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Dynamics of skew-products tangent to the identity

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 $dP_0=\id$. 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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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03633131
Contributor : Matthieu Astorg Connect in order to contact the contributor
Submitted on : Wednesday, April 6, 2022 - 5:34:46 PM
Last modification on : Tuesday, April 12, 2022 - 3:40:12 AM

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  • HAL Id : hal-03633131, version 1

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Matthieu Astorg, Luka Boc Thaler. Dynamics of skew-products tangent to the identity. 2022. ⟨hal-03633131⟩

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