Some properties of topological pressure of a semigroup of continuous maps

Based on the work of Ma and Wu,[9] some equivalent definitions of topological pressure of a semigroup of continuous maps are given and several of their basic properties are provided by using separated sets and spanning sets. We also answer an open problem of Bi and Urbaski.[19] That is, letting f(i), i = 2, ... , k, be homeomorphisms acting on a compact metric space, G(1) = {id(X), f(2), ... , f(k)}, G(1)(- 1) = {id(X), f(2)(- 1), ..., f(k)(- 1)} and letting G and G(-1) denote the semigroups generated by G(1) and G(1)(- 1), respectively, we give an example showing that the topological entropy of G does not equal the topological entropy of G(-1).