标题： 马尔可夫随机映射的物理测度的有限性 显示英文标题

标题： Finitude of physical measures for Markovian random maps

摘要： 我们研究与由遍历马尔可夫链驱动的离散时间随机动力系统相关的斜积变换$F$的物理测度的有限性。 我们开发了一个框架，使用马尔可夫过程的独立同分布(i.i.d.)表示，该框架有助于将已广泛研究的伯努利(i.i.d.)情形下的结果转移到马尔可夫情境中。 具体而言，我们建立了存在有限多个遍历的$F$-不变测度的条件，这些测度相对于参考测度是绝对连续的，并且它们对于可测有界观测量的统计吸引域几乎处处覆盖相空间。 此外，我们研究了一个较弱的概念，即要求存在有限多个物理测度（不一定绝对连续），其弱$^*$吸引域几乎处处覆盖相空间。 我们证明，对于由有限状态空间上的马尔可夫链驱动的紧致度量空间上的随机映射，如果系统是大部分收缩的，即所有马尔可夫不变测度都有负的最大李雅普诺夫指数，则该性质成立。 该结果应用于在无不变概率测度或有限不变集条件下，圆周和区间上的随机$C^1$微分同胚。 我们还将我们的结果与霍尔德连续函数空间上柯尔莫哥洛夫算子的准紧性联系起来。 显示英文摘要

摘要： We study the finiteness of physical measures for skew-product transformations $F$ associated with discrete-time random dynamical systems driven by ergodic Markov chains. We develop a framework, using an independent and identically distributed (i.i.d.) representation of the Markov process, that facilitates transferring results from the well-studied Bernoulli (i.i.d.) setting to the Markovian context. Specifically, we establish conditions for the existence of finitely many ergodic, $F$-invariant measures, absolutely continuous with respect to a reference measure, such that their statistical basins of attraction for measurable bounded observables cover the phase space almost everywhere. Furthermore, we investigate a weaker notion, which demands finitely many physical measures (not necessarily absolutely continuous) whose weak$^*$ basins of attraction cover the phase space almost everywhere. We show that for random maps on compact metric spaces driven by Markov chains on finite state spaces, this property holds if the system is mostly contracting, i.e., if all the Markovian invariant measures have negative maximal Lyapunov exponents. This result is applied to random $C^1$ diffeomorphisms of the circle and the interval under conditions based on the absence of invariant probability measures or finite invariant sets, respectively. We also connect our result to the quasi-compactness of the Koopman operator on the space of H\"older continuous functions.