Hyperbolicity Properties of C sup 2 Multimodal Collet-Eckmann Maps without Schwarzian Derivative Assumptions

摘要

The dynamical properties of general C sup 2 maps f: (0,1) to (0,1) with quadratic critical points (and not necessarily unimodal) are studied. It is shown that if such maps satisfy the Collet-Eckmann conditions, then one has: hyperbolicity on the set of periodic points; nonexistence of wandering intervals; sensitivity on initial conditions; and exponential decay of branches (intervals of monotonicity) of f sup n, as n tends to infinity. For these results, no assumptions are made on the Schwarzian derivative f. The return-time of points that start near critical points is estimated.