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 log λ. 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 λ. Also, f multiplies the natural measure on unstable sets of generic points by a positive unit of the ring generated by 1/λ and the algebraic integers of Q[λ]. Related results hold for bounded to one homomorphisms of sofic systems. © 1986 The Weizmann Science Press of Israel.