On a kind of self-similar sets with complete overlaps

Derong Kong; Yuanyuan Yao

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arXiv · 2005.03280

On a kind of self-similar sets with complete overlaps

Derong Kong·Yuanyuan Yao

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Abstract

Let $E$ be the self-similar set generated by the {\it iterated function system} {\[ f_0(x)=\frac{x}{\beta},\quad f_1(x)=\frac{x+1}{\beta}, \quad f_{\beta+1}=\frac{x+\beta+1}{\beta} \]}with $\beta\ge 3$. {Then} $E$ is a self-similar set with complete {overlaps}, i.e., $f_{0}\circ f_{\beta+1}=f_{1}\circ f_1$, but $E$ is not totally self-similar. We investigate all its generating iterated function systems, give the spectrum of $E$, and determine the Hausdorff dimension and Hausdorff measure of $E$ and of the sets which contain all points in $E$ having finite or infinite different triadic codings.

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Derong Kong, Yuanyuan Yao. 2020-05-07. On a kind of self-similar sets with complete overlaps. https://arxiv.org/abs/2005.03280

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