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arXiv · 1407.2563

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

Boris Solomyak
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Abstract

We study the connectedness locus 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 were obtained in joint work with P. Shmerkin (2006). Here we establish rigorous bounds for the set N based on the study of power series of special form. We also derive some bounds for the region of "*-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 is connected and locally connected, and conjecture that the entire connectedness locus is connected. We also prove that the set N has many zero angle "cusp corners," at certain points with algebraic coordinates.

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Boris Solomyak. 2014-07-09. Connectedness locus for pairs of affine maps and zeros of power series. https://arxiv.org/abs/1407.2563

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