ABSTRACT:

In this talk I want to discuss results with Mike Todd on the existence and uniqueness of equilibrium states for multimodal maps $f:I \to I$ beyond the Collet-Eckmann class. The class of potentials include the $-t log|f'|$, $t \approx 1$, for which we prove the analyticity of the pressure function except for the phase transition occurring at $t=1$ for non-Collett-Eckmann maps.