Topological $R$-pressure and topological pressure of free semigroup actions

Abstract

In this paper we introduce the definition of topological $r$-pressure of free semigroup actions on compact metric space and provide some properties of it. Through skew-product transformation into a medium, we can obtain the following two main results. 1. We extend the result that the topological pressure is the limit of topological $r$-pressure in\cite{C} to free semigroup actions ($r\to 0$). 2. Let $f_i,$ $i=0, 1, \cdots, m-1$, be homeomorphisms on a compact metric space. For any continuous function, we verify that the topological pressure of $f_0, \cdots, f_{m-1}$ equals the topological pressure of $f_0^{-1}, \cdots, f_{m-1}^{-1}.$