Fractal Structure of Parametric Cantor Sets With a Common Point

For $\lambda>0$, let $E_{\lambda}$ be the self-similar set generated by the iterated function system (IFS) $\left \{ \frac{x}{3}, \frac{x+\lambda}{3} \right \}$. In this paper we study the structure of parameters $\lambda$ in which $E_\lambda$ contains a common point. $E_{\lambda}$. More precisely, for a given point $x>0$ we consider the topology of the parameter set $\Lambda \left ( x \right ) =\left \{ \lambda >0:x\in E_{\lambda } \right \}$. We show that $\Lambda \left ( x \right )$ is a Lebesgue null set contains neither interior points nor isolated points, and the Hausdorff dimension of $\Lambda \left ( x \right )$ is $\log 2/ \log 3$. Furthermore, we consider the set $\Lambda_{\mathrm {not}}(x)$ which consists of all parameters $\lambda$ that the digit frequency of $x$ in base $\lambda$ does not exist. We also consider the set $\Lambda_p(x)$ consisting of all $\lambda$ in which the digit frequency of $2$ in the base $\lambda$ expansion of $x$ is $p$. We show that the Hausdorff dimension of $\Lambda _{\mathrm {not}} \left( x \right)$ is $\log2 /\log 3$ and the lower bound Hausdorff dimension of $\Lambda _{p} \left( x \right)$ is $-p\log_3 p-(1-p)\log_3(1-p)$.