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

Nowicki, Tomasz; van Strien, Sebastian

doi:10.1090/S0002-9947-1990-0994169-6

Hyperbolicity properties of $C^ 2$ multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions
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by Tomasz Nowicki and Sebastian van Strien PDF

Trans. Amer. Math. Soc. 321 (1990), 793-810 Request permission

Abstract:

In this paper we study the dynamical properties of general ${C^2}$ maps $f:[0,1] \to [0,1]$ 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 ${f^n}$ as $n \to \infty ;$ For these results we will not make any assumptions on the Schwarzian derivative $f$. We will also give an estimate of the return-time of points that start near critical points.

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