中介因子熵的普遍性
Ubiquity of entropies of intermediate factors
摘要：
考虑拓扑动力系统$(X,T)$，其中$X$是紧致可度量化空间，$T$表示可数顺从群$G$通过同胚作用在$X$上.对于两个这样的系统$(X,T)$和$(Y,S)$以及因子映射$π: X \rightarrow Y$，中间因子是拓扑动力学系统$(Z,R)$，对于该拓扑动力学系统$π$可以被写为因子映射$ψ: X \rightarrow Z$和$φ: Z \rightarrow Y$的合成。本文证明了对任意可数顺从群$G$，对任意$G$ -子移位$(X,T)$和$(Y,S)$，对任意因子映射$ π:X \rightarrow Y$，中间子移位因子的熵集在区间$[h(Y,S), h(X,T)]$中是稠密的.作为推论，我们还证明了如果$(X,T)$和$(Y,S)$是零维$G$ -系统，则中间零维因子的熵的集合等于区间$[h(Y,S), h(X,T)]$ .我们的证明依赖于一个广义标记引理，可能是独立的利益。
英文摘要：
We consider topological dynamical systems $(X,T)$, where $X$ is a compact metrizable space and $T$ denotes an action of a countable amenable group $G$ on $X$ by homeomorphisms. For two such systems $(X,T)$ and $(Y,S)$ and a factor map $π: X \rightarrow Y$, an intermediate factor is a topological dynamical system $(Z,R)$ for which $π$ can be written as a composition of factor maps $ψ: X \rightarrow Z$ and $φ: Z \rightarrow Y$. In this paper we show that for any countable amenable group $G$, for any $G$-subshifts $(X,T)$ and $(Y,S)$, and for any factor map $ π:X \rightarrow Y$, the set of entropies of intermediate subshift factors is dense in the interval $[h(Y,S), h(X,T)]$. As a corollary, we also prove that if $(X,T)$ and $(Y,S)$ are zero-dimensional $G$-systems, then the set of entropies of intermediate zero-dimensional factors is equal to the interval $[h(Y,S), h(X,T)]$. Our proofs rely on a generalized Marker Lemma that may be of independent interest.
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