An invariant for continuous factors of Markov shifts

Kitchens, Bruce

doi:10.1090/S0002-9939-1981-0630029-8

An invariant for continuous factors of Markov shifts
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by Bruce Kitchens

Proc. Amer. Math. Soc. 83 (1981), 825-828

DOI: https://doi.org/10.1090/S0002-9939-1981-0630029-8

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Abstract:

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

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