Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner (with an appendix by Emilio Corso)

Davide Ravotti

doi:10.5802/jep.219

Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner (with an appendix by Emilio Corso)
[Asymptotiques et théorèmes limites pour intégrales ergodiques des les flots horocycliques à la Ratner]

Davide Ravotti ¹

¹ Monash University, School of Mathematics Clayton Campus, 3800 Victoria, Australia

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 305-334.

Résumé
Abstract

Nous appliquons une méthode inspirée du travail de Ratner sur le mélange quantitatif pour le flot géodésique [29] et développée par Burger [11] pour étudier les intégrales ergodiques pour les flots horocycliques. Nous en déduisons un développement asymptotique explicite pour les moyennes horocycliques, retrouvant ainsi un résultat célèbre de Flaminio et Forni [15], et nous montrons que les coefficients dans le développement asymptotique sont Hölder continus par rapport au point de base. En outre, nous fournissons des preuves courtes et simplifiées des théorèmes limites spatiaux de Bufetov et Forni [10] et, dans un appendice d’Emilio Corso, d’un théorème limite temporel de Dolgopyat et Sarig [12].

We apply a method inspired by Ratner’s work on quantitative mixing for the geodesic flow [29] and developed by Burger [11] to study ergodic integrals for horocycle flows. We derive an explicit asymptotic expansion for horocycle averages, recovering a celebrated result by Flaminio and Forni [15], and we show that the coefficients in the asymptotic expansion are Hölder continuous with respect to the base point. Furthermore, we provide short and streamlined proofs of the spatial limit theorems of Bufetov and Forni [10] and, in an appendix by Emilio Corso, of a temporal limit theorem by Dolgopyat and Sarig [12].

Reçu le : 2022-03-11
Accepté le : 2023-02-09
Publié le : 2023-02-22

DOI : 10.5802/jep.219

Classification : 37D40, 37A17, 37A25, 37A50
Keywords: Horocycle flow, ergodic averages, distributional limit theorems
Mot clés : Flot horocyclique, moyennes ergodiques, théorèmes limites distributionnels

Affiliations des auteurs :

Davide Ravotti ¹

¹ Monash University, School of Mathematics Clayton Campus, 3800 Victoria, Australia

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Davide Ravotti. Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner (with an appendix by Emilio Corso). Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 305-334. doi : 10.5802/jep.219. https://jep.centre-mersenne.org/articles/10.5802/jep.219/

Bibliographie
Cité par

[1] A. Adam & V. Baladi - “Horocycle averages on closed manifolds and transfer operators”, Tunis. J. Math. 4 (2022) no. 3, p. 387-441 | DOI | MR | Zbl

[2] A. Avila, D. Dolgopyat, E. Duryev & O. Sarig - “The visits to zero of a random walk driven by an irrational rotation”, Israel J. Math. 207 (2015) no. 2, p. 653-717 | DOI | MR | Zbl

[3] A. Avila, G. Forni, D. Ravotti & C. Ulcigrai - “Mixing for smooth time-changes of general nilflows”, Adv. Math. 385 (2021), article ID 107759, 65 pages | DOI | MR | Zbl

[4] V. Bargmann - “Irreducible unitary representations of the Lorentz group”, Ann. of Math. (2) 48 (1947), p. 568-640 | DOI | MR | Zbl

[5] J. Beck - “Randomness of the square root of 2 and the giant leap, Part 1”, Period. Math. Hungar. 60 (2010) no. 2, p. 137-242 | DOI | MR | Zbl

[6] J. Beck - “Randomness of the square root of 2 and the giant leap, Part 2”, Period. Math. Hungar. 62 (2011) no. 2, p. 127-246 | DOI | MR | Zbl

[7] P. Billingsley - Convergence of probability measures, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1999 | DOI

[8] M. Bromberg & C. Ulcigrai - “A temporal central limit theorem for real-valued cocycles over rotations”, Ann. Inst. H. Poincaré Probab. Statist. 54 (2018) no. 4, p. 2304-2334 | DOI | MR | Zbl

[9] A. I. Bufetov - “Limit theorems for translation flows”, Ann. of Math. (2) 179 (2014) no. 2, p. 431-499 | DOI | MR | Zbl

[10] A. I. Bufetov & G. Forni - “Limit theorems for horocycle flows”, Ann. Sci. École Norm. Sup. (4) 47 (2014) no. 5, p. 851-903 | DOI | MR | Zbl

[11] M. Burger - “Horocycle flow on geometrically finite surfaces”, Duke Math. J. 61 (1990) no. 3, p. 779-803 | DOI | MR | Zbl

[12] D. Dolgopyat & O. Sarig - “Temporal distributional limit theorems for dynamical systems”, J. Statist. Phys. 166 (2017) no. 3-4, p. 680-713 | DOI | MR | Zbl

[13] G. K. Eagleson - “Some simple conditions for limit theorems to be mixing”, Teor. Veroyatnost. i Primenen. 21 (1976) no. 3, p. 653-660 | MR | Zbl

[14] S. C. Edwards - “On the rate of equidistribution of expanding translates of horospheres in $Γ ∖ G$ ”, Comment. Math. Helv. 96 (2021) no. 2, p. 275-337 | DOI | MR | Zbl

[15] L. Flaminio & G. Forni - “Invariant distributions and time averages for horocycle flows”, Duke Math. J. 119 (2003) no. 3, p. 465-526 | DOI | MR | Zbl

[16] H. Furstenberg - “The unique ergodicity of the horocycle flow”, in Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972), Lect. Notes in Math., vol. 318, Springer, Berlin, 1973, p. 95-115 | MR | Zbl

[17] I. M. Gelfand & S. V. Fomin - “Geodesic flows on manifolds of constant negative curvature”, Uspehi Mat. Nauk 7 (1952) no. 1(47), p. 118-137 | MR

[18] B. M. Gurevič - “The entropy of horocycle flows”, Dokl. Akad. Nauk SSSR 136 (1961), p. 768-770 | MR

[19] B. Hasselblatt & A. Katok - “Principal structures”, in Handbook of dynamical systems, vol. 1A, North-Holland, Amsterdam, 2002, p. 1-203 | DOI | Zbl

[20] G. A. Hedlund - “Fuchsian groups and transitive horocycles”, Duke Math. J. 2 (1936) no. 3, p. 530-542 | DOI | MR | Zbl

[21] B. Marcus - “The horocycle flow is mixing of all degrees”, Invent. Math. 46 (1978) no. 3, p. 201-209 | DOI | MR | Zbl

[22] R. McCutcheon - “The Gottschalk-Hedlund theorem”, Amer. Math. Monthly 106 (1999) no. 7, p. 670-672 | DOI | MR | Zbl

[23] E. Paquette & Y. Son - “Birkhoff sum fluctuations in substitution dynamical systems”, Ergodic Theory Dynam. Systems 39 (2019) no. 7, p. 1971-2005 | DOI | MR | Zbl

[24] O. S. Parasyuk - “Flows of horocycles on surfaces of constant negative curvature”, Uspehi Mat. Nauk 8 (1953) no. 3(55), p. 125-126 | MR

[25] M. Ratner - “The central limit theorem for geodesic flows on $n$ -dimensional manifolds of negative curvature”, Israel J. Math. 16 (1973), p. 181-197 | DOI | MR

[26] M. Ratner - “Factors of horocycle flows”, Ergodic Theory Dynam. Systems 2 (1982) no. 3-4, p. 465-489 (1983) | DOI | MR | Zbl

[27] M. Ratner - “Rigidity of horocycle flows”, Ann. of Math. (2) 115 (1982) no. 3, p. 597-614 | DOI | MR | Zbl

[28] M. Ratner - “Horocycle flows, joinings and rigidity of products”, Ann. of Math. (2) 118 (1983) no. 2, p. 277-313 | DOI | MR | Zbl

[29] M. Ratner - “The rate of mixing for geodesic and horocycle flows”, Ergodic Theory Dynam. Systems 7 (1987) no. 2, p. 267-288 | DOI | MR | Zbl

[30] D. Ravotti - “Quantitative equidistribution of horocycle push-forwards of transverse arcs”, Enseign. Math. (2) 66 (2020) no. 1-2, p. 135-150 | DOI | MR | Zbl

[31] A. Strömbergsson - “On the deviation of ergodic averages for horocycle flows”, J. Modern Dyn. 7 (2013) no. 2, p. 291-328 | DOI | MR | Zbl

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