The Decay Rate of Patterson-Sullivan Measures with Potential Functions and Critical Exponents

Basing upon the recent development of the Patterson-Sullivan measures with a H"older 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 H"older continuous potential function and the corresponding critical exponent.