Slow Entropy for Noncompact Sets and Variational Principle

This paper defines and discusses the dimension notion of topological slow entropy of any subset for Zd-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Z}}}^d-$$\end{document}actions. Also, the notion of measure-theoretic slow entropy for Zd-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^d-$$\end{document}actions is presented, which is modified from Brin and Katok (Geometric Dynamics, Springer, Berlin 1983). Relations between Bowen topological entropy Bowen (Trans Am Math, 184:125–136, 1973), and topological slow entropy are studied in this paper, and several examples of the topological slow entropy in a symbolic system are given. Specifically, a variational principle is proved.