Mike Todd

Title: Escape rates for multimodal maps with holes

Abstract:  I’ll present recent work with Mark Demers on multimodal interval maps with holes.  We make various generalisations of the current results on this topic, allowing the hole to be placed anywhere in the interval and getting escape rates, conditionally invariant measures and a variational principle, even when critical points have dense orbits, and with no slow approach conditions required.  The measures we deal with are equilibrium states, either for Hölder potentials or for `geometric potentials’, so, for example, we include acips here in the Collet-Eckmann case.  Under an extra (though much weaker than previously) assumption, we also prove `zero-hole limits’.