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On–off intermittency and chaotic walks

Part of: Smooth dynamical systems: general theory Random dynamical systems Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  30 January 2019

ALE JAN HOMBURG Open the ORCID record for ALE JAN HOMBURG [Opens in a new window]  and
VAHATRA RABODONANDRIANANDRAINA
ALE JAN HOMBURG
Affiliation:
KdV Institute for Mathematics, University of Amsterdam, Science park 107, 1098 XG Amsterdam, The Netherlands email v.f.rabodonandrianandraina@uva.nl Department of Mathematics, VU University Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands email a.j.homburg@uva.nl
VAHATRA RABODONANDRIANANDRAINA
Affiliation:
KdV Institute for Mathematics, University of Amsterdam, Science park 107, 1098 XG Amsterdam, The Netherlands email v.f.rabodonandrianandraina@uva.nl
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Abstract

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We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on–off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map.

Keywords

intermittencylow-dimensional dynamicsrandom dynamicsrandom walks

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37H20: Bifurcation theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1805 - 1842
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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