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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.

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