Quadratic Polynomials with Julia Sets of Hausdorr Dimension Close to 2

We consider certain families of quadratic polynomials which admit pa-rameterisations in a neighbourhood of the boundary of the Mandelbrot set. We show that the associated Julia sets are of Hausdorr dimension arbitrarily close to 2. Our construction clariies related work by Shishikura. 0 Introduction During the last decade much attention has been paid to the fractal aspects of degeneration processes naturally arising in Complex Dynamics. In this paper we study certain families of parabolic quadratic polynomials over C. We show that their associated Julia sets are of Hausdorr dimension arbitrarily close to 2. Our construction clariies related work by Shishikura concerning the boundary of the Mandelbrot set (see Shi91]). More precisely, for small values of t, we consider a family ff t g of maps of the form z 7 ! exp(?2it) z + z 2. For all of these maps the origin is a parabolic 1 xed point. We show that in a suitable neighbourhood of this common parabolic xed point the tree structure of the individual backward iteration admits a decomposition arbitrarily close to a semi-group action generated by three linear independent maps. The maps are called the stabiliser map, translation map and return map. The structure of the paper is as follows. In section 1 we give the construction of our basic fractal model. We show that this model gives rise to a family of iterated function systems whose associated limit sets have Hausdorr dimension arbitrarily close to 2. Subsequently, in section 2, we give a realisation of this fractal model in terms of quadratic maps over C. The construction is split into three stages. In the rst stage we consider the map f 0 , describing how to derive the sta-biliser, translation and return maps in this context, and giving the crucial estimates for their derivatives. In particular, we obtain a formula for the Hausdorr dimension of the limit set induced by the semi-group generated by the translation map together with the inverse branches of f 0. In the second stage we generalise this formula to the case of the map f 1=p , where p denotes the number of repelling parabolic petals. We show that the Hausdorr dimension of the corresponding limit set is bounded from below by 2 ? 2=(p + 1). In the third stage we deduce that, for a sensitive choice of parameter values t, the inverse branches of f t provide …