Computability of topological pressure on compact shift spaces beyond finite type
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by
Michael Burr, Suddhasattwa Das, Christian Wolf, Yun Yang
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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