Computability at zero temperatureBurr was partially supported by the National Science Foundation Grants CCF-1527193 and DMS-1913119. Wolf was partially supported by Grants from the PSC-CUNY (TRADB-49-253 to Christian Wolf) and the Simons Foundation (#637594 to Christian Wolf)

• Published in: Nonlinearity Vol. 33; no. 11; pp. 6157 - 6175
• Main Authors: Burr, Michael, Wolf, Christian
• Format: Journal Article
• Language: English
• Published: IOP Publishing 06.10.2020
• Subjects: computability; entropy; ground states; residual entropy; thermodynamic formalism; zero-temperature measures
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/ab9c71

We investigate the computability of thermodynamic invariants at zero temperature for one-dimensional subshifts of finite type. In particular, we prove that the residual entropy (i.e., the joint ground state entropy) is an upper semi-computable function on the space of continuous potentials, but it is not computable. Next, we consider locally constant potentials for which the zero-temperature measure is known to exist. We characterize the computability of the zero-temperature measure and its entropy for potentials that are constant on cylinders of a given length k. In particular, we show the existence of an open and dense set of locally constant potentials for which the zero-temperature measure can be computationally identified as an elementary periodic point measure. Finally, we show that our methods do not generalize to treat the case when k is not given.