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Dynamical profile of a class of rank-one attractors

Published online by Cambridge University Press:  08 May 2012

QIUDONG WANG  and
LAI-SANG YOUNG
QIUDONG WANG
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA (email: [email protected])
LAI-SANG YOUNG
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA (email: [email protected])
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Abstract

This paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1221 - 1264
Copyright
Copyright © 2012 Cambridge University Press 

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