Uniqueness of equilibrium states for nonpositive-magnetic-curvature flows

Determine whether the equilibrium state of a hyperbolic magnetic flow on a closed negatively curved surface with nonpositive magnetic curvature is unique.

Background

The paper proves existence of equilibrium states for Bowen-bounded potentials by establishing entropy-expansivity for the time-one map of the magnetic flow. It does not establish uniqueness, including for the measure of maximal entropy, in the nonuniformly hyperbolic setting considered. The authors explicitly identify uniqueness as unresolved in their context.

References

Whether such an equilibrium state is unique in our context remains an open question.

— Surfaces with nonpositive magnetic curvature  (2608.13534 - Hasselblatt et al., 13 Aug 2026) in Section 2, Subsection “Topological pressure and equilibrium states”