Markov partitions and sofic codings for Anosov diffeomorphisms of nilmanifolds

Abstract

Given an Anosov diffeomorphism of a compact manifold, the existence of a Markov partition and the associated conjugate symbolic dynamical system has been known for over fifty years by a celebrated result of Sinai, subsequently extended by Bowen. Building upon the work done by many authors in the context of hyperbolic toral automorphisms, we give an explicit arithmetic construction of sofic codings and Markov partitions for Anosov diffeomorphisms of nilmanifolds. Arising as quotients of nilpotent Lie groups by discrete and co-compact subgroups (lattices), nilmanifolds are conjecturally the only manifolds admitting Anosov diffeomorphisms, up to a finite covering.