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We obtain estimates for derivative and cross-ratio distortion for $C^{2+\eta }$ (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that $g_T:T\to ℝ$ is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove that for...

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 ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ for $t$ close to $1$, and also that the pressure function $t\mapsto P\left({\phi }_t\right)$ is analytic on an appropriate interval near $t=1$.