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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.28577 |
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| _version_ | 1866915899882078208 |
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| author | Astorg, Matthieu López-Hernanz, Lorena Raissy, Jasmin |
| author_facet | Astorg, Matthieu López-Hernanz, Lorena Raissy, Jasmin |
| contents | In this paper, we extend the theory of parabolic implosion in complex dimension 2 to the case of holomorphic maps tangent to the identity at order 2. We investigate the bifurcation phenomena that occur when a fully parabolic fixed point is perturbed. Under the assumption of a non-degenerate characteristic direction with a formal invariant curve and director $α$ satisfying $\reα> 2$, we establish the existence of Lavaurs maps as limits of iterates $f_{ε_n}^n$ for specific sequences of the perturbation parameter $ε_n$. Finally, we apply these results to prove the discontinuity of the Julia sets $J_1$ and $J_2$ for holomorphic endomorphisms of $\mathbb{P}^2$, generalizing classical one-dimensional results to this higher-dimensional setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28577 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Parabolic implosion in dimension 2 Astorg, Matthieu López-Hernanz, Lorena Raissy, Jasmin Dynamical Systems In this paper, we extend the theory of parabolic implosion in complex dimension 2 to the case of holomorphic maps tangent to the identity at order 2. We investigate the bifurcation phenomena that occur when a fully parabolic fixed point is perturbed. Under the assumption of a non-degenerate characteristic direction with a formal invariant curve and director $α$ satisfying $\reα> 2$, we establish the existence of Lavaurs maps as limits of iterates $f_{ε_n}^n$ for specific sequences of the perturbation parameter $ε_n$. Finally, we apply these results to prove the discontinuity of the Julia sets $J_1$ and $J_2$ for holomorphic endomorphisms of $\mathbb{P}^2$, generalizing classical one-dimensional results to this higher-dimensional setting. |
| title | Parabolic implosion in dimension 2 |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2603.28577 |