On Reversible Topological Markov Shifts

Abstract

A reversal system is a topological Markov shift $(X,\sigma)$ with an automorphism of $X$ (called a reversal) intertwining $\sigma$ and $\sigma^{-1}$. Two reversal systems are said to be conjugate if there is a topological conjugacy between their underlying topological Markov shifts that intertwines the reversals. We give an equivalent condition for a topological conjugacy between the underlying shifts of two reversal systems to be a conjugacy between the reversal systems. Also we show that the class of reversal systems satisfies an analogue of the decomposition theorem in the class of topological Markov shifts.