Entropy of self-homeomorphisms of statistical pseudo-metric spaces (original) (raw)

1974, Pacific Journal of Mathematics

A pseudo-Menger space is a set X together with a function ^:IxI-»S, the set of distribution functions, satisfying certain natural axioms similar to those of a pseudo-metric space. Let T: J-> J be a bijection and let θ τ denote the topology generated by {TWip, e, λ): i e Z, p6 X, ε > 0, λ > 0} where U(p, ε, λ) = {q: θ(p, q)(ε) > 1-λ}. Assume that θ τ is compact. Let h(T,θ) denote the topological entropy of T with respect to the θ τ topology. The purpose of this note is to show that if one is given a sequence {#"} of pseudo-Menger structures on X satisfying θ n (p, q) Ξ> θ(p, q) and θ n (p, q)-» θ(p, q) in distribution for all p,qeX then h(T,θ n)->h(T 9 θ). A counterexample is then given to show that, in general, the condition θ n (p, q) i> θ{p, q) cannot be removed.