An invariant for continuous factors of Markov shifts

Abstract

Let Σ A {\Sigma _A} and Σ B {\Sigma _B} be subshifts of finite type with Markov measures ( p , P ) (p,P) and ( q , Q ) (q,Q) . It is shown that if there is a continuous onto measure-preserving factor map from Σ A {\Sigma _A} to Σ B {\Sigma _B} , then the block of the Jordan form of Q Q with nonzero eigenvalues is a principal submatrix of the Jordan form of P P . If Σ A {\Sigma _A} and Σ B {\Sigma _B} are irreducible with the same topological entropy, then the same relationship holds for the matrices A A and B B . As a consequence, ζ B ( t ) / ζ A ( t ) {\zeta _B}(t)/{\zeta _A}(t) , the ratio of the zeta functions, is a polynomial. From this it is possible to construct a pair of equalentropy subshifts of finite type that have no common equal-entropy continuous factor of finite type, and a strictly sofic system that cannot have an equal-entropy subshift of finite type as a continuous factor.