论文标题
用于分散台球的热力学形式主义
Thermodynamic formalism for dispersing billiards
论文作者
论文摘要
对于两道托上的任何有限的Horize Horize Sinai台球图T t，我们发现T _*> 1，以便对于（0，t_*）中的每个t，都存在一个唯一的平衡状态$μ_t$，for $ - t \ log j^ut $，$μ_t$ t-opapted。 （特别是，SRB度量是$ - \ log j^ut $的独特平衡状态。）我们表明，$μ_t$指数为持有人可观察到，而压力函数$ p（t）= \sup_μ\ \ {h_μ-\ \ \ \ \ f int t \ int t \ log j^ut j^ut j^ut j^ut \dμ\} $是Analytic or Analytic or Analytic or Analytic or Analytic or tim TIC**********此外，且仅当$ \ log j^ut $不是$μ_t$ a.e时，p（t）是严格凸出的。同时出现常数，而如果存在$ t_a \ ne t_b $，则$μ_{t_a} =μ__{t_b} $，则P（t）在（0，t_*）上。附加的稀疏复发条件给出$ \ lim_ {t \ to 0} p（t）= p（0）$。
For any finite horizon Sinai billiard map T on the two-torus, we find t_*>1 such that for each t in (0,t_*) there exists a unique equilibrium state $μ_t$ for $- t\log J^uT$, and $μ_t$ is T-adapted. (In particular, the SRB measure is the unique equilibrium state for $- \log J^uT$.) We show that $μ_t$ is exponentially mixing for Holder observables, and the pressure function $P(t)=\sup_μ\{h_μ-\int t\log J^uT d μ\}$ is analytic on (0,t_*). In addition, P(t) is strictly convex if and only if $\log J^uT$ is not $μ_t$ a.e. cohomologous to a constant, while, if there exist $t_a\ne t_b$ with $μ_{t_a}= μ_{t_b}$, then P(t) is affine on (0,t_*). An additional sparse recurrence condition gives $\lim_{t\to 0} P(t)=P(0)$.
