Equilibrium measures of manifolds without conjugate points having visibility covering
Edhin Mamani
TL;DR
The paper develops a thermodynamic formalism for geodesic flows on closed manifolds without conjugate points with visibility universal coverings. By constructing a factor flow on a quotient space, the authors prove uniqueness of equilibrium measures for Bowen potentials constant on horospheric classes under a weak pressure gap, and they establish K-mixing, Gibbs properties, weighted periodic-orbit equidistribution, and large deviations/entropy-density results. The framework generalizes previous results by relaxing Green-bundle continuity and emphasizing global geometric hypotheses to obtain a robust variational theory. Key techniques include the factor flow approach, the projection of equilibrium measures, and transfer of dynamical properties between the original flow and its quotient. The results have implications for understanding the statistical behavior of geodesic flows in nonpositive curvature settings with visibility, highlighting the role of global geometry in thermodynamic-type phenomena.
Abstract
In this paper we study the equilibrium measures of geodesic flows of closed manifolds without conjugate points which have a visibility universal covering. Specifically, the uniqueness problem for Bowen potentials which are constants on some sets--intersection of horospheres-- and satisfy a weak pressure gap. Moreover, we study some ergodic properties of these measures such as the K-mixing property, weighted equidistribution of closed geodesics, the Gibbs property, large deviations and the entropy density of ergodic measures. Assuming, furthermore continuity of Green bundles, existence of a hyperbolic closed geodesic and a Gromov hyperbolic universal covering we prove that the above potentials always satisfy the weak pressure gap.