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TITLE: [GS_M_DT] Positive recurrence of a certain Gibbs measure on finite-area surfaces with pinched negative curvature

ABSTRACT: Thermodynamic formalism in ergodic theory studies invariant measures that maximize the pressure of a given potential function on the phase space. The simplest and most important example is the measure of maximal entropy (MME). For the geodesic flow on manifolds with pinched negative curvature, a thermodynamic formalism enables the construction of a Gibbs measure for any Hölder continuous potential. When the Gibbs measure is finite, it defines a unique pressure-maximizing measure; otherwise, no measure of maximal pressure exists. Unfortunately, Gibbs measures are not always finite for non-compact manifolds, even for the measure of maximal entropy on finite-area surfaces. In this talk, I will focus on the Gibbs measure describing the asymptotic behavior of Brownian motion on a finite-area surface with pinched negative curvature and prove that it is always finite.