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Faculty of Graduate Studies

Arantha Ranu

  • BS-MS Dual Degree (IISER Kolkata, 2021)
Notice of the Final Oral Examination for the Degree of Doctor of Philosophy

Topic

Failure of the Gibbs inequality for continuous potentials

Department of Mathematics and Statistics

Date & location

  • Tuesday, August 11, 2026
  • 1:00 P.M.
  • Clearihue Building, Room B007

Examining Committee

Supervisory Committee

  • Dr. Anthony Quas, Department of Mathematics and Statistics, University of Victoria (Supervisor)
  • Dr. Christopher Bose, Department of Mathematics and Statistics, UVic (Member)
  • Dr. Yun Lu, Department of Computer Science, UVic (Outside Member)

External Examiner

  • Dr. Dan Thompson, Department of Mathematics, Ohio State University

Chair of Oral Examination

  • Dr. Daniela Constantinescu, Department of Mechanical Engineering, UVic

Abstract

A classical result of Bowen shows that Hölder continuous potentials on shift spaces have unique equilibrium states. Moreover, these equilibrium states satisfy the Gibbs inequality, i.e. the corresponding Gibbs ratio is bounded above and below by positive constants. However, Hofbauer showed that this conclusion can fail for continuous potentials by constructing a continuous potential with two equilibrium states, one of which does not satisfy the Gibbs inequality. In this thesis we introduce thermodynamic formalism, and we construct a continuous potential with a unique equilibrium state for which even a very weak version of the classical Gibbs inequality fails.

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