On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps
| Autor: | Viviane Baladi, Mark F. Demers |
|---|---|
| Přispěvatelé: | Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Laboratoire de Probabilités, Statistique et Modélisation (LPSM UMR 8001), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Computer Science, Fairfield University, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)) |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
General Mathematics
[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] FOS: Physical sciences Dynamical Systems (math.DS) Topological entropy Space (mathematics) [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] 01 natural sciences Measure (mathematics) Multiplier (Fourier analysis) Mathematics - Spectral Theory Variational principle 0103 physical sciences FOS: Mathematics 0101 mathematics Mathematics - Dynamical Systems Spectral Theory (math.SP) Mathematical Physics Mathematical physics Mathematics Applied Mathematics 010102 general mathematics Mathematical Physics (math-ph) Nonlinear Sciences - Chaotic Dynamics 37D50 (Primary) 37C30 37B40 37A25 46E35 47B38 (Secondary) Functional Analysis (math.FA) Mathematics - Functional Analysis Domain (ring theory) [NLIN.NLIN-CD]Nonlinear Sciences [physics]/Chaotic Dynamics [nlin.CD] 010307 mathematical physics Invariant measure Chaotic Dynamics (nlin.CD) Dynamical billiards [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] |
| Zdroj: | Journal of the American Mathematical Society Journal of the American Mathematical Society, American Mathematical Society, In press Journal of the American Mathematical Society, American Mathematical Society, 2020, 33 (2), pp.381-449. ⟨10.1090/jams/939⟩ |
| ISSN: | 0894-0347 |
| DOI: | 10.1090/jams/939⟩ |
| Popis: | The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure $\mu_*$ of maximal entropy for $T$ (i.e., $h_{\mu_*}(T)=h_*$), we show that $\mu_*$ has full support and is Bernoulli, and we prove that $\mu_*$ is the unique measure of maximal entropy, and that it is different from the smooth invariant measure except if all non grazing periodic orbits have multiplier equal to $h_*$. Second, $h_*$ is equal to the Bowen--Pesin--Pitskel topological entropy of the restriction of $T$ to a non-compact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map $T$ has at least $C e^{nh_*}$ periodic points of period $n$ for all large enough $n \in \mathbb{N}$. Comment: 69 pages, 3 Figures. Version v3 is the electronic copy of the published version. The last lines of section 2.3 should be replaced by "if $h _* > s_0 \log 2$ then there exists $C > 0$ and $M\ge 2$ so that $\#Fix T^m \ge C \exp(h _*m)$ for all $m \ge M$ [Bu, Theorem 1.5]". The last sentence of the abstract holds for all large enough $n$ |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|