On Strange Attractors in a Class of Pinched Skew Products

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 +∞. 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.