TITLE:
On strange attractors in a class of pinched skew products

AUTHOR:
Ālex Haro

     Departament de Matemātica Aplicada i Anālisi
     Universitat de Barcelona
     Gran Via 585
     08007 Barcelona, Spain 

     E-mail: alex@maia.ub.es
     
     
ABSTRACT:
In this note we construct strange attractors in a class of skew product 
dynamical systems. 

A dynamical system of the class is a bundle map of a trivial bundle 
whose base is a compact metric space and the fiber is the non-negative 
half real line. The map on the base is a homeomorphism preserving 
an ergodic measure. The fiber maps either are strictly monotone and 
strictly concave or collapse at zero (pinching condition).  
The points on the base space whose fibers collapse are the pinched points 
of the skew product. We also assume that the set of pinched points has zero 
measure, and that there is a pinched point whose orbit is dense in the base 
space. Moreover, we assume that the zero-section is a super-repeller, 
in the sense that it is invariant and its  Lyapunov exponent is $+\infty$. 

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 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.