Computability of topological pressure on compact shift spaces beyond finite type* *Burr was partially supported by grants from the National Science Foundation (CCF-1527193 and DMS-1913119). Das was partially supported by ONR MURI Grant N00014-19-1-242 and ONR YIP Grant N00014-16-1-264. Wolf was partially supported by grants from the Simons Foundation (#637594 to Christian Wolf) and PSC-CUNY (TRADB-51-63715 to Christian Wolf). Yang

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 . 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, generalised gap shifts, and particular 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 generalised 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.