Measure of maximal entropy for finite horizon Sinai billiard flows

Baladi, Viviane; Carrand, Jérôme; Demers, Mark F.

Annales Henri Lebesgue, Tome 7 (2024), pp. 727-747

Using recent work of Carrand on equilibrium states for the billiard map, and adapting techniques from Baladi and Demers, we construct the unique measure of maximal entropy (MME) for two-dimensional finite horizon Sinai (dispersive) billiard flows ${\Phi }^1$ (and show it is Bernoulli), assuming the bound $h_{\text{top}}\left({\Phi }^1\right){\tau }_{min}>s_{0}log2$, where $s_0\in (0,1)$ quantifies the recurrence to singularities. This bound holds in many examples (it is expected to hold generically).

En combinant un travail récent de Carrand sur les états d’équilibre de l’application billard avec des techniques dues à Baladi et Demers, nous construisons l’unique mesure d’entropie maximale des flots billards de Sinai (dispersifs) à horizon fini en dimension deux (nous montrons aussi que cette mesure est Bernoulli) sous l’hypothèse $h_{\text{top}}\left({\Phi }^1\right){\tau }_{min}>s_{0}log2$, où $s_0\in (0,1)$ mesure le taux de récurrence aux singularités. Cette hypothèse est vérifiée dans de nombreux exemples (on s’attend à ce qu’elle soit génériquement satisfaite).

Classification : 37C83, 37D40, 37D35, 37B40, 37C30
Keywords: Sinai billiard flow, finite horizon, measure of maximal entropy, equilibrium state

[BCFT18] Burns, Keith; Climenhaga, Vaughn; Fisher, Todd L.; Thompson, Daniel J. Unique equilibrium states for geodesic flows in nonpositive curvature, Geom. Funct. Anal., Volume 28 (2018), pp. 1209-1259 | MR | Zbl | DOI

[BD20] Baladi, Viviane; Demers, Mark F. On the measure of maximal entropy for finite horizon Sinai billiard maps, J. Am. Math. Soc., Volume 33 (2020), pp. 381-449 | MR | DOI | Zbl

[BD22] Baladi, Viviane; Demers, Mark F. Thermodynamic formalism for dispersing billiards, J. Mod. Dyn., Volume 18 (2022), pp. 415-493 | DOI | Zbl

[Bow72a] Bowen, Rufus Entropy-expansive maps, Trans. Am. Math. Soc., Volume 164 (1972), pp. 323-331 | MR | DOI | Zbl

[Bow72b] Bowen, Rufus Periodic orbits for hyperbolic flows, Am. J. Math., Volume 94 (1972), pp. 1-30 | MR | DOI | Zbl

[Bow74] Bowen, Rufus Maximizing entropy for a hyperbolic flow, Math. Syst. Theory, Volume 7 (1974), pp. 300-303 | MR | DOI | Zbl

[BT08] Bálint, Peter; Tóth, Imre P. An application of Young’s tower method: Exponential decay of correlations in multidimensional dispersing billiards (2008) (Preprints of the Erwin Schrödinger International Institute for Mathematical Physics, https://math.bme.hu/~walzer/lecturenotes/balinttoth08esi.pdf)

[BW72] Bowen, Rufus; Walters, Peter Expansive one-parameter flows, J. Differ. Equations, Volume 12 (1972), pp. 180-193 | MR | DOI | Zbl

[Car22] Carrand, Jérôme A family of natural equilibrium measures for Sinai billiard flows (2022) (to appear in Annales de l’Institut Henri Poincaré, Probabilités et Statistiques) | arXiv

[Car23] Carrand, Jérôme Ergodic properties of low dimensional flows including dispersive billiards, Ph. D. Thesis, Sorbonne University, Paris, France (2023)

[Che01] Chernov, Nikolai I. Sinai billiards under small external forces, Ann. Henri Poincaré, Volume 2 (2001), pp. 197-236 | MR | DOI | Zbl

[CKW21] Climenhaga, Vaughn; Knieper, Gerhard; War, Khadim Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points, Adv. Math., Volume 376 (2021), 107452, 45 pages | MR | DOI | Zbl

[CM06] Chernov, Nikolai I.; Markarian, Roberto Chaotic Billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, 2006 | MR | Zbl | DOI

[CT21] Climenhaga, Vaughn; Thompson, Daniel J. Beyond Bowen’s specification property, Thermodynamic formalism. CIRM Jean-Morlet chair, fall 2019 (Pollicott, Mark et al., eds.) (Lecture Notes in Mathematics), Volume 2290, Springer, 2021, pp. 3-82 | MR | DOI | Zbl

[DK24] Demers, Mark F.; Korepanov, Alexey Rates of mixing for the measure of maximal entropy of dispersing billiard maps, Proc. Lond. Math. Soc., Volume 128 (2024), e12578, 38 pages | MR | DOI | Zbl

[DZ11] Demers, Mark F.; Zhang, Hong-Kun Spectral analysis for the transfer operator for the Lorentz gas, J. Mod. Dyn., Volume 5 (2011), pp. 665-709 | MR | DOI | Zbl

[Fra77] Franco, Ernesto Flows with unique equilibrium states, Am. J. Math., Volume 99 (1977), pp. 486-514 | MR | DOI | Zbl

[Kni98] Knieper, Gerhard The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. Math., Volume 148 (1998) no. 1, pp. 291-314 | DOI | Zbl

[LM18] Lima, Yuri; Matheus, Carlos Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities, Ann. Sci. Éc. Norm. Supér., Volume 51 (2018), pp. 1-38 | MR | Numdam | Zbl | DOI

[Mar69] Margulis, Grigoriĭ A. Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkts. Anal. Prilozh., Volume 3 (1969), pp. 89-90 | DOI | Zbl

[Mar04] Margulis, Grigoriĭ A. On some Aspects of the Theory of Anosov systems, Springer Monographs in Mathematics, Springer, 2004 (with a survey by R. Sharp: Periodic orbits of hyperbolic flows) | MR | Zbl | DOI

[Mis73] Misiurewicz, Michał Diffeomorphism without any measure with maximal entropy, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., Volume 21 (1973), pp. 903-910 | MR | Zbl