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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hyperbolicity properties of $ C\sp 2$ multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions


Authors: Tomasz Nowicki and Sebastian van Strien
Journal: Trans. Amer. Math. Soc. 321 (1990), 793-810
MSC: Primary 58F08; Secondary 58F13
MathSciNet review: 994169
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0994169-6
PII: S 0002-9947(1990)0994169-6
Article copyright: © Copyright 1990 American Mathematical Society