Endomorphisms of irreducible subshifts of finite type

Introduction. The properties of endomorphisms of the full shift dynamical system are described by Hedlund in [9]. His proofs are based on the very nice combinatorial properties of the full shift. However, the combinatorial structure of a symbolic flow is, in general, not susceptible to the kind of analysis done in [9]. For this reason, there are relatively few results about endomorphisms of symbolic flows other than full shifts. In this paper, we investigate the properties of endomorphisms of a class of symbolic flows known as irreducible subshifts of finite type. This class, which contains all full shifts, is in some sense a more appropriate class to study than the class of full shifts. Irreducible subshifts of finite type occur naturally in the work of the Smale school on Axiom A diffeomorphisms (see [3], [4], [12]). They were introduced by Parry [I 1], who called them intrinsic Markov chains. They are examples of intrinsically ergodic flows, i.e., flows having a unique invariant measure such that the topological entropy of the flow is finite and equal to the measure-theoretic entropy with respect to the distinguished measure. For an irreducible subshift of finite type, the value of this measure on a basic cylinder set is easily computed. We first establish some properties of intrinsically ergodic symbolic flows and their endomorphisms. Then we prove the equivalence of certain properties of an endomorphism of an irreducible subshift of finite type (e.g., being onto, being finite-to-one, preserving the distinguished measure). Finally, we show how to extend Hedlund's results on inverses of onto endomorphisms to endomorphisms of irreducible subshifts of finite type. The authors wish to thank G. A. Hedlund for many hours of stimulating discussion.