Factorization of the quadratic Misiurewicz-Thurston polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912443890925568 |
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| author | Mihalache, Nicolae Vigneron, Francois |
| author_facet | Mihalache, Nicolae Vigneron, Francois |
| contents | This note provides the complete factorization of the Misiurewicz-Thurston polynomial $q_{\ell,n}=p_{\ell+n}(z) - p_\ell(z)$ over $\mathbb{C}$, which plays a central role in the study of the Mandelbrot set, where \[ p_0(z) = 0, \qquad p_{n+1}(z) = p_n(z)^2 + z. \] The roots can be classified into two categories. First, there are hyperbolic points $\operatorname{hyp}(k)$ for any divisor $k$ of $n$, which are parameters whose critical orbits are of exact period $k$. Those are roots of $q_{\ell,n}$ with multiplicity $\left\lfloor \frac{\ell -1}{k} \right\rfloor + 2$. Next are the points $\operatorname{mis}(j,k)$ for $2\leq j\leq \ell$ whose critical orbits are pre-periodic of exact period $k$ with an exact pre-period $j$. Those are simple roots of $q_{\ell,n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17662 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Factorization of the quadratic Misiurewicz-Thurston polynomials Mihalache, Nicolae Vigneron, Francois Dynamical Systems 37F10, 37F20 This note provides the complete factorization of the Misiurewicz-Thurston polynomial $q_{\ell,n}=p_{\ell+n}(z) - p_\ell(z)$ over $\mathbb{C}$, which plays a central role in the study of the Mandelbrot set, where \[ p_0(z) = 0, \qquad p_{n+1}(z) = p_n(z)^2 + z. \] The roots can be classified into two categories. First, there are hyperbolic points $\operatorname{hyp}(k)$ for any divisor $k$ of $n$, which are parameters whose critical orbits are of exact period $k$. Those are roots of $q_{\ell,n}$ with multiplicity $\left\lfloor \frac{\ell -1}{k} \right\rfloor + 2$. Next are the points $\operatorname{mis}(j,k)$ for $2\leq j\leq \ell$ whose critical orbits are pre-periodic of exact period $k$ with an exact pre-period $j$. Those are simple roots of $q_{\ell,n}$. |
| title | Factorization of the quadratic Misiurewicz-Thurston polynomials |
| topic | Dynamical Systems 37F10, 37F20 |
| url | https://arxiv.org/abs/2506.17662 |