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Exponents, attractors and Hopf decompositions for interval maps

Published online by Cambridge University Press:  19 September 2008

Gerhard Keller
Gerhard Keller
Affiliation:
Mathematisches Institut, Universitat Erlangen-Nurnberg, Bismarckstraße 1½, D–8520 Erlangen, FRG
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Abstract

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Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following:

(1) There is a set CT such that the ω-limit of Lebesgue-a.e. point equals CT. CT is a finite union of closed intervals or it coincides with the closure of the critical orbit.

(2) There is a constant λT such that for Lebesgue-a.e. x.

(3) λT > 0 if and only if T has an absolutely continuous invariant measure of positive entropy.

(4) , i.e. uniform hyperbolicity on periodic points implies λT > 0, and λT < 0 implies the existence of a stable periodic orbit.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 4 , December 1990 , pp. 717 - 744
Copyright
Copyright © Cambridge University Press 1990

References

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