 1 P F then the sets tx 0 , . . . , x m´1 u and tx 1 0 , . . . , x 1 m´1 u are distinct. Indeed, if the two sets coincide then x 1 i " x σpiq for some permutation σ of v0, m´1w. By the previous claim, this permutation is not the identity; therefore there exists ℓ P v0, m´2w such that k :" σpℓq ą σpℓ`1qj. Then dpf ℓ py 1 q, f k pyqq ă ε and dpf ℓ`1 py 1 q, f j pyqq ă ε. Thereforè f j pyq, f j`1 pyq, . . . , f k´1 pyq, f ℓ py 1 q˘is a periodic pε, dq-pseudoorbit of period k´j`1 ď m ă Rpε, dq, contradiction.
We conclude that the number of elements of the set F cannot exceed the number of subsets of the set E with exactly m elements, that is, Taking log's, recalling that |E| " Spd, ε 2 q and |F | " Spd m,f , ε 2 q, and rearranging, we obtain the inequality stated in the lemma.
Recall that a homeomorphism f : Y Ñ Y is called expansive if there is a uniform separation between every pair of distinct orbits. In that case, the topological entropy h top pf q is finite; furthermore, for every sufficiently small ε ą 0, the limit lim nÑ8 log Spε, d n , f q n exists and equals h top pf q. (A.27) (see [Wal,p. 174,177]).
Lemma A.8. If f is expansive then for every sufficiently small ε ą 0 we have: lim nÑ8 log Rpε, d n,f , f q n " 0 .
Proof. Fix a small ε ą 0 and a large integer n. Write R n :" Rpε, d n,f , f q and m n :" tpR n´1 q{nu. Assume that m n ą 0, otherwise plog R n q{n is already small. Using (A.26), we have Rpε, d n,f , f n q ą m n . Applying Lemma A.7, we obtain that log m n n ď log Spε{2, d n,f q n´l og Spε, d nmm q nm n`1 n .
By (A.27), the right-hand side is small: the first two terms essentially cancel each other. It follows that plog R n q{n is small.
Proof of Theorem 7.3. Given the hyperbolic homeomorphism T and the compact T -invariant set Y ‰ ∅, let f be the restriction of T to Y . Hyperbolic homeomorphisms are expansive (recall Remark 2.5), so f is expansive as well.
Fix ε ą 0 small enough so that Lemma A.8 applies. Note that if px i q iPZ{kZ is a periodic pε, d n , f q-pseudoorbit then, letting y i :" f rn{2s px i q, we have, for all i P Z{kZ, max jPv´rn{2s,rn{2s´1w dpf j`1 py i q, f j py i`1 qq ă ε .
Hyperbolicity implies that dpf py i q, y i`1 qq ă Ce´λ n εε n , where C and λ are positive constants. That is, py i q iPZ{kZ is a periodic pε n , d, f q-pseudoorbit. So Rpε, d f,n , f q ě Rpε n , d, f q -N n . In particular, log N n n and log N n log ε´1 n also tend to 0 as n Ñ 8.
Therefore, for any given τ ą 0, if n is large enough then ε n ă N´τ n . By definition, there exists a periodic pε n , d, T q-pseudoorbit of period N n in the set Y . By the Lipschitz shadowing lemma [Sak,Thrm. 2], there exist a periodic orbit for T of period N n within distance Opε n q " OpN´τ n q. This proves the theorem.

B Appendix: Examples
Here we present examples that show some of the limits of our results.

B.1 Examples of sets of calibrated vectors with exceptional behavior
The following two examples show that the set K of calibrated vectors defined by (6.7) can have exceptional fibers, justifying Remark 6.8.
Example B.1. Let T : X Ñ X be a hyperbolic homeomorphism having a fixed point x0. Let f be a non-negative Hölder function vanishing only at x0. Consider the cocycle Apxq :"ˆ1 0 0 e´f pxq˙.
Example B.2. Suppose T : X Ñ X is a homeomorphism admitting two nonempty compact invariant sets X1, X2 such that: • each Xi equals the support of some T -invariant probability measure µi; • X1 Y X2 " X; • X1 X X2 " tx0u Y tT n y0 ; n P Zu where x0 is a fixed point and y0 ‰ x0 is an homoclinic point.
Let f be a non-negative continuous function vanishing only at X1 X X2. Consider the cocycle Apxq :"ˆ1 0 0 e´f pxq˙i f x P X1, Apxq :"ˆe´f pxq 0 0 1˙i f x P X2.
Then the corresponding automorphism Φ on the trivial bundle E :" XˆR 2 has βpΦq " 0, and its Mather sets are M1pΦq " X and M2pΦq " tx0u. The euclidian norm is extremal. Consider the corresponding set K of calibrated vectors, defined by (6.7). Then Ky 0 " R 2 despite the fact that y0 P M1pΦq M2pΦq.
Remark B.3. One may contend that "correctly" defined Mather sets should not lie in the base X, but instead in the bundle E, or in its projectivizationÊ. Fix a norm }¨} on E and define a function f :Ê Ñ R by f prusq :" logp}Φu}{}u}q. LetM be the union of the supports of all probability measures onÊ that are invariant under the automorphismΦ and that maximize the integral of the function f . Let M :" tu P E ; u " 0 or rus PMu. This is a closed subset of E that projects down on the Mather set M pΦq Ď X. Given an extremal norm, it is clear that the fibers of M are calibrated, i.e. Mx Ď Kx for every x P M pΦq. A stronger property actually holds: spanpMxq Ď Kx for every x P M pΦq; we omit the proof. However, Mx may fail to be a subspace. Indeed, in Example B.2 the set My 0 is a union of two lines.

B.2 On Riemannian extremal norms
After having established the existence of extremal Finsler norms (under appropriate hypotheses), one naturally wonders about the existence of extremal Riemannian norms. Let us begin with a weak positive result: Proposition B.4. In the situation of Theorem 6.5, there exists a Riemannian norm }¨} 1 such that for all x P Y , the spaces F x and G x are orthogonal, and }Φpvq} 1 " e βpΦq ą }Φpwq} 1 for all unit vectors v P F x , w P G x .
Proof. As usual, assume βpΦq " 0. For each x P Y , consider the restriction of the extremal norm to the space F x , and let B x Ď F x be the unit ball. Let E x be the John ellipsoid of B x , namely the unique ellipsoid of maximal volume contained in B x (see e.g. [Ball]). This field of ellipsoids is continuous, since finding the John ellipsoid is a continuous operation (as a consequence of its uniqueness). Consider the Riemannian norm on the bundle F whose unit balls are the E x 's. Since Φ x pB x q " B T x and the John ellipsoid is equivariant with respect to linear isomorphisms, we obtain Φ x pE x q " E T x . This means that the Riemannian norm just constructed on the bundle F is preserved by Φ. 11 In the bundle G, we use the standard construction of Lyapunov norms (see e.g. [KH,p. 667]). Fix a small positive ε, and for each x P M p and w P G x , let: }w} 1 x :"˜8 ÿ n"0 e 2εn }Φ n x pvq} 2¸1
As a consequence of domination, the series converges exponentially, so the formula yields a well-defined continuous Riemannian norm on the bundle G. It is immediate that }Φ x pwq} 1 ď e´ε}w} 1 , so the norm along G is uniformly contracted. Finally, we extend the Riemannian norm to the fibers E x for x P Y by declaring F x and G x to be orthogonal. This completes the construction.
The Riemannian norm }¨} 1 provided by Proposition B.4 is extremal over the restricted subbundle E Y " π´1pY q. Can one extend this Riemannian norm to the whole bundle, keeping it extremal? The answer is no, as we will see next.
We will present an example of an irreducible fiber bunched automorphism in dimension 2 that admits no Riemannian extremal norm.
Since the spectral radius of A 0 is 1, it follows that βpΦq " 0, and so |||¨||| is a (constant) extremal norm, as claimed. Also note that δ p is the unique Lyapunovmaximizing measure.
Proof. Assume for a contradiction that Φ admits a Riemannian extremal norm t}¨} x u xPX . Since e βpΦq " 1 is the spectral radius of F ppq " A 0 , we must have: }A 0 } pÐp ď 1 , in the operator norm notation (1.2). This means that if D Ă R 2 denotes the unit ball in the norm }¨} p , we have A 0 pDq Ď D. Since the norm is assumed to be Riemannian, D is a (filled) ellipse, and since A 0 is a rotation, this ellipse must be a disk. Rescaling the norm if necessary, we can assume that D is the unit disk. Equivalently, }¨} pÐp is the usual euclidian operator norm, which for emphasis we will write }¨} eucl . Consider the homoclinic point q -p. . . 0, 0, 9 1, 0, 0 . . . q, i.e., the sequence that has a unique symbol 1 at position 0. Note that for any k ą 0 we have the identity: In particular, taking k " 1, by triviality of local holonomies (5.13) we obtain: Using that A 0 preserves euclidian norm, applying the extremal Riemannian norm to (B.2): Taking k Ñ 8 we obtain that }A 1 } eucl ď 1. This is a contradiction: actually }A 1 } eucl " 0.8 ? 2 ą 1 (see Fig. 5). Let us comment on other properties of our example. We claim that there are perturbationsΦ of Φ for which the measure δ p ceases to be Lyapunovmaximizing; so the "locking property" (verrouillage) is not satisfied. Indeed, let k " 1 be an integer, let m :" 4k`2, and letÃ 0 be rotation matrix of angle π 2´π 4m . Then:Ã m 0 A 1 "ˆ´0 .8 ? 2 0 0´0.1 ? 2˙.
Consider the associated one-step cocycleF , and the associated automorphism Φ. Then the probability measureμ supported on the orbit of the periodic point p :" p. . . , 9 1, 0, . . . , 0 loomoon m , 1, 0, . . . , 0 loomoon m , . . . q, i.e.,p n " 1 iff m`1 divides n, has Lyapunov exponent χ 1 pΦ,μq " logp0.8 ? 2q m`1 ą 0 " χ 1 pΦ, δ p q , showing that δ p was "unlocked". Therefore the argument of the proof of Proposition B.5 does not apply to the perturbationΦ, and it is possible that these perturbationsΦ admit Riemannian extremal norms (though there is no obvious candidate). So the main property of our example Φ, namely not to possess Riemannian extremal norms, may be fragile. Going beyond this specific example, we ask: Question B.6. Let T : X Ñ X be a hyperbolic automorphism. Let E be a 2dimensional θ-Hölder vector bundle over X. Let B Ă Aut θ pE, T q be the set of fiber bunched irreducible automorphisms, endowed with the θ-Hölder topology. Let R Ă B be the subset of automorphisms that admit a Riemannian extremal norm. Is R dense in B? Is the interior of R dense in B?