ON THE MEASURE OF MAXIMAL ENTROPY FOR FINITE HORIZON SINAI BILLIARD MAPS
ON THE MEASURE OF MAXIMAL ENTROPY FOR FINITE HORIZON SINAI BILLIARD MAPS
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DOI：
10.1090/jams/939
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发表时间：
2020-04-01
影响因子：
3.9
通讯作者：
Demers, Mark F.
中科院分区：
文献类型：
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作者：
Baladi, Viviane;Demers, Mark F.
两环上的西奈台球映象，即周期洛伦兹气体，是一个不连续的映象。在有限视界的情况下，给出了的拓扑熵的定义。我们证明了它不小于变分原理给出的值，并且它等于用支撑集或分离集的Bowen定义。在稀疏递归到奇点的温和条件下，我们得到了更多的结果：首先，利用作用于各向异性分布空间的转移算子，我们构造了(即)的最大熵的不变概率测度，证明了它是完全支撑的且是Bernoulli的，并且证明了它是最大熵的唯一测度，并且它不同于光滑不变测度，除非所有非掠食周期轨道的乘子都等于。其次，等于对连续的非紧域的限制的Bowen-Pesin-Pitskel拓扑熵。最后，应用Lima和Matheus的结果，Buzzi对其进行了改进，该地图至少在周期点上适用于所有人。参考文献
The Sinai billiard mapon the two-torus, ie, the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definitionfor the topological entropy of. We prove thatis 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 measureof maximal entropy for(ie,), we show thathas full support and is Bernoulli, and we prove thatis the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to. Second,is equal to the Bowen–Pesin–Pitskel topological entropy of the restriction ofto a noncompact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the maphas at leastperiodic points of periodfor all. References