An Invariant for Continuous Factors of Markov Shifts

Let <!-- MATH ${\Sigma _A}$ --> and <!-- MATH ${\Sigma _B}$ --> be subshifts of finite type with Markov measures and . It is shown that if there is a continuous onto measure-preserving factor map from <!-- MATH ${\Sigma _A}$ --> to <!-- MATH ${\Sigma _B}$ --> , then the block of the Jordan form of with nonzero eigenvalues is a principal submatrix of the Jordan form of . If <!-- MATH ${\Sigma _A}$ --> and <!-- MATH ${\Sigma _B}$ --> are irreducible with the same topological entropy, then the same relationship holds for the matrices and . As a consequence, <!-- MATH ${\zeta _B}(t)/{\zeta _A}(t)$ --> , 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.