Equilibrium states for interval maps: the potential $-t\log \mathrm{\mid }Df\mathrm{\mid }$

Let $f:I\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $\phi_t:x\mapsto -t\log|Df(x)|$for$t$closeto1,andalsothatthepressurefunction$t \mapsto P(\phi_t)$isanalyticonanappropriateintervalnear$t = 1$.