AI 中文总结
该研究解决了Huang等人2014年关于拓扑动力系统遗传可下性的问题与猜想，证明有限熵系统遗传可下，而存在无限熵遍历不变测度的系统不具有该性质。
AI 中文摘要
设$(X,T)$为拓扑动力系统，$h(T,K)$表示紧集$K\subset X$的拓扑熵。我们解决了Huang、Ye和Zhang（2014）提出的关于遗传可下性的一个问题与一个猜想。首先，我们证明每个具有有限拓扑熵的系统都是遗传可下的：对任意非空紧集$K\subset X$及任意$0\leq h\leq h(T,K)$，存在紧集$K_h\subset K$使得$h(T,K_h)=h$，这否定了他们的问题2'。其次，我们证明若$(X,T)允许具有无限熵的遍历不变测度，则$(X,T)$不是遗传可下的；更确切地说，我们构造了一个具有无限熵的紧集$K$，使得$K$的每个紧子集的熵要么为0要么为无限，这证明了问题2'之后陈述的猜想。
英文摘要
Let $(X,T)$ be a topological dynamical system and let $h(T,K)$ denote the topological entropy of a compact set $K\subset X$. We settle a question and a conjecture raised by Huang, Ye, and Zhang (2014) concerning hereditary lowerability. First, we show that every system with finite topological entropy is hereditarily lowerable: for every nonempty compact set $K\subset X$ and every $0\leq h\leq h(T,K)$, there is a compact set $K_h\subset K$ such that $h(T,K_h)=h$. This gives a negative answer to their Question 2'. Second, we prove that if $(X,T)$ admits an ergodic invariant measure with infinite entropy, then $(X,T)$ is not hereditarily lowerable. More precisely, we construct a compact set $K$ with infinite entropy such that every compact subset of $K$ has entropy either zero or infinity. This proves the conjecture stated immediately after Question 2'.
