Fractal Structure of Parametric Cantor Sets With a Common Point
Xinyi Meng
TL;DR
The paper studies, for each fixed $x>0$, the set of parameters $\lambda$ for which $x$ lies in the self-similar Cantor-type set $E_{\lambda}$ generated by the IFS $\{x/3,(x+\lambda)/3\}$. It constructs a bijection $\Phi_x$ between the parameter set $\Lambda(x)$ and the symbolic space, and proves that $\Lambda(x)$ is a Cantor-type set of Lebesgue measure zero with $\min\Lambda(x)=2x$, $\sup\Lambda(x)=\infty$, and $\dim_H\Lambda(x)=\log 2/\log 3$. The paper further analyzes auxiliary parameter sets: $\Lambda_{\mathrm{not}}(x)$, where the digit-frequency in the $\lambda$-adic expansion of $x$ does not exist, also has $\dim_H=\log 2/\log 3$, while $\Lambda_p(x)$ has a universal lower bound $\dim_H(\Lambda_p(x))\ge \mathrm{h}(p,1-p)/\log 3$. The techniques combine symbolic dynamics, bi-Lipschitz correspondences, Moran-set constructions, and Billingsley’s dimension theory, advancing understanding of fractal structures arising from parameter variation and digit-frequency phenomena in Cantor-type constructions and beta expansions.
Abstract
For $λ>0$, let $E_λ$ be the self-similar set generated by the iterated function system (IFS) $\left \{ \frac{x}{3}, \frac{x+λ}{3} \right \}$. In this paper we study the structure of parameters $λ$ in which $E_λ$ contains a common point. $E_λ$. More precisely, for a given point $x>0$ we consider the topology of the parameter set $Λ\left ( x \right ) =\left \{ λ>0:x\in E_{λ} \right \}$. We show that $Λ\left ( x \right )$ is a Lebesgue null set contains neither interior points nor isolated points, and the Hausdorff dimension of $Λ\left ( x \right ) $ is $ \log 2/ \log 3 $. Furthermore, we consider the set $Λ_{\mathrm {not}}(x)$ which consists of all parameters $λ$ that the digit frequency of $x$ in base $λ$ does not exist. We also consider the set $Λ_p(x)$ consisting of all $λ$ in which the digit frequency of $2$ in the base $λ$ expansion of $x$ is $p$. We show that the Hausdorff dimension of $ Λ_{\mathrm {not}} \left( x \right) $ is $\log2 /\log 3$ and the lower bound Hausdorff dimension of $ Λ_{p} \left( x \right) $ is $-p\log_3 p-(1-p)\log_3(1-p)$.