Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature

K. Burns, J. Buzzi, T. Fisher*, N. Sawyer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study the one parameter family of potential functions (Formula presented.) associated with the unstable Jacobian potential (or geometric potential) (Formula presented.) for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For q<1, it is known that there is a unique equilibrium state associated with (Formula presented.), and it has full support. For q>1 it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value q = 1 and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure or measures supported on the singular set. In particular, when q = 1, there is a unique ergodic equilibrium state that gives positive measure to the regular set.

Original languageEnglish (US)
Pages (from-to)527-535
Number of pages9
JournalDynamical Systems
Volume36
Issue number3
DOIs
StatePublished - 2021

Keywords

  • Dynamical systems
  • equilibrium states
  • geodesic flow

ASJC Scopus subject areas

  • Mathematics(all)
  • Computer Science Applications

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