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An invariant for continuous factors of Markov shifts


Author: Bruce Kitchens
Journal: Proc. Amer. Math. Soc. 83 (1981), 825-828
MSC: Primary 28D20; Secondary 54H20, 58F20
DOI: https://doi.org/10.1090/S0002-9939-1981-0630029-8
MathSciNet review: 630029
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0630029-8
Keywords: Subshift of finite type, Markov shift, zeta function, factor maps
Article copyright: © Copyright 1981 American Mathematical Society

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