论文标题
西奈台球的自然平衡度量家族
A family of natural equilibrium measures for Sinai billiard flows
论文作者
论文摘要
西奈台球在两道弯曲（即周期性洛伦兹气体）上流动是连续的流动，但并非无处可区分。假设有限的视野，我们将流量的平衡状态与西奈台球地图$ t $的平衡状态联系起来，这是一张不连续的地图。我们提出了与潜在$ g $相关的拓扑压力$ p _*（t，g）$的定义。我们证明，对于满足温和假设的任何分段hölder潜在的$ g $，$ p _*（t，g）$等于使用跨度或分离集的Bowen的定义。我们提供了足够的条件，在该条件下，潜力引起了西奈台球地图的平衡状态。我们证明，在这种情况下，均衡状态$μ_g$是独特的，伯努利，适应并为所有非空的开放式设置提供了积极的措施。为此，我们利用作用于各向异性Banach空间的正确选择的转移操作员，并通过配对其最大特征向量来构建该度量。最后，我们证明流动不变概率度量$ \barμ_g$，通过将$μ_g$的乘积与沿轨道的Lebesgue Measure一起获得，是Bernoulli和Flow Adpopted。我们给出了台球表的例子，其中存在一套满足这些足够条件的开放式潜力。
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.
