标题： 拓扑慢熵，序列熵，以及广义的$[T,T^{-1}]$系统 显示英文标题

标题： Topological slow entropy, sequence entropy, and generalized $[T,T^{-1}]$ systems

摘要： 我们考虑由斜积给出的拓扑动力系统$S\rtimes_{\tau} T$，其中$S\colon Y\to Y$是一个子移位，$\tau\colon Y\to\mathbb{Z}$是一个连续的上同调，$T$是一个任意的可逆拓扑系统。 对于固定的$(Y,S,\tau)$，可能会发生所有形式为$S\rtimes_\tau T$的系统具有相同的拓扑熵，因此出现了区分这两种系统的問題。 我们证明，如果$T_1$和$T_2$是具有不同拓扑熵的可逆拓扑动力系统，那么可以使用 Katok 和 Thouvenot 引入的慢熵来区分$S\rtimes_{\tau} T_1$和$S\rtimes_{\tau} T_1$。 我们在纤维系统在某个尺度上具有不同慢熵的假设下，证明了类似的结果（如果$T_1$和$T_2$都有零熵，或者有相同的熵，这可以应用）。 这些结果对$(Y,S,\tau)$的假设相当温和，并且可以应用于基系统中的一些零熵系统。 我们推广了 Goodman 在额外有限拓扑维数假设下证明的序列熵理论中的经典结果。 我们证明了任何系统的测度论序列熵都受其拓扑序列熵的限制。 在序列的额外假设下，我们建立了变分原理，并证明了系统的拓扑序列熵等于其拓扑熵乘以一个仅依赖于序列的正常数。 显示英文摘要

摘要： We consider topological dynamical systems given by skew products $S\rtimes_{\tau} T$, where $S\colon Y\to Y$ is a subshift, $\tau\colon Y\to\mathbb{Z}$ is a continuous cocycle, and $T$ is an arbitrary invertible topological system. For fixed $(Y,S,\tau)$ it may happen that all systems of the form $S\rtimes_\tau T$ have the same topological entropy, and thus it arises the problem of distinguishing two such systems. We show that if $T_1$ and $T_2$ are invertible topological dynamical systems with different topological entropy then $S\rtimes_{\tau} T_1$ and $S\rtimes_{\tau} T_1$ can be distinguished using slow entropy as introduced by Katok and Thouvenot. We prove a similar result under the assumption that the fiber systems have different slow entropy at some scale (this can be applied if $T_1$ and $T_2$ have both zero entropy, or have the same entropy). These results require rather mild assumptions on $(Y,S,\tau)$, and can be applied to some entropy-zero systems in the base. We generalize classical results in the theory of sequence entropy, which were proved by Goodman with the additional assumption of finite topological dimension. We show that the measure-theoretic sequence entropy of any system is bounded by its topological sequence entropy. Under an extra assumption on the sequence we establish a variational principle, and prove that the topological sequence entropy of a system equals its topological entropy multiplied by a positive constant that depends only on the sequence.