Baby Mandelbrot sets and Spines in some one-dimensional subspaces of the parameter space for generalized McMullen Maps
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908412933046272 |
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| author | Boyd, Suzanne Hoeppner, Matthew |
| author_facet | Boyd, Suzanne Hoeppner, Matthew |
| contents | For the family of complex rational functions of the form $R_{n,c,a}(z) = z^n + \dfrac{a}{z^n}+c$,
known as ``Generalized McMullen maps'', for $a\neq 0$ and $n \geq 3$ fixed, we study the boundedness locus in some one-dimensional slices of the $(a,c)$-parameter space, by fixing a parameter or imposing a relation.
First, if we fix $c$ with $|c|\geq 6$ while allowing $a$ to vary, assuming a modest lower bound on $n$ in terms of $|c|$, we establish the location in the $a$-plane of $n$ ``baby" Mandelbrot sets, that is, homeomorphic copies of the original Mandelbrot set. We use polynomial-like maps, introduced by Douady and Hubbard and applied for the subfamily $R_{n,a,0}$ by Devaney.
Second, for slices in which $c=ta$, we again observe what look like baby Mandelbrot sets within these slices, and begin the study of this subfamily by establishing a neighborhood containing the boundedness locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Baby Mandelbrot sets and Spines in some one-dimensional subspaces of the parameter space for generalized McMullen Maps Boyd, Suzanne Hoeppner, Matthew Dynamical Systems 37F10, 32A20 (Primary) 32A19 (Secondary) For the family of complex rational functions of the form $R_{n,c,a}(z) = z^n + \dfrac{a}{z^n}+c$, known as ``Generalized McMullen maps'', for $a\neq 0$ and $n \geq 3$ fixed, we study the boundedness locus in some one-dimensional slices of the $(a,c)$-parameter space, by fixing a parameter or imposing a relation. First, if we fix $c$ with $|c|\geq 6$ while allowing $a$ to vary, assuming a modest lower bound on $n$ in terms of $|c|$, we establish the location in the $a$-plane of $n$ ``baby" Mandelbrot sets, that is, homeomorphic copies of the original Mandelbrot set. We use polynomial-like maps, introduced by Douady and Hubbard and applied for the subfamily $R_{n,a,0}$ by Devaney. Second, for slices in which $c=ta$, we again observe what look like baby Mandelbrot sets within these slices, and begin the study of this subfamily by establishing a neighborhood containing the boundedness locus. |
| title | Baby Mandelbrot sets and Spines in some one-dimensional subspaces of the parameter space for generalized McMullen Maps |
| topic | Dynamical Systems 37F10, 32A20 (Primary) 32A19 (Secondary) |
| url | https://arxiv.org/abs/2411.04938 |