The interior of randomly perturbed self-similar sets on the line
Michel Dekking (Centrum Wiskunde & Informatica (CWI), TU Delft - Applied Probability)
Károly Simon (Budapest University of Technology and Economics, Alfréd Rényi Institute)
Balázs Székely (Budapest University of Technology and Economics)
Nóra Szekeres (Budapest University of Technology and Economics)
More Info
expand_more
Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.
Abstract
Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper by Jordan et al. (2007) [6]. For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure.
help_outline