The Decay Rate Of Patterson-Sullivan Measures With Potential Functions And Critical Exponents

For a simply connected negatively curved Riemannian manifold X,there are various families of measures on X(?)indexed by points of X and the members of each family belong to a same measure class.Among these,three kinds of measures,the Lebesgue measures,the harmonic measures and the Patterson-Sullivan measures,are particularly important.This thesis is devoted to the study of the properties of Patterson-Sullivan measures with a Holder continuous nonzero potential function on X and the relations between these Patterson-Sullivan measures and dynamics of geodesic flows,especially the lower bound of the corresponding critical exponent,which is a basic problem in geometric group theory.Patterson-Sullivan measures play a critical role in the study of geodesic flows on negatively curved manifolds and ergodic theory.Basing upon the recent development of the Patterson-Sullivan measures with a Holder continuous nonzero potential function,we use tools of both dynamics of geodesic flows and geometric properties of negatively curved manifolds to present a new formula illustrating the relation between the exponential decay rate of Patterson-Sullivan measures with a Holder continuous potential function and the corresponding critical exponent.Meanwhile,we generalize some results of Kaimanovich's from classical Patterson-Sullivan measures to Patterson-Sullivan measures with a Holder continuous nonzero potential function.