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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2503.10082 |
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| _version_ | 1866917954977792000 |
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| author | Meng, Xinyi |
| author_facet | Meng, Xinyi |
| contents | 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 $-p\log_3 p-(1-p)\log_3(1-p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractal Structure of Parametric Cantor Sets With a Common Point Meng, Xinyi Dynamical Systems 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 $-p\log_3 p-(1-p)\log_3(1-p)$. |
| title | Fractal Structure of Parametric Cantor Sets With a Common Point |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2503.10082 |