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

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.