Infinite Iterated Function Systems Depending on a Parameter
Volume 55 / 2007
Abstract
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 $J_{0,\sigma}$.
