A well-known theorem due to Mañé-Sad-Sullivan and Lyubich asserts that J-stable maps
are dense in any holomorphic family of rational maps in dimension 1. In this paper we show …

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Let $ T_1,\ldots, T_m $ be a family of $ d\times d $ invertible real matrices with $\| T_i\|< 1/2$
for $1\leq i\leq m $. We provide some sufficient conditions on these matrices such that the …

We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical
parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension …

This paper examines thresholds for certain properties of the attractor of a general one-
parameter affine family of iterated functions systems. As the parameter increases, the …

Let $ C $ be the middle-third Cantor set, and $ f $ a continuous function defined on an open
set $ U\subset\mathbb {R}^{2} $. Denote the image\begin {equation*} f_ {U}(C, C)=\{f (x, y):(x …

Let K be the attractor of the following iterated function system (IFS){f 1 (x)= λ x, f 2 (x)= λ x+ c−
λ, f 3 (x)= λ x+ 1− λ}, where f 1 (I)∩ f 2 (I)≠∅,(f 1 (I)∪ f 2 (I))∩ f 3 (I)=∅, and I=[0, 1] is the …

If $ X $ is the attractor set of a conformal IFS (iterated function system) in dimension two or
three, we prove that there exists a quasiregular semigroup $ G $ with a Julia set equal to $ X …

Let $ M $ be a $ d\times d $ contracting matrix. In this paper we consider the self-affine
iterated function system $\{Mv-u, Mv+ u\} $, where $ u $ is a cyclic vector. Our main result is …

Let 0< λ< μ< 1 and λ+ μ> 1. In this paper, we prove that for the vast majority of such
parameters the top of the planar attractor A λ, μ of the IFS {(λ x, μ y),(μ x+ 1− μ, λ y+ 1− λ)} is …