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The boundary of a simply connected, negatively curved manifold carries two natural types of measures: on one hand, Gibbs measures such as the Lebesgue measure and the Patterson-Sullivan measure. On the other hand, harmonic measures arising from random walks. We prove that the absolute continuity between a harmonic measure and a Gibbs measure is equivalent to a relation between entropy, drift and critical exponent, extending the previous “fundamental inequality” of Guivarc’h, Ledrappier, and Blachere-Haissinsky-Mathieu. This shows that if the manifold is geometrically finite but not compact, harmonic measures are singular with respect to Gibbs measures. Joint with I. Gekhtman.
Giulio Tiozzo is an Assistant Professor of Mathematics at the University of Toronto. He obtained his PhD from Harvard in 2013, under the supervision of C.T. McMullen. Prior to joining University of Toronto, he was a Gibbs Assistant Professor at Yale University, and in 2018 he was awarded the Alfred P. Sloan Fellowship. His field of research is dynamical systems and ergodic theory, with applications to complex analysis, probability, and geometric group theory.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai