A family of natural equilibrium measures for Sinai billiard flows
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909143913201664 |
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| author | Carrand, Jérôme |
| author_facet | Carrand, Jérôme |
| contents | The Sinai billiard flow on the two-torus, i.e., the periodic Lorentz gas, is a continuous flow, but it is not everywhere differentiable. Assuming finite horizon, we relate the equilibrium states of the flow with those of the Sinai billiard map $T$ -- which is a discontinuous map. We propose a definition for the topological pressure $P_*(T,g)$ associated to a potential $g$. We prove that for any piecewise Hölder potential $g$ satisfying a mild assumption, $P_*(T,g)$ is equal to the definitions of Bowen using spanning or separating sets. We give sufficient conditions under which a potential gives rise to equilibrium states for the Sinai billiard map. We prove that in this case the equilibrium state $μ_g$ is unique, Bernoulli, adapted and gives positive measure to all nonempty open sets. For this, we make use of a well chosen transfer operator acting on anisotropic Banach spaces, and construct the measure by pairing its maximal eigenvectors. Last, we prove that the flow invariant probability measure $\bar μ_g$, obtained by taking the product of $μ_g$ with the Lebesgue measure along orbits, is Bernoulli and flow adapted. We give examples of billiard tables for which there exists an open set of potentials satisfying those sufficient conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_14444 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A family of natural equilibrium measures for Sinai billiard flows Carrand, Jérôme Dynamical Systems 37C83 (Primary) 37C30, 37D35, 37A25, 46E35, 47B38 (Secondary) The Sinai billiard flow on the two-torus, i.e., the periodic Lorentz gas, is a continuous flow, but it is not everywhere differentiable. Assuming finite horizon, we relate the equilibrium states of the flow with those of the Sinai billiard map $T$ -- which is a discontinuous map. We propose a definition for the topological pressure $P_*(T,g)$ associated to a potential $g$. We prove that for any piecewise Hölder potential $g$ satisfying a mild assumption, $P_*(T,g)$ is equal to the definitions of Bowen using spanning or separating sets. We give sufficient conditions under which a potential gives rise to equilibrium states for the Sinai billiard map. We prove that in this case the equilibrium state $μ_g$ is unique, Bernoulli, adapted and gives positive measure to all nonempty open sets. For this, we make use of a well chosen transfer operator acting on anisotropic Banach spaces, and construct the measure by pairing its maximal eigenvectors. Last, we prove that the flow invariant probability measure $\bar μ_g$, obtained by taking the product of $μ_g$ with the Lebesgue measure along orbits, is Bernoulli and flow adapted. We give examples of billiard tables for which there exists an open set of potentials satisfying those sufficient conditions. |
| title | A family of natural equilibrium measures for Sinai billiard flows |
| topic | Dynamical Systems 37C83 (Primary) 37C30, 37D35, 37A25, 46E35, 47B38 (Secondary) |
| url | https://arxiv.org/abs/2208.14444 |