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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 321 (1990), 793-810
  • MSC: Primary 58F08; Secondary 58F13
  • DOI: https://doi.org/10.1090/S0002-9947-1990-0994169-6
  • MathSciNet review: 994169