Ergodic optimization and zero temperature limits in negative curvature (Q2159082)

The authors deal with some aspects of the ergodic theory of the geodesic flow on a non-compact negatively curved manifold. More precisely, zero temperature limits of a particular class of potentials are studied. Let \(\mathcal{F}\) be the family of Hölder-continuous potentials \(\varphi\) vanishing at infinity and such that \(\beta (\varphi)>0\). The following result is proven: For any \(\varphi\in\mathcal{F}\), every ground state at zero temperature of \(\varphi\) is a maximizing measure. Moreover, they all have the same measure-theoretic entropy. As consequences of the above result, the authors obtain the following two claims: (1) The geodesic flow verifies the intermediate entropy property; (2) The set of equilibrium states of potentials in \(\mathcal{F}\) is dense in \(\mathcal{M}_{\le 1}(g)\), relative to the vague topology. Finally, potentials in \(\mathcal{F}\) for which the zero temperature limits converge and diverge are constructed.