On the periodic points of topological Markov chains

Abstract

In this paper, our aim is to prove that for every aperiodic and irreducible topological Markov chain (X,S) one can in fact find another aperiodic and irreducible topological Markov chain (X,S) such that (1) holds. A point in X is said to be a doubly transitive point of the topological Markov chain (X, S) if its forward orbit and its backward orbit under S enter every open set infinitely often. Given two topological Markov chains (X, S) and (X', S'), a continuous mapping (p of X onto X' is said to be an almost homeomorphic factor map if ~0 S = S'cp and if every doubly transitiv point of(X', S') has only one preimage under (p. Almost homeomorphic factor maps appear in the paper by Adler and Marcus [11 in which they prove that for aperiodic and irreducible topological Markov chains (X, S) and (X, S) of equal entropy one can find an aperiodic and irreducible Markov chain (X, S) and almost homeomorphic factor maps ~0: (J~, S) -+ (X, S) and 0: (J~, S*) --+ (X, S). Here we obtain for a given (X, S) the ()~, S) such A A that (1) holds by constructing an (X, S) together with almost homeomorphic factor maps ~o: (X, S)-+ (X, S) and 0: (X, S)--+ (X, S). In this way we give an alternate method for carrying out that part of the development of Adler and Marcus where they are concerned with producing a fixed point.