Topological entropy zero and asymptotic pairs

Abstract

Let T be a continuous map on a compact metric space (X, d). A pair of distinct points x, y ∈ X is asymptotic if lim n→∞ d(T n x, T n y) = 0. We prove the following four conditions to be equivalent: 1. h top(T) = 0; 2. (X, T) has a (topological) extension (Y,S) which has no asymptotic pairs; 3. (X, T) has a topological extension (Y ′, S′) via a factor map that collapses all asymptotic pairs; 4. (X, T) has a symbolic extension (i.e., with (Y ′, S′) being a subshift) via a map that collapses asymptotic pairs. The maximal factors (of a given system (X, T)) corresponding to the above properties do not need to coincide.