On strange attractors in a class of pinched skew products

Àlex Haro

doi:10.3934/dcds.2012.32.605

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2012, Volume 32, Issue 2: 605-617. Doi: 10.3934/dcds.2012.32.605

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On strange attractors in a class of pinched skew products

Àlex Haro^1,

1.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona

Received: October 2010

Revised: July 2011

Published: February 2012

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Abstract

In this paper we construct strange attractors in a class of pinched skew product dynamical systems over homeomorphims on a compact metric space. We assume that maps between fibers satisfy Inada conditions and that the base space is a super-repeller (it is invariant and its vertical Lyapunov exponent is $+\infty$). In particular, we prove the existence of a measurable but non-continuous invariant graph, whose vertical Lyapunov exponent is negative. %We will refer to such an object as a strange attractor.
Since the dynamics on the strange attractor is the one given by the base homeomorphism, we will say that it is a strange chaotic attractor or a strange non-chaotic attractor depending on the fact that the dynamics on the base is chaotic or non-chaotic. The results complement the paper by G. Keller on rigorous proofs of existence of strange non-chaotic attractors.

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Mathematics Subject Classification: Primary: 37C60, 37C70; Secondary: 37B55.

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