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arXiv 2403.13626math.DS
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有限视界 Sinai 台球存在最大熵测度的直接证明

A direct proof of the existence of MME for finite horizon Sinai billiards

Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlev\'e, F-59000 Lille, France

Jérôme Carrand

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Version v3 is the electronic copy of the version to appear in J. Eur. Math. Soc
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AI总结：

本文通过引入可忽略奇点条件，直接证明有限视界 Sinai 台球的度量压力上半连续性及 MME 存在性，并建立周期轨道等分布结果。

摘要：

二维环面上的 Sinai 台球映射 $T$，即周期 Lorentz 气体，是一个不连续映射。在有限视界以及我们引入的另一条件——即“可忽略奇点”——的假设下，我们证明了与台球映射 $T$ 相关的度量压力映射是上半连续的，并且 $T$-不变测度集合具有紧性。特别地，对于势函数 $g \equiv 0$ 和 $g = -h_{\rm top}(Φ^1) τ$，我们重新得到了最近关于台球映射和流 $Φ^t$ 均存在最大熵测度（MME）的结果；其中关于 $T$ 的结果归功于 Baladi 和 Demers，关于 $Φ^t$ 的结果则由他们与 Carrand 共同得到。我们证明，在具有 $C^{3+α}$ 边界的台球桌中，就 $C^{3+α}$ 拓扑而言，可忽略奇点条件是通有的。对于有限视界 Sinai 台球，我们给出了度量压力映射的上半连续性亏损以及拓扑尾部熵的界。在假设奇点可忽略且稀疏回归成立的情况下，我们推出周期轨道关于唯一 MME 的等分布性。我们还给出了同时满足这两个条件的台球桌示例。

英文摘要：

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 $G_δ$) among billiard tables with $C^{3+α}$ boundaries, in the $C^{3+α}$ 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 $g = -h_{\rm top}(Φ^1) τ$, we recover and extend the recent results of the existence of measures of maximal entropy (MME) for both the map $T$ and the flow $Φ^t$, due to Baladi and Demers for $T$, jointly with Carrand for $Φ^t$. 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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