Topological entropy for noncompact sets

Abstract

The topological entropy of a continuous map on a compact space was defined by Adler, Konheim and McAndrew [l]. In the present paper we will define entropy for subsets of compact spaces in a way which resembles Hausdorff dimension.This will be used to genetalize known results about the Hausdorff dimension of the quasiregular points of certain measures and to define a notion of conjugacy that is a cross between the topological and measure theoretic ones.In [5] we gave a definition of entropy for uniformly continuous maps on metric spaces. That definition was motivated by different examples (linear maps on R"and calculating entropy on T") and it sometimes differs from the definition given here.We wish to thank Karl Sigmund who pointed us in the direction this paper takes and Ben Weiss who helped us formulate §4.

Slow Entropy for Noncompact Sets and Variational Principle

“…This quantity h S top (T , •) is defined in way which resembles the Hausdorff dimension, also satisfies most properties like Bowen entropy [6] for Z + action. For convenience, we use M(Z , s, N , ), M(Z , s, ) instead of M T (Z , s, N , ), M T (Z , s, ) without any confusion.…”

“…Bowen [6] gave the definition of topological entropy for non-compact subset for Z + -action. Here, we present an equivalent definition for convenience.…”

“…We now give a equivalent definition of h S top (T, •) for the situation d = 1, which comes from Bowen [6]. We let (X, T ) be a TDS, C o X be the set of all finite open covers of X.…”

“…In 1973 Bowen [6] introduced the topological entropy h B top (T, K ) for any set K in a TDS (X, T ) resembling Hausdorff dimension. He also proved the remarkable result that h B top (T, G μ ) = h μ (T ) for ergodic measure μ, where G μ denotes the set of generic points of μ, and h μ (T ) is the measure-theoretical entropy.…”

Pairwise nearly compact and pairwise nearly paracompact spaces and it application

“…There are several definitions of topological entropy in literature since 1965 till now, proposed such as by Adler et al [9], Bowen [10], Cánovas and Rodríguez [11], Yang and Bai [12], etc. In section 7, we review some concepts and properties of topological entropy in the sense of Adler et al [9].…”