Computability at zero temperature*Burr 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). (6th October 2020)

• Record Type: Journal Article
• Authors: Burr, Michael; Wolf, Christian
• Is Part Of: Nonlinearity. Volume 33:Number 11(2020)
• Page Start: 6157
• Page End: 6175
• Publication Date: 2020-10-06
• Subjects: zero-temperature measures -- residual entropy -- ground states -- entropy -- thermodynamic formalism -- computability; Primary 37D35 -- 37E45 -- 03D15 -- Secondary37B10 -- 37L40 -- 03D80; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab9c71
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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.