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

TL;DR

This paper establishes a new formula linking the exponential decay rate of Patterson-Sullivan measures with a H"older continuous potential to the critical exponent, using tools from dynamical systems and geometry.

Contribution

It introduces a novel formula connecting decay rates of measures with critical exponents in negatively curved manifolds with potential functions.

Findings

Derived a new relation between decay rate and critical exponent

Utilized dynamics of geodesic flows and geometric properties

Extended Patterson-Sullivan measures to H"older continuous potentials

Abstract

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.