| Title: | Dynamics of skew-products tangent to the identity |
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| Authors: | ID Astorg, Matthieu (Author)
ID Boc Thaler, Luka (Author) |
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| Files: |  PDF - Presentation file, download (2,71 MB)
MD5: 44E0091CAD94C11342B7A72C48200979
 URL - Source URL, visit https://ems.press/journals/jems/articles/14298412
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | We study the local dynamics of generic skew-products tangent to the identity, i.e. maps of the form $P(z,w)=(p(z), q(z,w))$ with $dP_0=\mathrm{Id}$. More precisely, we focus on maps with non-degenerate second differential at the origin; such maps have local normal form $P(z,w)=(z-z^2+O(z^3),w+w^2+bz^2+O(\|(z,w)\|^3))$. We prove the existence of parabolic domains, and prove that inside these parabolic domains the orbits converge non-tangentially if and only if $b \in (\frac{1}{4},+\infty)$. Furthermore, we prove the existence of a type of parabolic implosion, in which the renormalization limits are different from previously known cases. This has a number of consequences: under a diophantine condition on coefficients of $P$, we prove the existence of wandering domains with rank $1$ limit maps. We also give explicit examples of quadratic skew-products with countably many grand orbits of wandering domains, and we give an explicit example of a skew-product map with a Fatou component exhibiting historic behaviour. Finally, we construct various topological invariants, which allow us to answer a question of Abate. |
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| Keywords: | skew-products, germs tangent to identity, parabolic implosion, wandering domains |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 559-618 |
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| Numbering: | Vol. 28, no. 2 |
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| PID: | 20.500.12556/DiRROS-27564  |
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| UDC: | 517 |
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| ISSN on article: | 1435-9855 |
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| DOI: | 10.4171/JEMS/1566 |
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| COBISS.SI-ID: | 223408643 |
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| Note: |
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| Publication date in DiRROS: | 13.02.2026 |
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| Views: | 554 |
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| Downloads: | 188 |
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| Metadata: |    |
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