非平衡热力学形式主义 第二部分：半Ruelle算子、共形测度与有效扩张及拟紧性
Thermodynamic Formalism Out of Equilibrium Part II: Semi-Ruelle Operator, Conformal Measures, and Effective Expansion and Quasi-Compactness
AI总结：
本文引入半Ruelle算子，构造共形测度与调和函数，证明POE变分原理中极大化测度的非平衡熵与半吉布斯估计，并在有效扩张条件下证明平均半Ruelle算子的拟紧性及谱间隙与维数界。
AI中文摘要：
我们引入了一套通用机制来研究非平衡热力学形式主义：一个拓扑马尔可夫移位（记为$\Sigma^-$）的热力学，其中势能由紧致度量空间$X$上的随机游走给出（随机性由吉布斯过程驱动）。我们引入了半Ruelle算子，它作用于$C(\Sigma^-\times X)$上。我们为半Ruelle算子构造了共形测度和调和函数。我们给出几个应用：（1）我们为POE变分原理提供了新的证明（POE代表与非平衡压力相关的非平衡压力），并证明了POE变分原理中的极大化测度具有正的“非平衡熵”，并满足“半吉布斯估计”。（2）在纤维$X$上的随机游走由$C^{1+}$微分同胚（允许非常耗散）给出，并且满足“平均有效扩张”的开条件的设定下，我们证明了平均半Ruelle算子在Sobolev函数空间上作用时是拟紧的。一个应用包括在假设体积衰减相关性时证明谱间隙，以及证明平稳测度的维数受相似维数约束的界。
英文摘要：
We introduce a general machinery to study {\em thermodynamic formalism out of equilibrium}: The thermodynamics of a topological Markov shift (denoted by $Σ^-$) where the potential is given by a random walk on a compact metric space $X$ (and the randomness is driven by a Gibbs process). We introduce the {\em semi-Ruelle operator}, which acts on $C(Σ^-\times X)$. We construct conformal measures and harmonic functions for the semi-Ruelle operator. We present a few applications: (1) We provide a new proof to the POE variational principle (POE stands for the {\em pressure out of equilibrium} which is associated with the process), and we show that maximizing measures in the POE variational principle admit positive {\em entropy out of equilibrium}, and satisfy {\em semi-Gibbs estimates}. (2) In the setting where the random walk on the fiber $X$ is given by $C^{1+}$ diffeomorphisms (which are allowed to be very dissipative), and it satisfies the open condition of {\em effective expansion on average}, we show that the {\em averaged semi-Ruelle operator} is quasi-compact when acting on a Sobolev function space. An application includes proving a spectral gap when assuming volume decay of correlations, and proving bounds on the dimension of stationary measure in terms of similarity dimension.
