Fractal Structure of Parametric Cantor Sets With a Common Point

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 $-p\log_3 p-(1-p)\log_3(1-p)$.