论文标题
中间因素熵的普遍性
Ubiquity of entropies of intermediate factors
论文作者
论文摘要
我们认为拓扑动力系统$（x，t）$，其中$ x $是一个紧凑的Metrizable Space，$ T $表示同型同构$ x $上的可数amenable amenable tobs $ g $的动作。对于两个这样的系统，$（x，t）$和$（y，s）$和一个因子映射$π：x \ rightarrow y $，一个中间因素是拓扑动力系统$（z，r）$，可以将$π$写为因子映射$ψ：x \ rightarrow z $和$ rightarrow z $ and $ ostoum y和z \ rightarrow y y y y y y y y y y y y y y y $ y $。在本文中，我们表明，对于任何可数的$ g $ g $，对于任何$ g $ -subshifts $（x，t）$和$（y，s）$，以及对于任何因素映射$π：x \ rightarrow y $，中等次级shift shift y $的熵在此间隔$ [h（y，s），h（y，x，x，x，x，x，t）中是密集的。作为推论，我们还证明，如果$（x，t）$和$（y，s）$是零维$ g $ -Systems，则中等零维因子的熵集等于间隔$ [H（y，s），h（x，x，x，t）] $。我们的证明依赖于可能具有独立利益的广义标记引理。
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.
