作为后临界集不变概率测度空间的Choquet单纯形
Choquet simplices as spaces of invariant probability measures of post-critical sets
摘要：
遍历分解定理的一个众所周知的结论是：赋予弱$^*$拓扑的拓扑动力系统的不变概率测度空间是非空的可度量化Choquet单形。证明了每个非空的可度量化Choquet单形都是Logistic映射的后临界集上的不变概率测度空间。这里，Logistic映射的后临界集是其唯一临界点的$ω$ -极限集.事实上，我们证明了Logistic映射$f$可以这样取，使得它的后临界集是一个Cantor集，其中$f$是最小的，并且使得这个集上的每个不变概率测度的李雅普诺夫指数为零，并且是势$- \ln |f'|$的平衡态.
英文摘要：
A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak$^*$ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-critical set of a logistic map. Here, the post-critical set of a logistic map is the $ω$-limit set of its unique critical point. In fact we show the logistic map $f$ can be taken in such a way that its post-critical set is a Cantor set where $f$ is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential $- \ln |f'|$.
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