Hyperbolicity properties of 𝐶² multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions

Abstract

In this paper we study the dynamical properties of general C 2 {C^2} maps f : [ 0 , 1 ] → [ 0 , 1 ] f:[0,1] \to [0,1] with quadratic critical points (and not necessarily unimodal). We will show that if such maps satisfy the well-known Collet-Eckmann conditions then one has (a) hyperbolicity on the set of periodic points; (b) nonexistence of wandering intervals; (c) sensitivity on initial conditions; and (d) exponential decay of branches (intervals of monotonicity) of f n {f^n} as n → ∞ ; n \to \infty ; For these results we will not make any assumptions on the Schwarzian derivative f f . We will also give an estimate of the return-time of points that start near critical points.