Projections of four corner Cantor set: total self-similarity, spectrum and unique codings
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910372120756224 |
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| author | Kong, Derong Sun, Beibei |
| author_facet | Kong, Derong Sun, Beibei |
| contents | Given $ρ\in (0,1/4]$, the four corner Cantor set $E\subset \mathbb{R}^{2}$ is a self-similar set generated by the iterated function system
\[
\left\{(ρx, ρy), \quad(ρx, ρy+1-ρ),\quad (ρx+1-ρ, ρy),\quad(ρx+1-ρ,ρy+1-ρ)\right\}.
\]
For $θ\in[0,π)$ let $E_θ$ be the orthogonal projection of $E$ onto a line with an angle $θ$ to the $x$-axis. In this paper we give a complete characterization on which the projection $E_θ$ is totally self-similar. We also study the spectrum of $E_θ$, which turns out that the spectrum of $E_θ$ achieves its maximum value if and only if $E_θ$ is totally self-similar. Furthermore, when $E_θ$ is totally self-similar, we calculate its Hausdorff dimension and study the subset $U_θ$ which consists of all $x\in E_θ$ having a unique coding. In particular, we show that $\dim_H U_θ=\dim_H E_θ$ for Lebesgue almost every $θ\in[0,π)$. Finally, for $ρ=1/4$ we describe the distribution of $θ$ in which $E_θ$ contains an interval. It turns out that the possibility for $E_θ$ to contain an interval is smaller than that for $E_θ$ to have an exact overlap. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11241 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Projections of four corner Cantor set: total self-similarity, spectrum and unique codings Kong, Derong Sun, Beibei Dynamical Systems Number Theory Given $ρ\in (0,1/4]$, the four corner Cantor set $E\subset \mathbb{R}^{2}$ is a self-similar set generated by the iterated function system \[ \left\{(ρx, ρy), \quad(ρx, ρy+1-ρ),\quad (ρx+1-ρ, ρy),\quad(ρx+1-ρ,ρy+1-ρ)\right\}. \] For $θ\in[0,π)$ let $E_θ$ be the orthogonal projection of $E$ onto a line with an angle $θ$ to the $x$-axis. In this paper we give a complete characterization on which the projection $E_θ$ is totally self-similar. We also study the spectrum of $E_θ$, which turns out that the spectrum of $E_θ$ achieves its maximum value if and only if $E_θ$ is totally self-similar. Furthermore, when $E_θ$ is totally self-similar, we calculate its Hausdorff dimension and study the subset $U_θ$ which consists of all $x\in E_θ$ having a unique coding. In particular, we show that $\dim_H U_θ=\dim_H E_θ$ for Lebesgue almost every $θ\in[0,π)$. Finally, for $ρ=1/4$ we describe the distribution of $θ$ in which $E_θ$ contains an interval. It turns out that the possibility for $E_θ$ to contain an interval is smaller than that for $E_θ$ to have an exact overlap. |
| title | Projections of four corner Cantor set: total self-similarity, spectrum and unique codings |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2211.11241 |