STRANGE ATTRACTORS FOR THE LOZI MAPPINGS

into itself. Therefore. one can hope that ( 1 ) i t will be easier to investigate the Lozi mappings than the Hiinon ones and (2) for certain values of parameters. the Hknon mappings will be conjugate to some Lozi mappings. Although it may happen that the last statement is false, there should be some connection between piecewise linear and quadratic mappings of the plane. In the present paper. I prove that, for some set of values of the parameters, the Lozi mappings have strange attractors. To do this, I find a trapping region and then prove that the mapping has a hyperbolic structure. These facts, along with some geometrical properties of the mapping, enable me to prove that the intersection of the images of the trapping region is ;I strange attractor. Since the mapping is not everywhere differentiable. its hyperbolic structure can be understood only as the existence of a hyperbolic splitting a t those points a t which it may exist. This splitting cannot be extended to a continuous one on the whole plane. The usual meaning of the notions of stable and unstable manifolds also has to be changed a little bit. They are broken lines, and therefore not manifolds. Nevertheless, I prefer to call them (un)stable manifolds rather than (un)stable sets. I show that they exist a t almost all points of the trapping region. In the last section, I discuss briefly what happens for other values of parameters.