On the structure of λ-Cantor set with overlaps

2018/10/19 by Dajani, Karma, Kong, Derong, Yao, Yuanyuan · 1 citation
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.08324

Abstract

Given λ∈(0, 1), let E_λ be the self-similar set generated by the iterated function system \x/3,(x+λ)/3,(x+2)/3\. Then E_λ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for E_λ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When E_λ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of E_λ and show that the spectrum of E_λ vanishes if and only if λ is irrational.

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