Infinite Iterated Function Systems Depending on a Parameter

Volume 55 / 2007

Ludwik Jaksztas Bulletin Polish Acad. Sci. Math. 55 (2007), 105-122 MSC: Primary 37F45; Secondary 37D35. DOI: 10.4064/ba55-2-2


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 $J_{0,\sigma}$.


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

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