FRACTAL PARAMETERS FOR THE STUDY OF HÉNON-LIKE ATTRACTORS

Abstract

For invertible, area-contracting planar maps, much work has been done on the study of the crucial role played by certain unstable periodic orbits, distinguished by being "accessible", in understanding some global bifurcations such as metamorphoses of basin boundaries and crisis of attractors. In this paper we concentrate on a one-parameter family of Hénon maps whose attractors present a type of discontinuous change characterized by a sudden replacement of the accessible orbits as the parameter is varied. The change is quantified by a jump in the corresponding accessible rotation number. The rotation rate describes a devil's staircase as a function of the parameter. We estimate the Hausdorff dimensions of the Hénon attractors — via the Mendès France fractal dimension — for the same interval of parameter values. Our purpose is to show the strong connection between these two functions: the variation of the dimension stays minimal for the parameters under the same plateau in the staircase, while it is markedly greater when the parameter moves from one plateau to another.