Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity

Abstract

A dynamical system is a pair (X,G), where X is a compact metrizable space and G is a countable group acting by homeomorphisms of X. An endomorphism of (X,G) is a continuous selfmap of X which commutes with the action of G. One says that a dynamical system (X,G) is surjunctive provided that every injective endomorphism of (X,G) is surjective (and therefore is a homeomorphism). We show that when G is sofic, every expansive dynamical system (X,G) with nonnegative sofic topological entropy and satisfying the weak specification and the strong topological Markov properties, is surjunctive.