  In this paper we investigate the range of validity of Ruelle's property.
First, we show that every finitely-generated Fuchsian group has Ruelle's
property. We also prove the existence of an infinitely-generated Fuchsian group
satisfying Ruelle's property. Concerning the negative results we first
generalize Astala-Zinsmeister's results by proving that all convergence
Fuchsian groups of the first kind fail to have Ruelle's property. At last, we
also give some results about the second kind Fuchsian groups.
