A finite dimensional proof of a result of Hutchings about irrational pseudo-rotations

We prove that the Calabi invariant of a $C^1$ pseudo-rotation of the unit disk, that coincides with a rotation on the unit circle, is equal to its rotation number. This result has been shown some years ago by Michael Hutchings (under very slightly stronger hypothesis). While the original proof used Embedded Contact Homology techniques, the proof of this article uses generating functions and the dynamics of the induced gradient flow.


Introduction
1.1.Statement of the main theorem.-We denote by D the closed unit disk of the Euclidean plane and by S the unit circle.We will furnish D with the standard area form ω = dx∧dy and will denote by Diff 1 ω (D) the group of diffeomorphisms of class C 1 that preserve ω (we will say that f is symplectic).Note that every element of Diff 1 ω (D) preserves the orientation and therefore is isotopic to the identity.More precisely, the set Diff 1 ω (D) is path-connected when furnished with the C 1 -topology.We will also denote by Homeo * (D) the group of orientation preserving homeomorphisms of D (which is also path-connected when furnished with the C 0 -topology).
If f ∈ Diff 1 ω (D) fixes every point in a neighborhood of S, its Calabi invariant Cal(f ) ∈ R is a well-studied object that has several interpretations (see [Cal70], [Fat80], [GG97] for instance).It admits a natural extension to Diff 1 ω (D).We will explain this extension, as described by Benoit Joly in his thesis [Jol21].If κ is a primitive of ω, and A : D → R a primitive of f * (κ) − κ, then S A dµ does not depend on µ, where µ is a Borel probability measure invariant by f |S . (1)Consequently, there exists a unique primitive A f,κ of f * (κ) − κ satisfying S A f,κ dµ = 0 for such µ.Moreover, the quantity Cal(f ) = D A f,κ ω does not depend on the choice of κ.For example we have Cal(f ) = 0 if f is an Euclidean rotation.Note that Cal(f ) is the usual Calabi invariant in case f fixes every point in a neighborhood of S (in that case, A f,κ vanishes on a neighborhood of S).
We can also define a real function Cal on the universal covering space Diff 1 ω (D) of Diff 1 ω (D) as follows.Every identity isotopy of f in Diff 1 ω (D), meaning every continuous path I = (f s ) s∈[0,1] in Diff 1 ω (D) joining the identity map of D to f , is homotopic, relative to its endpoints, to a Hamiltonian isotopy I ′ = (f ′ s ) s∈[0,1] .It means that there exists a time dependent divergence free vector field (X s ) s∈[0,1] of class C 1 such that for every z ∈ D it holds that d ds f ′ s (z) = X s (f s (z)).There exists a (uniquely defined) family (H s ) s∈[0,1] of functions of class C 2 vanishing on S, such that for every s ∈ [0, 1], every z ∈ D and every v ∈ R 2 it holds that dH s (z).v = ω(v, X s (z)).The quantity Cal(I) = 1 0 D H s ω ds depends only on the homotopy class [I] of I and we can also denote it Cal([I]).Let us give a simple example.For every α ∈ R, denote R α the rotation of angle 2πα and T α the isotopy (R sα ) s∈[0,1] .It holds that Cal([T α ]) = π 2 α.As stated in [Jol21], there is a link between Cal and Cal that we will explain now.Every identity isotopy I = (f s ) s∈[0,1] of f in Diff and developed by Gambaudo and Ghys [GG97].This interpretation is still valid in this more general situation.Consider the usual angular form dθ = xdy − ydx x 2 + y 2 and define Consider f ∈ Diff 1 ω (D).For every identity isotopy I = (f s ) s∈[0,1] of f in Homeo * (D) and every (z, z ′ ) ∈ W , one gets a path Generalizing a proof due to Shelukhin [She15] the following equality is proved in [Jol21].Let us add that this interpretation of the Calabi invariant has permitted to Gambaudo and Ghys [GG97] to prove that two elements of Diff 1 ω (D) fixing every point in a neighborhood of S and conjugate by an orientation and area preserving homeomorphism have the same Calabi invariant.To conclude, just note the following: -for every [I] ∈ Diff 1 ω (D) and every p, q in Z we have Cal( We will say that f ∈ Diff 1 ω (D) is an irrational pseudo-rotation if it fixes 0 and does not possess any other periodic point.Using an extension of Poincaré-Birkhoff theorem due to Franks [Fra88], we know that for every lift , and that α = α+Z ∈ R/Z is the Poincaré rotation number of f |S .We will say that f is a pseudo-rotation of rotation number α.The goal of the article is to prove the following: or equivalently, we have Cal(f ) = 0.
The fact that the equalities Cal([I]) = π 2 rot([I] |S ) and Cal(f ) = 0 are equivalent is due to the equation It is not difficult to prove that a C 1 diffeomorphism of S can be extended to a C 1 symplectic diffeomorphism of D. So, it is sufficient to prove the theorem in case f coincides with a rotation on the unit circle.This theorem was already known and due to Michael Hutchings (with a very slightly stronger hypothesis), appearing as an easy consequence of a theorem we will recall now.Suppose that f ∈ Diff 1 ω (D) coincides with a rotation R α in a neighborhood of S. Fix a primitive κ of ω and note that A f,κ vanishes on S whether α is rational or irrational.If O is a periodic orbit of f , one proves easily that z∈O A f,κ (z) does not depend on the choice of κ.So, one can define the mean action of O as being One has the following ( [Hut16]). (2)  Theorem 1.2.-Suppose that f ∈ Diff ∞ ω (D) coincides with a rotation in a neighborhood of S and denote O the set of periodic orbits of f .Then it holds that: (2) The proof is stated for smooth diffeomorphisms but should be possibly extended supposing a low differentiability condition, as Michael Hutchings explained to us.

J.É.P. -M., 2023, tome 10
It is easy to prove that if f ∈ Diff 1 ω (D) coincides with a rotation on S, then we have act({0}) = 0. Consequently, by Theorem 1.2, it holds that Cal(f ) = 0 if f is an irrational pseudo-rotation.It must be noticed that Hutchings' theorem has been recently improved by Pirnapasov [Pir21] (still in the smooth category) that uses a preliminary extension result.A much weaker hypothesis on the boundary is needed, even weaker than being a rotation on the boundary.A nice corollary, is the fact that Cal(f ) = 0 for every smooth irrational pseudo-rotation f , regardless of any condition on the boundary.
Let us continue with a nice consequence of Theorem 1.1.Let D ⊂ D be a closed disk that does not contain 0 and does not meet S. One constructs easily f ′ ∈ Diff ∞ ω (D) arbitrarily close to the identity map for the C ∞ -topology, fixing every point outside D and such that Cal(f This implies that f • f ′ is not an irrational pseudo-rotation, which means that it has at least one periodic orbit different from {0}.Such a periodic orbit must meet D. The proof of Hutchings is based on a reduction to a problem of contact geometry and then the applications of methods of Embedded Contact Homology theory.The previous perturbation result is true in a more general situation.Asaoka and Irie [AI16], using Embedded Contact Homology as well, have proved that it remains true provided f ∈ Diff 1 ω (D) has no periodic point inside D. Other striking applications of contact or symplectic geometry to dynamics on surfaces have appeared recently that use elaborated tools of contact or symplectic geometry (for instance see Cristofaro-Gardiner, Humilière, Seyfaddini [CGHS20], Cristofaro-Gardiner, Prasad, Zhang [CGPZ21], or Edtmair, Hutchings [EH21]).The proof of Theorem 1.1 that will be given in the present article does not use Floer homology but only generating functions.Is there any hope to find proofs of these deep results using such classical tools?1.2.Idea of the proof.-Let us state first the following result of Bramham [Bra15]: is the limit, for the C 0 topology, of a sequence of finite order C ∞ diffeomorphisms.
Write α = rot(f ).The proof says something more precise: if (q n ) n⩾0 is a sequence of positive integers such that (q n α) n⩾0 converges to 0 in R/Z, then there exists a sequence of homeomorphisms (f n ) n⩾0 fixing 0 and satisfying (f n ) qn = Id, that converges to f for the C 0 topology (to obtain a sequence of smooth approximations one needs a simple additional argument of approximation).Theorem 1.1 would have been an easy consequence of Theorem 1.3 if the stronger following properties were true: -the f n are symplectic diffeomorphisms of class C 1 ; -the sequence (f n ) n⩾0 converges to f in the C 1 topology.
Unfortunately, there is no reason why the f n appearing in the construction of the sequence satisfy these properties.
The original proof of Theorem 1.3 uses pseudoholomorphic curve techniques from symplectic geometry.In [LC16] we succeeded to find a finite dimensional proof by using generating functions, like in Chaperon's broken geodesics method [Cha84].Let us remind the ideas of this last proof.The hypothesis were slightly different, the result was stated for an irrational pseudo-rotation f ∈ Diff 1 ω (D) coinciding with a rotation on S.This last property permits us to extend our map, also denoted f , to a piecewise C 1 diffeomorphism of the whole plane, being an integrable polar twist map with increasing rotation number in an annulus {z ∈ R 2 | 1 ⩽ |z| ⩽ r 0 } and equal to an irrational rotation in {z ∈ R 2 | |z| ⩾ r 0 }.Let I be an identity isotopy of f |D in Diff 1 ω (D) that fixes 0. We write α = rot([I] |S ).One can extend I to an identity isotopy (f s ) s∈[0,1] of f , also denoted I, where each f s is an integrable polar map in n .The map f being piecewise C 1 , one can write it as the composition of m maps "close to Id R 2 ", where m ⩾ 1, and then construct a m-periodic family (h i ) i∈Z of generating real functions that are C 1 with Lipschitz derivatives.One knows that for every b ⩾ 1, the fixed point set of f b corresponds to the singular point set of a A-Lipschitz vector field ζ defined on a 2mb-dimensional space E b , furnished with a natural scalar product.This vector field is the gradient flow of a function h defined in terms of the h i , i ∈ Z, and the constant A is independent of b.In particular each circle S a/b ⊂ R 2 corresponds to a curve Σ a ⊂ E b of singularities of ζ.A fundamental result is the fact that Σ a bounds a disk ∆ a ⊂ E b that contains the singular point corresponding to the fixed point 0 and that is invariant by the flow and by a natural Z/bZ action on E b .Moreover, the dynamics of the flow of ζ on ∆ a is north-south and the non trivial orbits have the same energy.This energy can be explicitly computed and is small if a/b is a convergent of α.Using the independence of the Lipschitz constant A, one can deduce that ζ is "uniformly small" on ∆ a .The disk ∆ a projects homeomorphically on the disk bounded by S a/b , denoted D a/b , by an explicit map q 1 .The Z/bZ action on ∆ a defines, by projection on D a/b , a homeomorphism f of order b that coincides with f on S a/b .Using what has been said above, in particular the fact that ζ is small on ∆ a , one can prove that f is uniformly close to f on the disk D a/b if a/b is a convergent of α.This is the way we prove the approximation result.
The orbits of ζ |∆a define a radial foliation on ∆ a that projects by q 1 onto a topological radial foliation F 1 of D a/b that is invariant by f .It is natural to ask whether one can compute the Calabi invariant of f by using the fact that f is arbitrarily close to f .The main problem is that f and F 1 are continuous.There is no differentiability anymore, and differentiability is crucial while dealing with Calabi invariants, regardless of the approach we choose.Nevertheless, the fact that we are dealing with Lipschitz maps close to Id (the maps f 1 , . . ., f m appearing in the decomposition of f ) permit us to state some quantitative results.The three key-points that will be used to get the proof are related to the foliation F 1 : (1) the energy of the non trivial orbits of ζ |∆a is bounded by K|bα − a|, where K does not depend on a and b; (2) the leaves of The assertion (3) says that the foliation f −b (F 1 ) winds relative to F 1 no more than 4mq.A precise definition will be given in the next section.Just say that if F is a radial foliation of class C 1 , there exists K > 0 such that for every b > 0 the winding distance between the foliations F and f −b (F) is bounded by Kb.The foliation that appears in our construction depends on a and b, is not differentiable, possesses interesting dynamical properties stated in (1) and (2) and nevertheless satisfies a "differentiallike" property.
The properties (1), (2) and (3) will permit us to bound the Calabi invariant of each map f |D a/b .Note that we will use the third definition, as the map is only piecewise C 1 on D a/b .The Calabi invariant will be small if a/b is a convergent that is close to α.By a limit process, we will prove that Cal(f |D ) = 0.
The rest of the article is divided into three sections.Section 2 will be dedicated to the study of topological radial foliations defined on a disk.We will introduce different objects on the set of such foliations, in particular the winding distance between two foliations.We will see how to compute the Calabi invariant of a symplectic diffeomorphism by using such a foliation.
In Section 3 we will recall the construction done in [LC16] and quickly explained in the introduction.In particular we will define F 1 and prove the properties (1), (2) and (3) stated above.
We will prove Theorem 1.1 in the short section 4.
many conversations we had on this subject some years ago and would like to thank him warmly.Finally, I would like to thank the referee, the many comments have permit to improve the redaction of the article.

Radial foliations and Calabi invariant
We will introduce a formal setting to study radial foliations on a disk, based on a "discretization of the angles", that seems pertinent for studying topological foliations.Classical objects related to disk homeomorphisms, like rotation numbers or linking numbers, will be investigated in this formalism.One can look at Bechara [Bec20] for similar questions in a differentiable context.
A radial foliation is an oriented topological foliation on D * such that every leaf ϕ is a ray, meaning that it satisfies The sets α(ϕ) and ω(ϕ) are defined as follows: if t → ϕ(t) is a real parametrization of ϕ compatible with the orientation of the leaf, then we have where the notation Y means the closure of Y in R 2 .In other words, ϕ tends to 0 in the past and to ∂D in the future.We denote F the set of radial foliations.The group Homeo * (D * ) of homeomorphisms of D * that are isotopic to the identity coincides with the group of orientation preserving homeomorphisms that fix the two ends of D * .It acts naturally on F: if F ∈ F and f ∈ Homeo * (D * ), then the foliation f (F), whose leaves are the images by f of the leaves of F, is a radial foliation.Moreover, Homeo * (D * ) acts transitively on F. If Homeo * (D * ) is endowed with the C 0 topology, meaning the compact-open topology applied to maps and the inverse maps, the stabilizer means the lift of F to the universal covering space D. Note that F is a trivial foliation.
In particular, every leaf ϕ of F is an oriented line of D that separates D into two connected open sets, the component of D ∖ ϕ lying on the left of ϕ will be denoted L( ϕ) and the component lying on the right will be denoted R( ϕ).We get a total order ⪯ F on the set of leaves of F as follows: We can also define a partial order ⩽ F on D: denote ϕ z the leaf of F that contains z and write and if the segment of ϕ z beginning at z and ending at z ′ inherits the orientation of ϕ z .
2.2.Topological angles in the universal covering space.-Consider the natural projection π : If Z/4Z is endowed with the topology whose open sets are and Z with the topology generated by the sets {2k +1} and {2k −1, 2k, 2k +1}, k ∈ Z, then π is a covering map.
(3) Note that both sets Z/4Z and Z are non Hausdorff but path connected.If k, ℓ are two integers, we will define Note that for every integers k, ℓ, m we have The quantity λ(k, ℓ) measures the "algebraic intersection number" of a continuous path γ : For every ( z, z ′ ) ∈ W , we can define a continuous function θ z, z ′ : F → Z/4Z as follows: The space F being simply connected, the Lifting Theorem asserts that there exists a continuous function, θ z, z ′ : F → Z, uniquely defined up to an additive constant in 4Z, such that π In particular, if F and F ′ are two radial foliations, the numbers do not depend on the choice of the lift θ.

□
The next result will be useful later: Lemma 2.2.-For every compact set K ⊂ W and every F, F ′ in F, there exists M > 0 such that for every ( z, z ′ ) ∈ K it holds that Proof.-For every ( z, z ′ ) ∈ W , there exists a neighborhood d z, z ′ × d ′ z, z ′ ⊂ W of ( z, z ′ ), where d z, z ′ and d ′ z, z ′ are topological closed disks.One can cover K by a finite family of such neighborhoods.So it is sufficient to prove the result when K = d × d ′ is the product of two topological closed disks.The map being continuous and the space d × d ′ × F being simply connected, one can lift this last map to a continuous map For every F in F, the function ( z, z ′ ) → θ z, z ′ (F) is continuous on d × d ′ and so is bounded by compactness of d × d ′ .This last affirmation comes from the fact that for every ℓ ⩾ 1, the set as the preimage of an open set by a continuous map.So, for every F, F ′ in F, the function Let us conclude with the following: Moreover, the equality Proof.
-One proves immediately that d is symmetric and satisfies the triangular inequality.The fact that d(F, F) = 0 for every F ∈ F is also obvious.It remains to prove that d(F, F ′ ) ̸ = 0 if F and F ′ are distinct radial foliations.In that case, one can find ( z, z ′ ) ∈ W such that z and z ′ belong to the same leaf of F and to different leaves of We will call d the winding distance between two radial foliations.
∈ Z and the difference between the value at 1 and the value at 0 of this last map is nothing but τ ( z, z ′ , F, f −1 (F)).Of course it depends neither on the choice of I, nor on the choice of the lift I of I. Suppose now that z and z ′ project onto two different points of D * .Note that if k is large enough, then for every s ∈ [0, 1], it holds that and so the functions Consequently, if F and F ′ are two radial foliations, then for every each sum being independent of the choice of the lifts z, z ′ of z, z ′ .

Note that we have the following:
Lemma 2.4.-For every compact subset K ⊂ W , for every F, F ′ in F, there exists M > 0 such that for every (z, z ′ ) ∈ K it holds that Proof.-Like in the proof of Lemma 2.2, it is sufficient to study the case where K = d × d ′ is the product of two topological closed disks.Choose a lift d of d and a lift d ′ of d ′ in D. Choose f ∈ Homeo * (D * ) such that F ′ = f −1 (F) and an identity isotopy (f s ) s∈[0,1] of f .Lift this isotopy to an identity isotopy ( f s ) s∈[0,1] on D. There exists k 0 > 0 such that if k ⩾ k 0 , then for every s ∈ [0, 1], every z ∈ d and every and this implies that To conclude, it remains to apply Lemma 2.2 to the compact sets d×T k ( d ′ ), |k| < k 0 .□ Let us conclude this section by stating an analogous of Lemma 2.1.Define first what is a displacement function.For every f ∈ Homeo * (D * ), every lift f of f to D and every ray ϕ we can define a function m f ,ϕ : D * → Z as follows: we choose a lift ϕ of ϕ, then for every z ∈ D, we consider the lift z of z such that z ∈ L( ϕ)∩R(T ( ϕ)) and we denote m f ,ϕ (z) the integer m such that f ( z) ∈ L(T m ( ϕ)) ∩ R(T m+1 ( ϕ)), noting that it does not depend on the choice of ϕ.Observe that m T k • f ,ϕ = m f ,ϕ + k, for every k ∈ Z.One proves easily the following result: Lemma 2.5.-For every compact subset K ⊂ D * and every rays ϕ, ϕ ′ , there exists M > 0 such that for every f ∈ Homeo * (D * ) and every lift f of f to D, it holds that The following result clarifies the lack of symmetry of (z, Proof.-Fix (z, z ′ ) ∈ W .During the proof, we will lighten the notations by writing By using Lemma 2.1 we get Recalling that z = z * and z ′ = z ′ * , we have and J.É.P. -M., 2023, tome 10 only on the ray ϕ and on f .
(2) The equality proved in Lemma 2.6 can be written (3) For every k ∈ Z it holds that (1) there exists a compact set (2) if ϕ is a ray and if K ⊂ D * is a compact set containing z, then for every ε > 0, there exists n 0 ⩾ 0 such that for every n ⩾ n 0 it holds that z).Using Lemma 2.5, one deduces that if the second assertion is true for a ray ϕ, it is true for every ray ϕ ′ .J.É.P. -M., 2023, tome 10 (1) Say that µ has a rotation number rot f (µ) ∈ R if µ-almost every point z has a rotation number and if the function rot f is µ-integrable, and in that case set (2) Say that µ has a self-linking number link f (µ) ∈ R, if µ is non atomic, µ has a rotation number, µ × µ-almost every pair (z, z ′ ) ∈ W has a linking number and the function link f is µ × µ-integrable, and in that case set

Remarks
(1) If there exists a ray ϕ such that m f ,ϕ is µ-integrable, then by Birkhoff Ergodic Theorem, one knows that µ has a rotation number and it holds that (2) If there exists a radial foliation F and a leaf ϕ of F such that m f ,ϕ is µ-integrable and λ f,F is µ × µ-integrable, then µ has a self-linking number and it holds that Suppose that f ∈ Diff 1 ω (D).Define D * = D ∖ (S ∪ {0}).Choose an identity isotopy I = (f s ) s∈[0,1] of f in Homeo * (D) that fixes 0 and write f for the lift of f |D * naturally defined by the restriction of I to D * .Extend the isotopy to a family (f s ) s∈R such that f s+1 = f s • f for every s ∈ [0, 1].Denote µ ω the finite measure naturally defined by ω.Consider the Euclidean radial foliation F * on D * , whose leaves are the paths (0, 1) ∋ t → te 2iπα , α ∈ [0, 1).Note now that for every leaf ϕ of F * , every n ⩾ 1 and every z ∈ D * it holds that This implies that m f ,ϕ is bounded and so, µ ωω has a rotation number (according to Definition 2.9) and we have Note now that for every n ⩾ 1 and every (z, z ′ ) ∈ W it holds that To get this inequality, one must consider a frame at z moving with time: at time n it is the image of the original frame at z by the rotation R ang I n (0,z) .Indeed, we have the two following properties: -the difference between the variation of angle of the vector f s (z ′ )−f s (z), s ∈ [0, n], in the moving frame and λ f n ,F * (z, z ′ ) is smaller than 1.
One deduces that µ ω has a self-linking number (according to Definition 2.9) and we have Let us conclude with a proposition, that will useful to prove Theorem 1.1, and that summarizes what has been done in this section.
Proposition 2.10.-Suppose that f ∈ Diff 1 ω (D) fixes 0. Let I be an identity isotopy of f and f be the lift of f |D * naturally defined by

Construction of a good radial foliation for an irrational pseudo-rotation.
The three first sections of this chapter come from [LC16].All proofs can be found there.The fourth one, concerning properties of projected foliations is mainly new.The last proposition of the fourth section and the isotopy defined in the last section already appeared in [LC99] but in a slightly different context (twist maps instead of untwisted maps).
3.1.Generating functions.-Let us denote π 1 : (x, y) → x and π 2 : (x, y) → y the two projections defined on the Euclidean plane R 2 .An orientation preserving homeomorphism f of R 2 will be called untwisted if the map (x, y) −→ (π 1 (f (x, y)), y) is a homeomorphism, which means that there exist two continuous functions g, g ′ on R 2 such that In this case, the maps X → g(X, y) and y → g ′ (X, y) are orientation preserving homeomorphisms of R. If moreover, f is area preserving, the continuous form xdy + Y dX is exact: there exists a C 1 function h : R 2 → R such that The function h, defined up to an additive constant, is a generating function of f .We can make precise the definition by saying that f is a K Lipschitz untwisted homeomorphism, where K ⩾ 1, if (i) f is untwisted; J.É.P. -M., 2023, tome 10 (ii) f is K bi-Lipschitz; (iii) the maps X → g(X, y) and y → g ′ (X, y) are K bi-Lipschitz; (iv) the maps y → g(X, y) and X → g ′ (X, y) are K Lipschitz.
If f is a diffeomorphism of R 2 , denote Jac(f )(z) the Jacobian matrix at a point z.One proves easily, that for every K > 1, there exists a neighborhood U of the identity matrix in the space of square matrices of order 2, such that every Suppose that f is a C 1 orientation preserving diffeomorphism of D that fixes 0 and coincides with a rotation R α on S. Fix β > α.We can extend our map to a homeomorphism of the whole plane (also denoted f ) such that: Using the fact that the group of orientation preserving C 1 diffeomorphisms of D (and the group of symplectic diffeomorphisms of D) that fix 0 and every point of S, when furnished with the C 1 topology, is path connected, one can prove the following: where each f i is a K Lipschitz untwisted homeomorphism that fixes 0 and induces a rotation on every circle of origin 0 and radius r ⩾ 1.Moreover, if f is area preserving, one can suppose that each f i preserves the area.
We fix K > 1 and a decomposition f = f m • • • • • f 1 given by Proposition 3.1.We define two families (g i ) 1⩽i⩽m , (g ′ i ) 1⩽i⩽m of continuous maps as follows For every i ∈ {1, . . ., m} one can find an identity isotopy , where each f i,s is an untwisted map such that , and a natural isotopy In that case every map is area preserving and it holds that is a generating function of f i,s .
J.É.P. -M., 2023, tome 10 3.2.The vector field associated to a decomposition.-We consider in this section a C 1 orientation preserving diffeomorphism f of D that fixes 0 and coincides with a rotation R α on S. We suppose that it is extended and then decomposed into untwisted maps as in Section 3.1.We keep the same notations.We extend the families to m periodic families i∈Z , and the family (h i ) 1⩽i⩽m to a m periodic family (h i ) i∈Z in case the f i are area preserving.
We fix an integer b ⩾ 1 and consider the finite dimensional vector space furnished with the scalar product where Observe that ζ is invariant by the (b periodic) shift Let us state some facts about ζ.
One deduces that the associated differential system defines a flow on E b .We will denote by z t the image at time t of a point z ∈ E b by this flow.As an application of Gronwall's lemma, one gets: In the case where the f i are area preserving, observe that ζ is the gradient vector field of the function and that h is invariant by φ.One can define the energy of an orbit (z t ) t∈R to be As a consequence of Lemma 3.2 it holds that Lemma 3.4.-For every z ∈ E b , one has For every i ∈ Z, define the maps Let us state the main properties of these maps: for every i ∈ Z, -Q 1 induces a bijection between the singular set of ζ and the fixed point set of f b , -the sequence 0 = (0) i∈Z is a singular point of ζ that is sent onto 0 by each Q i , 3.3.The case of an irrational pseudo-rotation.-In this section we keep the notation of Section 3.2 but we suppose than f is an irrational pseudo-rotation.We suppose moreover that β ̸ ∈ Q and that (α, β) ∩ Z = ∅.The extension f is a piecewise C 1 area preserving transformation that satisfies the following properties: -0 is the unique fixed point of f ; -there is no periodic point of period b if (bα, bβ) ∩ Z = ∅; -if (bα, bβ) ∩ Z ̸ = ∅, the set of periodic points of period b can be written α<a/b<β S a/b , where S a/b is the circle of center 0 and radius 1 + a/b − α.In that case, we have the additional following properties for the vector ζ defined on E b : -the singular set consists of the constant sequence 0 and of finitely many smooth closed curves (Σ a ) a∈(bα,bβ)∩Z ; -the curve Σ a is sent homeomorphically onto S a/b by each Proposition 3.8.-If z, z are two distinct points of E b satisfying z ′ − z ̸ ∈ V ′ , then there exists ε > 0 such that for every t ∈ (0, ε], it holds that: This result admits an infinitesimal version (see [LC99]): Proposition 3.9.-If z ∈ E b is non singular and satisfies ζ(z) ̸ ∈ W , then there exists ε > 0 such that for every t ∈ (0, ε], it holds that: As an immediate corollary of the second assertion, one gets the following result (not explicitly stated in [LC16]) The assertion (iii) tells us that the maps Q i | ∆a and Q ′ i | ∆a , i ∈ Z, induce homeomorphisms from ∆ a to D a/b denoted respectively q i and q ′ i (5) .The assertion (iv) tells us that the maps P i | ∆a , i ∈ Z, induce homeomorphisms from ∆ a to D a/b denoted p i .Note that p i •q −1 i is a homeomorphism which let invariant every horizontal segment of D a/b and induces on this segment an increasing homeomorphism and similarly p i •q ′−1 i is a homeomorphism which let invariant every vertical segment of D a/b and induces on this segment an increasing homeomorphism.We denote F the radial foliation defined on ∆ * a whose leaves are the gradient lines of ζ included in ∆ * a .For every s ∈ [0, 1], and every i ∈ Z we define q s i = (1 − s)q i + sp i , q ′s i = (1 − s)p i + sq ′ i .The maps q s i and q ′s i are homeomorphisms from ∆ a to D a/b (all coinciding on Σ a ) and send F onto a radial foliation of D a/b .3.4.Projected radial foliations.-Recall that d denotes the winding distance between two radial foliations.Let us begin with the following result: Lemma 3.11.-For every s 1 , s 2 in [0, 1] and every i ∈ Z, we have Proof.-We fix i ∈ Z and will prove the first inequality, the proof of the second one being similar.The leaves of q s i (F) are the paths t → q s i (z t ), z ∈ ∆ * a .Note that the map t → π 2 (q s i (z t )) is C 1 and that d dt π 2 • q s i (z t ) |t=0 = η i (z) = x i − g i (x i+1 , y i ).
(5) We do not refer to a to lighten the notations.
J.É.P. -M., 2023, tome 10 Say that z ∈ D * a/b is horizontal if it can be written z = q i (z), where x i −g i (x i+1 , y i ) = 0. Denote H i the set of horizontal points. (6)For every point z ∈ D * a/b , define J(z) to be equal to {z} if z is horizontal and to the largest horizontal interval contained in D * a/b ∖ H i and containing z, if z is not horizontal.The sign of η i (q −1 i (z ′ )) does not depend on the choice of z ′ ∈ J(z).We orient J(z) with x increasing if this sign is positive and with x decreasing if this sign is negative.Note that the path s → u s i (z) = q s i • q −1 i (z) is constant if z is horizontal and is an oriented segment of J(z) inheriting the same orientation if z is not horizontal.The fact that d dt π 2 • q s i (z t ) |t=0 = η i (z), for every s ∈ [0, 1] and every z ∈ ∆ * a , implies that if z is not horizontal, the oriented interval J(z) is transverse to the foliations q s i (F), s ∈ [0, 1], crossing locally every leaf from the left to the right.
Denote D a/b the universal covering space of D * a/b and for every z ∈ D a/b that lifts z ∈ D * a/b , denote J( z) the lift of J(z) containing z, with the induced orientation in case z ̸ ∈ H i .Write H i for the lift of H i , write q s i (F) for the lift of q s i (F) and denote ( u s i ) s∈[0,1] the identity isotopy that lifts (u s i ) s∈[0,1] .Suppose that z ∈ D a/b is not in H i .For every s 0 ∈ [0, 1], the oriented interval I( z) is transverse to the foliation q s0 i (F), crossing locally every leaf from the right to the left.Moreover, if 0 ⩽ s 1 < s 2 , then u s2 i ( z) follows u s1 i ( z) on J( z).Consequently it holds that: Now, let us fix two different points z 0 and z 1 in D a/b and s ∈ [0, 1].Observe that the three sets are connected, whether the points belong to H i or not.Indeed, the paths J( z 0 ) and J( z 1 ) in D a/b draw intervals of leaves of the foliation q s i (F) whose intersection is an interval of leaves if not empty.In particular the set of pairs ( z ′ 0 , z ′ 1 ) ∈ J( z 0 )× J( z 1 ) such that z ′ 0 and z ′ 1 are distinct and belong to the same leaf of ) being continuous on W , takes at most three values on J( z 0 ) × J( z 1 ) (either 0 or 2 is missing).
In particular, if 0 ⩽ s 1 < s 2 ⩽ 1, the map 6) if η i (z) = 0, then ξ i (z) ̸ = 0, so every leaf of p i (F) is a C 1 embedded line and H i is nothing but the set of points where the foliation p i (F) is horizontal.
J.É.P. -M., 2023, tome 10 takes at most three values and is lifted to a map taking at most three values, which implies that The homeomorphism u s i sends the foliation q i (F) onto the foliation q s i (F).The leaves of q i (F) are pushed on the right by the isotopy ( u s i ) s∈[0,1] .More precisely, for every 0 ⩽ s 1 < s 2 ⩽ 1 and every leaf ϕ of q i (F), it holds that: , where y i −g i−1 (x i , y i−1 ) = 0. Denote V i the set of vertical points and V i its lift in D a/b .Define u ′s i = q ′s i • p −1 i and lift the isotopy (u ′s i ) s∈[0,1] to an identity isotopy ( u ′s i ) s∈[0,1] .Write q ′s i (F) for the lift of q ′s i (F).Then, for every 0 ⩽ s 1 < s 2 ⩽ 1 and every leaf ϕ of q ′ i (F), it holds that: where It is an isotopy from Id to q The map v s i sends the foliation q i (F) onto the foliation q s i (F) if 0 ⩽ s ⩽ 1 and onto the foliation q ′ s−1 i (F) if 1 ⩽ s ⩽ 2. Moreover: -for every 0 ⩽ s 1 < 1 < s 2 ⩽ 2 and every leaf ϕ of F 0 i , it holds that R( v s2 i ( ϕ)) ⊂ R( v s1 i ( ϕ).Indeed, the sets H i and V i are disjoint.

Its periodicity comes from the equalities
Note also that f i fixes 0 and coincides with f i on S a/b because In particular, we have Proposition 3.12.-For every s 1 , s 2 in [0, 2mb] we have d fs1 (q 1 (F)), fs2 (q 1 (F)) ⩽ 2|s 2 − s 1 | + 4.
To conclude, we refer to Corollary 3.13 that says that for every ( z, z We deduce that µ ω has a rotation vector (which was obviously known) and that We deduce from Lemma 4.2 that λ f b ,F1 is integrable on W a/b for the measure µ ω × µ ω and that 1 ω (D) defines by restriction an identity isotopy I |S = (f s|S ) s∈[0,1] of f |S in Diff 1 (S).Moreover, every homotopy class [I] ∈ Diff 1 ω (D) defines by restriction a homotopy class [I] |S ∈ Diff 1 * (S), where Diff 1 The functionangI : W −→ R (z, z ′ ) −→ 1 2π I z,z ′ dθ depends only on the homotopy class of I in Homeo * (D) and is bounded because f is a diffeomorphism of class C 1 .In particular, one naturally gets a function ang [I] for every [I] ∈ Diff 1 ω (D).If I = (f s ) s∈[0,1]) and I ′ = (f ′ s ) s∈[0,1] are two identity isotopies in Diff 1 ω (D), one can define an identity isotopy II ′ = (f ′′ s ) s∈[0,1] in Diff 1 ω (D) writing: The homotopy class of II ′ depends only on [I] and [I ′ ] and one gets a group structure on Diff 1 ω (D) by setting [I][I ′ ] = [II ′ ].Note that for every n ⩾ 1 we have

F 1 are
Brouwer lines of I b T −a|D a/b ; (3) the winding distance between F 1 and f −b (F 1 ) is bounded by 4mb.Let us clarify these three points.The non trivial orbits of ζ |∆a have the same energy.This quantity measures the area swept by a leaf of F 1 along the isotopy I b T −a|D a/b .By saying that the leaves of F 1 are Brouwer lines, we mean that they are pushed on the right along the isotopy I b T −a|D a/b .Equivalently, we can say the following: let D a/b be the universal covering space of D * a/b = D a/b ∖ (S a/b ∪ {0}), let I = ( f s ) s∈[0,1] be the identity isotopy on D a/b that lifts I |D * a/b and set f = f 1 .Denote F 1 the (non singular) foliation of D a/b that lifts F 1|D * a/b .Every leaf ϕ of F 1 is an oriented topological line of D a/b that separates its complement into two components, one on the left and one on the right.The time one map T of the identity isotopy of D a/b that lifts T 1|D * a/b generates the group of covering automorphisms.The assertion (2) states that f b • T −a ( ϕ) is on the right of ϕ for every leaf ϕ.The assertion (1) states that the area of the domain between ϕ and f b • T −a ( ϕ), which is equal to the energy of the corresponding orbit of ζ, is bounded by K|bα − a|.

2. 1 .
The space of radial foliations.-Let D be an open disk of the form D = {z ∈ R 2 | |z| < r}, where r > 0. We will set D * = D ∖ {0} and will denote D the universal covering space of D * .We will consider the sets is bijective and F can be furnished with a natural C 0 topology.It is the topology, induced from the quotient topology defined on Homeo * (D * )/ Homeo * ,F (D * ) by this identification map.Note that this topology does not depend on F. It is well known that the fundamental groups of Homeo * (D * ) and Homeo * ,F (D * ) are infinite cyclic and that the morphism i * : π 1 (Homeo * ,F (D * ), Id) → π 1 (Homeo * (D * ), Id) induced by the inclusion map i : Homeo * ,F (D * ) → Homeo * (D * ) is bijective.This implies that F is simply connected.In the whole article, when F is a radial foliation, the notation F J.É.P. -M., 2023, tome 10

3. 5 .
Construction of a good isotopy.-One gets a m-periodic family of homeomorphisms ( f i ) i∈Z of D a/b by writing: