JournalsjfgVol. 2, No. 3pp. 281–308

Connectedness locus for pairs of affine maps and zeros of power series

  • Boris Solomyak

    Bar-Ilan University, Ramat Gan, Israel
Connectedness locus for pairs of affine maps and zeros of power series cover
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Abstract

We study the connectedness locus N\mathcal N for the family of iterated function systems of pairs of affine-linear maps in the plane (the non-self-similar case). First results on the set N\mathcal N were obtained in joint work with P. Shmerkin [11]. Here we establish rigorous bounds for the set N\mathcal N based on the study of power series of special form. We also derive some bounds for the region of “\ast-transversality” which have applications to the computation of Hausdorff measure of the self-affine attractor. We prove that a large portion of the set N\mathcal N is connected and locally connected, and conjecture that the entire connectedness locus is connected. We also prove that the set N\mathcal N has many zero angle “cusp corners,” at certain points with algebraic coordinates.

Cite this article

Boris Solomyak, Connectedness locus for pairs of affine maps and zeros of power series. J. Fractal Geom. 2 (2015), no. 3, pp. 281–308

DOI 10.4171/JFG/22