On Deterministic Perturbations of Summability Maps

Abstract

This thesis contains two topics on perturbations of non-uniformly expanding interval maps. The first topic is to provide a strengthened version of the famous Jakobson's theorem. Consider an interval map $f$ satisfying a summability condition. For a generic one-parameter family $f_t$ of maps with $f_0=f$, we prove that $t=0$ is a Lebesgue density point of the set of parameters for which $f_t$ satisfies both the Collet-Eckmann condition and a strong polynomial recurrence condition. The second topic is to investigate the asymptotic distributions of the critical orbits. Consider a one-parameter family with some conditions and let E be the set of parameters $t$ for which $f_t$ satisfies a summability condition. We prove that for almost all $t\in E$, each critical points of $f_t$ belongs to one of the ergodic acips.