熵的累积阶数
Orders of accumulation of entropy
摘要：
对于具有有限拓扑熵的紧致可度量化空间$X$的连续映射$T$，$T$的熵的累积阶是在熵结构和符号扩张的背景下产生的可数序数.我们证明了每个可数序数都可以实现为某个动力系统的积累序。我们的证明依赖于可度量化的Choquet单纯形的泛函分析和Downarowicz和Serafin的实现定理。进一步地，如果$M$是可度量化的Choquet单形，则我们将不变测度的单形仿射同胚于$M$的动力系统的熵的累积阶作为序数。这些界限是根据$\overline{\ex(M)}$的Cantor-Bendixson秩、$M$极值点的闭包以及$\overline{\ex(M)}$相对于$\ex(M)$的Cantor-Bendixson秩给出的。我们还讨论了这些边界的最优性。
英文摘要：
For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if $M$ is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to $M$. These bounds are given in terms of the Cantor-Bendixson rank of $\overline{\ex(M)}$, the closure of the extreme points of $M$, and the relative Cantor-Bendixson rank of $\overline{\ex(M)}$ with respect to $\ex(M)$. We also address the optimality of these bounds.
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