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Infinite iterated function systems depending on a parameter

Jaksztas L.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 2

105-122

This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets Jo,σ for the map fo(z) = x2 + 1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of Jo,σ, given by Urbański and Zinsmeister. The closure of the limit set of our IFS {φ[...]} is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of Jo,σ. The parameter a determines the diameter of the largest circle, and therefore the diameters of other circles. We prove that for all parameters a except possibly for a set without accumulation points, for all appropriate t > 1 the sum of the tth powers of the diameters of the images of the largest circle under the maps of the IFS depends on the parameter σ. This is the first step to verifying the conjectured dependence of the pressure and Hausdorff dimension on a for our model and for Jo,σ.