# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## Infinite Iterated Function Systems Depending on a Parameter

### Tom 55 / 2007

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

#### Streszczenie

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 $J_{0,\sigma}$.

#### Autorzy

• Ludwik JaksztasInstitute of Mathematics
00-956 Warszawa, Poland
and
Faculty of Mathematics and Information Sciences
Warsaw University of Technology
Pl. Politechniki 1
00-661 Warszawa, Poland
e-mail

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek