Collinear Fractals and Bandt's Conjecture
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916779881660416 |
|---|---|
| author | Espigule, Bernat Juher, David Saldaña, Joan |
| author_facet | Espigule, Bernat Juher, David Saldaña, Joan |
| contents | For a complex parameter $c$ outside the unit disk and an integer $n\ge2$, we examine the $n$-ary collinear fractal $E(c,n)$, defined as the attractor of the iterated function system $\{\mbox{$f_k \colon \mathbb{C} \longrightarrow \mathbb{C}$}\}_{k=1}^n$, where $f_k(z):=1+n-2k+c^{-1}z$. We investigate some topological features of the connectedness locus $\mathcal{M}_n$, similar to the Mandelbrot set, defined as the set of those $c$ for which $E(c,n)$ is connected. In particular, we provide a detailed answer to an open question posed by Calegari, Koch, and Walker in 2017. We also extend and refine the technique of the covering property by Solomyak and Xu to any $n\ge2$. We use it to show that a nontrivial portion of $\mathcal{M}_n$ is regular-closed. When $n\ge21$, we enhance this result by showing that, in fact, the whole $\mathcal{M}_n\setminus\mathbb{R}$ lies within the closure of its interior, thus proving that the generalized Bandt's conjecture is true. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00160 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Collinear Fractals and Bandt's Conjecture Espigule, Bernat Juher, David Saldaña, Joan Dynamical Systems 28A80, 28A78, 37F45, 11R06, 26C10 For a complex parameter $c$ outside the unit disk and an integer $n\ge2$, we examine the $n$-ary collinear fractal $E(c,n)$, defined as the attractor of the iterated function system $\{\mbox{$f_k \colon \mathbb{C} \longrightarrow \mathbb{C}$}\}_{k=1}^n$, where $f_k(z):=1+n-2k+c^{-1}z$. We investigate some topological features of the connectedness locus $\mathcal{M}_n$, similar to the Mandelbrot set, defined as the set of those $c$ for which $E(c,n)$ is connected. In particular, we provide a detailed answer to an open question posed by Calegari, Koch, and Walker in 2017. We also extend and refine the technique of the covering property by Solomyak and Xu to any $n\ge2$. We use it to show that a nontrivial portion of $\mathcal{M}_n$ is regular-closed. When $n\ge21$, we enhance this result by showing that, in fact, the whole $\mathcal{M}_n\setminus\mathbb{R}$ lies within the closure of its interior, thus proving that the generalized Bandt's conjecture is true. |
| title | Collinear Fractals and Bandt's Conjecture |
| topic | Dynamical Systems 28A80, 28A78, 37F45, 11R06, 26C10 |
| url | https://arxiv.org/abs/2411.00160 |