Computability of topological pressure on compact shift spaces beyond finite type release_k4weffq54jdgblzuihn2e5ezme

by Michael Burr, Suddhasattwa Das, Christian Wolf, Yun Yang

Released as a article .

2020  

Abstract

We investigate the computability (in the sense of computable analysis) of the topological pressure P_ top(ϕ) on compact shift spaces X for continuous potentials ϕ:X→ R. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and Beta shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalized pressure function (X,ϕ)↦ P_ top(X,ϕ|_X) is not computable for a large set of shift spaces X and potentials ϕ. In particular, the entropy map X↦ h_ top(X) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.
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Date   2020-10-29
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arXiv  2010.14686v2
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