Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas¹

¹ Institute of Mathematics Polish Academy of Sciences ęniadeckich 8 00-956 Warszawa, Poland and Faculty of Mathematics and Information Sciences Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) no. 2, pp. 105-122

Résumé

This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia–Lavaurs sets $J_{0,\sigma}$ for the map $f_0(z)=z^2+1/4$ on the parameter~$\sigma$. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0,\sigma}$, given by Urbański and Zinsmeister. The closure of the limit set of our IFS $\{\phi^{n,k}_{\sigma,\alpha}\}$ is the closure of some family of circles, and if the parameter $\sigma$ varies, then the behavior of the limit set is similar to the behavior of $J_{0,\sigma}$. The parameter $\alpha$ determines the diameter of the largest circle, and therefore the diameters of other circles. We prove that for all parameters $\alpha$ except possibly for a set without accumulation points, for all appropriate $t>1$ the sum of the $t$th powers of the diameters of the images of the largest circle under the maps of the IFS depends on the parameter $\sigma$. This is the first step to verifying the conjectured dependence of the pressure and Hausdorff dimension on $\sigma$ for our model and for $J_{0,\sigma}$.

Détail

DOI : 10.4064/ba55-2-2

Keywords: paper motivated problem dependence hausdorff dimension julia lavaurs sets sigma map parameter sigma using homographies imitate construction iterated function system ifs whose limit set subset sigma given urba ski zinsmeister closure limit set ifs phi sigma alpha closure family circles parameter sigma varies behavior limit set similar behavior sigma parameter alpha determines diameter largest circle therefore diameters other circles prove parameters alpha except possibly set without accumulation points appropriate sum tth powers diameters images largest circle under maps ifs depends parameter sigma first step verifying conjectured dependence pressure hausdorff dimension sigma model sigma