Phase Transitions for Surface Diffeomorphisms

In this paper we consider \(C^1\) surface diffeomorphisms and study the existence of phase transitions, here expressed by the non-analiticity of the pressure function associated to smooth and geometric-type potentials. We prove that the space of \(C^1\)-surface diffeomorphisms admitting phase transitions is a \(C^1\)-Baire generic subset of the space of non-Anosov diffeomorphisms. In particular, if S is a compact surface which is not homeomorphic to the 2-torus then a \(C^1\)-generic diffeomorphism on S has phase transitions. We obtain similar statements in the context of \(C^1\)-volume preserving diffeomorphisms. Finally, we prove that a \(C^2\)-surface diffeomorphism exhibiting a dominated splitting admits phase transitions if and only if has some non-hyperbolic periodic point.