Cubic polynomials with a 2-cycle of Siegel disks
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912081549197312 |
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| author | Fu, Yuming Hu, Jun Muzician, Oleg |
| author_facet | Fu, Yuming Hu, Jun Muzician, Oleg |
| contents | Under conjugation by affine transformations, the dynamical moduli space of cubic polynomials $f$ with a $2$-cycle of Siegel disks is parameterized by a three-punctured complex plane as a degree-$2$ cover. Assuming the rotation number of $f^2$ on the Siegel disk is of bounded type, we show that on the three-punctured complex plane, the locus of the cubic polynomials with both finite critical points on the boundaries of the Siegel disks on the $2$-cycle is comprised of two arcs, corresponding to the cases with two critical points on the boundary of the same Siegel disk, and a Jordan curve, corresponding to the cases with two critical points on the boundaries of different Siegel disks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16728 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cubic polynomials with a 2-cycle of Siegel disks Fu, Yuming Hu, Jun Muzician, Oleg Dynamical Systems Complex Variables 37F10 Under conjugation by affine transformations, the dynamical moduli space of cubic polynomials $f$ with a $2$-cycle of Siegel disks is parameterized by a three-punctured complex plane as a degree-$2$ cover. Assuming the rotation number of $f^2$ on the Siegel disk is of bounded type, we show that on the three-punctured complex plane, the locus of the cubic polynomials with both finite critical points on the boundaries of the Siegel disks on the $2$-cycle is comprised of two arcs, corresponding to the cases with two critical points on the boundary of the same Siegel disk, and a Jordan curve, corresponding to the cases with two critical points on the boundaries of different Siegel disks. |
| title | Cubic polynomials with a 2-cycle of Siegel disks |
| topic | Dynamical Systems Complex Variables 37F10 |
| url | https://arxiv.org/abs/2410.16728 |