Infinite-to-one codes and Markov measures

Boyle, Mike; Tuncel, Selim

doi:10.1090/S0002-9947-1984-0752497-0

Infinite-to-one codes and Markov measures
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by Mike Boyle and Selim Tuncel

Trans. Amer. Math. Soc. 285 (1984), 657-684

DOI: https://doi.org/10.1090/S0002-9947-1984-0752497-0

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Abstract:

We study the structure of infinite-to-one continuous codes between subshifts of finite type and the behaviour of Markov measures under such codes. We show that if an infinite-to-one code lifts one Markov measure to a Markov measure, then it lifts each Markov measure to uncountably many Markov measures and the fibre over each Markov measure is isomorphic to any other fibre. Calling such a code Markovian, we characterize Markovian codes through pressure. We show that a simple condition on periodic points, necessary for the existence of a code between two subshifts of finite type, is sufficient to construct a Markovian code. Several classes of Markovian codes are studied in the process of proving, illustrating and providing contrast to the main results. A number of examples and counterexamples are given; in particular, we give a continuous code between two Bernoulli shifts such that the defining vector of the image is not a clustering of the defining vector of the domain.

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