Q-entropy for general topological dynamical systems

Yun Zhao,Chih-Chang Ho,Wen-Chiao Cheng

doi:10.3934/dcds.2019086

Abstract

The aim of this paper is to extend the $ q $-entropy from symbolic systems to a general topological dynamical system. Using a (weak) Gibbs measure as the reference measure, this paper defines $ q $-topological entropy and $ q $-metric entropy, then studies basic properties of these entropies. In particular, this paper describes the relations between $ q $-topological entropy and topological pressure for almost additive potentials, and the relations between $ q $-metric entropy and local metric entropy. Although these relations are quite similar to that described in [19], the methods used here need more techniques from the theory of thermodynamic formalism with almost additive potentials.

Highlights

A basic issue in the theory of dynamical systems is the study of the complexity of orbits
Along with the study in [19], this paper studies some basic properties of the q-topological and q-metric entropies, including the relations between q-topological entropy and almost additive topological pressure
We find that the Hentschel-Procaccia entropy spectrum is closely related to the q-topological entropy, and to the almost additive topological pressure

Summary

Introduction

A basic issue in the theory of dynamical systems is the study of the complexity of orbits. If the entropy map μ → hμ(f ) is upper semi-continuous and for any q > 0, the almost additive potentials qΦ = {qφn}n≥1 have a unique equilibrium Gibbs measure μq, the map q → Hq(f, X) is differentiable, where Hq denotes either hq or hq or hq. If μΦ is the unique Gibbsian equilibrium state for f with respect to Φ, it is ergodic and by statement (3) of the above proposition we have hq(f, μΦ) = hq(f, μΦ) = hq(f, μΦ) = hμΦ (f ) + qF∗(μΦ, Φ) − q(hμΦ (f ) + F∗(μΦ, Φ)) = (1 − q)hμΦ (f ) These two quantities are called the local lower and upper metric entropy at point x with respect to ν, respectively.

For any positive integer

One can find