The interior of randomly perturbed self-similar sets on the line

Journal article (2024)

Authors

F.M. Dekking Applied Probability - EEMCS , Centrum Wiskunde & Informatica (CWI)
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
Research Group
Applied Probability (EEMCS) (TU Delft)
Palis conjecture Difference of Cantor sets Multi type branching processes Random fractals Self-similar set
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Published Date

2024

Language

English

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

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.