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A direct proof of the existence of MME for finite horizon Sinai billiards
Dynamical Systems · Submitted 20 Mar 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Assuming finite horizon and another condition we introduce -- namely \emph{negligible singularities} -- we prove that the metric pressure map associated with the billiard map $T$ is upper semi-continuous, as well as the compactness of the set of $T$-invariant measures.
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The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gaz, is a discontinuous map. Assuming finite horizon and another condition we introduce -- namely \emph{negligible singularities} -- we prove that the metric pressure map associated with the billiard map $T$ is upper semi-continuous, as well as the compactness of the set of $T$-invariant measures. In particular, for the potentials $g \equiv 0$ and $g = -h_{\rm top}(\Phi^1) \tau$, we recover the recent results of the existence of measures of maximal entropy (MME) for both the billiard map and flow $\Phi^t$, due to Baladi and Demers for $T$, jointly with Carrand for $\Phi^t$. We prove that the negligible singularities condition is generic among the billiard table with $C^{3+\alpha}$ boundary, with respect to the $C^{3+\alpha}$ topology. For finite horizon Sinai billiards, we provide bounds on the defect of upper semi-continuity of the metric pressure map and on the topological tail entropy. Assuming that singularities are negligible and sparse recurrence hold, we deduce the equidistribution of the periodic orbits with respect to the unique MME. We provide examples of billiard tables satisfying both conditions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gaz, is a discontinuous map. | background 1.00 | 0.09 | 0.02 | asserted 1.00 |
| s1 | Assuming finite horizon and another condition we introduce -- namely \emph{negligible singularities} -- we prove that the metric pressure map associated with the billiard map $T$ is upper semi-continuous, as well as the compactness of the set of $T$-invariant measures. | result 0.82 | 0.95 | 0.43 | asserted 1.00 |
| s2 | In particular, for the potentials $g \equiv 0$ and $g = -h_{\rm top}(\Phi^1) \tau$, we recover the recent results of the existence of measures of maximal entropy (MME) for both the billiard map and flow $\Phi^t$, due to Baladi and Demers for $T$, jointly with Carrand for $\Phi^t$. | interpretation 0.53 | 0.50 | 0.29 | asserted 1.00 |
| s3 | We prove that the negligible singularities condition is generic among the billiard table with $C^{3+\alpha}$ boundary, with respect to the $C^{3+\alpha}$ topology. | result 0.89 | 0.95 | 0.10 | asserted 1.00 |
| s4 | For finite horizon Sinai billiards, we provide bounds on the defect of upper semi-continuity of the metric pressure map and on the topological tail entropy. | result 0.93 | 0.95 | 0.11 | asserted 1.00 |
| s5 | Assuming that singularities are negligible and sparse recurrence hold, we deduce the equidistribution of the periodic orbits with respect to the unique MME. | result 0.98 | 0.93 | 0.13 | asserted 1.00 |
| s6 | We provide examples of billiard tables satisfying both conditions. | result 0.53 | 0.94 | 0.04 | asserted 1.00 |
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The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. We provide two separate conditions for a finite horizon billiard to admit equilibrium states. The first one -- namely \emph{negligible singularities} -- only concerns the billiard table and implies upper semi-continuity for the metric pressure map of $T$. The condition of negligible singularities is shown to be generic (dense $G_\delta$) among billiard tables with $C^{3+\alpha}$ boundaries, in the $C^{3+\alpha}$ topology, and implies the sparse recurrence condition for the null potential. Compactness of the set of $T$-invariant measures is proved. For finite horizon Sinai billiards, we provide a bound on the defect of upper semi-continuity of the metric pressure map. From it, a condition of existence of equilibrium states only involving potentials is deduced. In both situations, for the potentials $g \equiv 0$ and $g = -h_{\rm top}(\Phi^1) \tau$, we recover and extend the recent results of the existence of measures of maximal entropy (MME) for both the map $T$ and the flow $\Phi^t$, due to Baladi and Demers for $T$, jointly with Carrand for $\Phi^t$. Upper semi-continuity of the metric entropy is shown. Assuming negligible singularities, equidistribution of the periodic orbits with respect to the unique MME is proved. A bound on the topological tail entropy is given.
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In Section [ref], we provide a bound the defect of upper semi-continuity, similar to the ones obtained in [ref] in other contexts, as well as its consequences, such that the equidistribution of the periodic orbits with respect to the MME (assuming both sparse recurrence and negligible singularities).
Thermodynamic formalism and the entropy at infinity of the geodesic flow · The case of Sinai Billiards · 0.92
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