Vol. 51, No. 2, 1974

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Entropy of self-homeomorphisms of statistical pseudo-metric spaces

Alan Saleski

Vol. 51 (1974), No. 2, 537–542

A pseudo-Menger space is a set X together with a function 𝜃;X × X →𝒟, the set of distribution functions, satisfying certain natural axioms similar to those of a pseudo-metric space. Let T : X X be a bijection and let 𝜃r denote the topology generated by {TiU(p,𝜖,λ);i Z,p X,𝜖 > 0,λ > 0} where U(p,𝜖,λ) = {q;𝜃(p,q)(𝜖) > 1 λ}. Assume that 𝜃r is compact. Let h(T,𝜃) denote the topological entropy of T with respect to the 𝜃r topology. The purpose of this note is to show that if one is given a sequence {𝜃n} of pseudo-Menger structures on X satisfying 𝜃n(p,q) 𝜃(p,q) and 𝜃n(p,q) 𝜃(p,q) in distribution for all p,q X then h(T,𝜃n) h(T,𝜃). A counterexample is then given to show that, in general, the condition 𝜃n(p,q) 𝜃(p,q) cannot be removed.

Mathematical Subject Classification 2000
Primary: 54H20
Received: 27 November 1972
Published: 1 April 1974
Alan Saleski