Sequence entropy and IT-tuples for minimal group actions

Abstract

Let G be an infinite discrete countable group and $(X,G)$ a minimal G-system. First, we prove that$h_{top}^{⁎}(X,G)\ge \log \sum _{\mu \in M^e(X,G)}{e}^{h_{\mu }^{⁎}(X,G)},$ where $h_{top}^{⁎}(X,G)$ and $h_{\mu }^{⁎}(X,G)$ are the supremum of the topological and metric sequence entropy, respectively. Additionally, if G is abelian, there exists $K\in N\cup \{\infty \}$ with $\log K\le h_{top}^{⁎}(X,G)$ such that it is a regular K-to-one extension of its maximal equicontinuous factor.

Furthermore, for any infinite countable discrete group G, we show that if the factor map from a minimal G-system to its maximal equicontinuous factor is regular $K_1$-to-one and almost $K_2$-to-one, then the system admits $\lceil K_1/K_2\rceil$-IT-tuples, where $K_1\in N\cup \{\infty \}$ and $K_2\in N$. As a corollary, we refine the upper bound on the number of ergodic measures for systems that are almost N-to-one extensions of their maximal equicontinuous factors and lack K-IT-tuples, thereby improving the result of Huang et al. (2021) [17].