Irreducibility of eventually $2$-periodic curves in the moduli space of cubic polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916900216242176 |
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| author | Patra, Niladri |
| author_facet | Patra, Niladri |
| contents | Consider the moduli space, $\mathcal{M}_{3},$ of cubic polynomials over $\mathbb{C}$, with a marked critical point. Let $\mathscr{S}_{k,n}$ be the set of all points in $\mathcal{M}_{3}$ for which the marked critical point is strictly $(k,n)$-preperiodic. Milnor conjectured that the affine algebraic curves $\mathscr{S}_{k,n}$ are irreducible, for all $k \geq 0, n>0$. In this article, we show the irreducibility of eventually $2$-periodic curves, i.e. $\mathscr{S}_{k,2},\; k\geq 0$ curves. We also note that the curves, $\mathscr{S}_{k,2},\; k\geq 0$, exhibit a possible splitting-merging phenomenon that has not been observed in earlier studies of $\mathscr{S}_{k,n}$ curves. Finally, using the irreducibility of $\mathscr{S}_{k,2}$ curves, we give a new and short proof of Galois conjugacy of unicritical points lying on $\mathscr{S}_{k,2}$, for even natural number $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19944 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Irreducibility of eventually $2$-periodic curves in the moduli space of cubic polynomials Patra, Niladri Dynamical Systems Number Theory 11R09 (Primary), 37F12, 37P45 (Secondary) Consider the moduli space, $\mathcal{M}_{3},$ of cubic polynomials over $\mathbb{C}$, with a marked critical point. Let $\mathscr{S}_{k,n}$ be the set of all points in $\mathcal{M}_{3}$ for which the marked critical point is strictly $(k,n)$-preperiodic. Milnor conjectured that the affine algebraic curves $\mathscr{S}_{k,n}$ are irreducible, for all $k \geq 0, n>0$. In this article, we show the irreducibility of eventually $2$-periodic curves, i.e. $\mathscr{S}_{k,2},\; k\geq 0$ curves. We also note that the curves, $\mathscr{S}_{k,2},\; k\geq 0$, exhibit a possible splitting-merging phenomenon that has not been observed in earlier studies of $\mathscr{S}_{k,n}$ curves. Finally, using the irreducibility of $\mathscr{S}_{k,2}$ curves, we give a new and short proof of Galois conjugacy of unicritical points lying on $\mathscr{S}_{k,2}$, for even natural number $k$. |
| title | Irreducibility of eventually $2$-periodic curves in the moduli space of cubic polynomials |
| topic | Dynamical Systems Number Theory 11R09 (Primary), 37F12, 37P45 (Secondary) |
| url | https://arxiv.org/abs/2305.19944 |