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 language | English (US) |
|---|---|
| Pages (from-to) | 527-535 |
| Number of pages | 9 |
| Journal | Dynamical Systems |
| Volume | 36 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2021 |
Funding
This work was supported by Agence Nationale de la Recherche [ANR-16-CE40-0013] and Simons Foundation [239708]. This work was carried out in a workshop at the American Institute of Mathematics. We thank AIM for their support and hospitality. We also thank the referees for many useful suggestions.
Keywords
- Dynamical systems
- equilibrium states
- geodesic flow
ASJC Scopus subject areas
- General Mathematics
- Computer Science Applications