The Collet-Eckmann Condition in One-Dimnesional Dynamics: a survey
by
Gregory Swiatek
Penn State University

The Collet-Eckmann condition, in a simplified form, is that lower Lyapunov exponents are positive at all critical values of the transformation. Uniquely to one-dimensional dynamics, this type of expansion assumed for just finitely many orbits has implications for the global picture of the dynamics. At the same time, the condition is robust in the sense that it holds for a set of positive measure of parameters in typical families. The theory of the Collet-Eckmann condition is particularly well developed for unimodal maps. More recently, close connections emerged with holomorphic dynamics and the theory is being developed for multi-modal interval transformations. In this survey we will present recent results, their inter-connections and key concepts behind them.

Date received: March 4, 2000