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Basing upon the recent development of the Patterson-Sullivan measures with aH\"older continuous nonzero potential function, we use tools of both dynamicsof geodesic flows and geometric properties of negatively curved manifolds topresent a new formula illustrating the relation between the exponential decayrate of Patterson-Sullivan measures with a H\"older continuous potentialfunction and the corresponding critical exponent.

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