Constraints on the Degree of a Sofic Homomorphism and the Induced Multiplication of Measures on Unstable Sets

Let f be an endomorphism of an irreducible sofic system S, where S has entropy Iog)t. The degree of f is the number d such that f is d to 1 almost everywhere. Then d divides a power of the greatest common divisor of the nonleading coefficients of the minimal polynomial of )t. Also, f multiplies the natural measure on unstable sets of generic points by a positive unit of the ring generated by 1/A and the algebraic integers of Q[A]. Related results hold for bounded to one homomorphisms of sofic systems. §0. Introduction Suppose S is an irreducible sofic shift (in particular, S may be of finite type), and f is a factor map from S onto a subshift T of equal entropy. It is well known that there is a number d such that f is d to 1 almost everywhere (every bilaterally transitive point of T has exactly d preimages under/) ; endorsing the terminology of [16], we call this number the degree of/. An old theorem of L. R. Welch (14.9 of [12]) shows that if f is an endomorphism of the full shift on n symbols, then the degree of f divides a power of n. Generalizing, we find that for any irreducible sofic shift, the entropy of the shift provides severe and computable constraints on the possible degrees of endomorphisms; other factor maps also have constrained degrees. Let S be irreducible sofic with entropy log A. Let U(A) be the set of positive integers which divide some power of the greatest common divisor of the nonleading coefficients of the minimal polynomial of ,L Then, given another sofic shift T of equal entropy, there is a finite set E of positive integers such that the Received November 1, 1984 and in revised form June 3, 1985