On Automorphisms of Markov Chains

We prove several theorems about automorphisms of Markov chains, using the weight-per-symbol polytope. Introduction The main purpose of this paper is to show how the weight-per-symbol polytope (WPS), introduced in [MT1, §3] can be used in the study of automorphisms (i.e., measure-preserving conjugacies) of Markov chains, in particular, with regard to the question of which permutations of finitely many periodic cycles can be realized by such automorphisms. In a Markov chain, the weight of a cycle is the product of the transition probabilities around the cycle. The weight-per-symbol (wps) of a cycle is its weight normalized by its length. We study the periods of the set of cycles with a given wps—in particular, with reference to how the given wps sits in WPS. In § 1, we establish the notion of a period for any wps which lies in the interior of WPS, and we compute this period. (The growth rate of the number of cycles with a given wps and length is computed in [MT2].) We also show that any given path is a subpath of some cycle with prescribed large least period and prescribed wps, provided the wps lies in the interior of WPS. The latter is used heavily later in the paper. While the results of §1 are also contained implicitly in [MT2], our proof here is a bit different and more direct. In [MT1, §3], a scaffold of Markov chains was associated to any given Markov chain: the scaffold consists of induced Markov chains, corresponding to the faces of WPS. In §2, we refine the scaffold by introducing boundary components, and using this we examine the set of (large) least periods of cycles with given wps in the boundary of WPS. In §3, we look at the permutation of boundary components induced by an automorphism of a Markov chain. We give a necessary condition in terms of shift equivalence for the existence of an automorphism that maps one boundary component to another. This gives a necessary and sufficient condition, though hard to check in general, for the existence of an "eventual" automorphism that realizes a given permutation of boundary components. We also explore inert automorphisms on Markov chains, introduced by Wagoner [Wa], and show that these automorphisms must fix each boundary component. From this, we see Received by the editors April 2, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 28D20, 54H20, 58F20. ©1992 American Mathematical Society 0002-9947/92 $1.00+ $.25 per page