On strange attractors in a class of pinched skew products

Àlex Haro

doi:10.3934/dcds.2012.32.605

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. &nbsp 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.

Highlights

The existence of attractive non-continuous invariant graphs in non-autonomous systems is a topic that has generated great interest, specially for the case of quasiperiodically forced dynamical systems, that is bundle maps over irrational rotations
For such a skew product dynamical system, we prove the existence of a measurable but non-continuous invariant graph, whose Lyapunov exponent is negative
We assume that the set of pinched points has zero measure, ant that there is a pinched point whose orbit is dense in the base space

Summary

Introduction

The existence of attractive non-continuous invariant graphs in non-autonomous systems is a topic that has generated great interest, specially for the case of quasiperiodically forced dynamical systems, that is bundle maps over irrational rotations In this context, such objects are known as strange non-chaotic attractors or SNA, since the dynamics is the one given by the external irrational rotation. The orbit of a pinched point is dense in Θ, the graph can not be continuous, just log-integrable, and we will say that ΓΦ is a Strange Attractor. Notice that in [Kel96], and most of the papers in the literature dealing with the so called Strange Non-Chaotic Attractors, the skew product map is defined over an irrational rotation on the torus Θ = R/Z, while here this assumption is considerably generalized.

Preliminary results

Orbits

Invariant graphs

Findings

A paradigmatic example