Lavaurs algorithm for cubic symmetric polynomials
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909677732757504 |
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| author | Blokh, Alexander Oversteegen, Lex G. Selinger, Nikita Timorin, Vladlen Vejandla, Sandeep Chowdary |
| author_facet | Blokh, Alexander Oversteegen, Lex G. Selinger, Nikita Timorin, Vladlen Vejandla, Sandeep Chowdary |
| contents | To investigate the degree $d$ connectedness locus, Thurston studied \emph{$σ_d$-invariant laminations}, where $σ_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f_c(z) = z^2 +c$. In the same spirit, we consider the space of all \emph{cubic symmetric polynomials} $f_λ(z)=z^3+λ^2 z$ in three articles. In the first one we construct the lamination $C_sCL$ together with the induced factor space $\mathbb{S}/C_sCL$ of the unit circle $\mathbb{S}$. As will be verified in the third paper, $\mathbb{S}/C_sCL$ is a monotone model of the \emph{cubic symmetric connectedness locus}, i.e., the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for constructing $C_sCL$ analogous to the Lavaurs algorithm for constructing a combinatorial model $\mathcal{M}^{comb}_2$ of the Mandelbrot set $\mathcal{M}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_06734 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Lavaurs algorithm for cubic symmetric polynomials Blokh, Alexander Oversteegen, Lex G. Selinger, Nikita Timorin, Vladlen Vejandla, Sandeep Chowdary Dynamical Systems Primary 37F20, Secondary 37F10 To investigate the degree $d$ connectedness locus, Thurston studied \emph{$σ_d$-invariant laminations}, where $σ_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f_c(z) = z^2 +c$. In the same spirit, we consider the space of all \emph{cubic symmetric polynomials} $f_λ(z)=z^3+λ^2 z$ in three articles. In the first one we construct the lamination $C_sCL$ together with the induced factor space $\mathbb{S}/C_sCL$ of the unit circle $\mathbb{S}$. As will be verified in the third paper, $\mathbb{S}/C_sCL$ is a monotone model of the \emph{cubic symmetric connectedness locus}, i.e., the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for constructing $C_sCL$ analogous to the Lavaurs algorithm for constructing a combinatorial model $\mathcal{M}^{comb}_2$ of the Mandelbrot set $\mathcal{M}_2$. |
| title | Lavaurs algorithm for cubic symmetric polynomials |
| topic | Dynamical Systems Primary 37F20, Secondary 37F10 |
| url | https://arxiv.org/abs/2202.06734 |