Equilibrium states, pressure and escape for multimodal maps with holes

MD was partially supported by NSF grants DMS 1101572 and DMS 1362420. MT was partially supported by NSF grants DMS 0606343 and DMS 0908093.For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of the escape rate in the zero-hole limit.PostprintPeer reviewe