Infinite iterated function systems depending on a parameter

• Autorzy: Jaksztas L.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Hausdorff dimension; iterated function system; Julia-Lavaurs sets

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2007
• Tom: Vol. 55, no 2
• Strony: 105--122
• Opis fizyczny: Bibliogr. 6 poz., rys.

Twórcy

autor

Jaksztas L.

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland,

Bibliografia

• [1] A. Douady, Does a Julia set depend continuously on the polynomial?, in: Proc. Sympos. Appl. Math. 49, Amer. Math. Soc., 1994, 91-135.
• [2] A. Douady, P. Sentenac et M. Zinsmeister, Implosion parabolique et dimension de Hausdorff, C. R. Acad. Sci. Paris Ser. I 325 (1997), 765-772.
• [3] E. Hille, Analytic Function Theory, Ginn and Co., Boston, 1962.
• [4] R. D. Mauldin and M. Urbański, Dimensions and measures for infinite iterated function systems, Proc. London Math. Soc. 3 (1996), 105-154.
• [5] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99-107.
• [6] M. Urbański et M. Zinsmeister, Continuity of Hausdorff dimension of Julia-Lavaurs sets as a function of the phase, Conform. Geom. Dyn. 5 (2001), 140-152.