Interesting examples in $\mathbb{C}^2$ of maps tangent to the identity without domains of attraction

二维恒等切触双全纯映射的局部动力学

Local dynamics of tangent to the identity biholomorphisms in dimension two

本文综述了二维恒等切触双全纯映射稳定流形的存在性，并探讨其与形式向量场不变曲线存在性的关联。

Invariants of identity-tangent diffeomorphisms: explicit formulae and effective computation

In this short Survey we revisit the subject of local, identity-tangent diffeomorphisms of $\doC$ and their analytic invariants, under two viewpoints: that of explicit expansions, which necessarily involve multitangents and multizetas; and that of effective computation. Along the way, we stress the difference between the \textup{collectors} (pre-invariant but of one piece) and the \textup{connectors} (invariant but mutually unrelated). We also attempt to streamline the nomenclature and notations.

Dynamics of skew-products tangent to the identity

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…

Birational properties of tangent to the identity germs without non-degenerate singular directions

We provide a family of isolated tangent to the identity germs $f:(\mathbb{C}^3,0) \to (\mathbb{C}^3,0)$ which possess only degenerate characteristic directions, and for which the lift of $f$ to any modification (with suitable properties) has only degenerate characteristic directions. This is in sharp contrast with the situation in dimension $2$, where any isolated tangent to the identity germ $f$ admits a modification where the lift of $f$ has a non-degenerate characteristic direction. We compare this situation with the resolution of singularities of the infinitesimal generator of $f$, showing…

Remarks on Automorphisms of $\mathbb{C}^* \times \mathbb{C}^*$ and their basins

We study basins of attraction of automorphisms of $\CC^2$ tangent to the identity that fix both axes. Our main result is that, if a well known conjecture about automorphisms of $\CC^*\times\CC^*$ holds, then there are no basins of attraction associated to the non-degenerate characteristic directions (in the sense of Hakim), and therefore we cannot find a Fatou-Bieberbach domain that does not intersect both axis with this method.

A p-adic approach to local analytic dynamics: analytic flows and analytic maps tangent to the identity

In this note, we will consider the question of local equivalence of analytic functions which fix the origin and are tangent to the identity, as well as the question of flows of analytic vector fields. All mappings and equivalences are considered in the non-archimedean context e.g. all norms can be considered $p$-adic norms. We show that any two mappings $f$ and $g$ which are formally equivalent are also analytically equivalent, and we show that analytic vector fields generate analytic flows. We consider the related questions of roots and centralizers for analytic mappings. In this setting, any…

Characteristic directions of two-dimensional biholomorphisms

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $\mathbb{C}^2$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

Parabolic curves of diffeomorphisms asymptotic to formal invariant curves

We prove that if $F$ is a tangent to the identity diffeomorphism at $0\in\mathbb{C}^2$ and $Γ$ is a formal invariant curve of $F$ then there exists a parabolic curve (attracting or repelling) of $F$ asymptotic to $Γ$. The result is a consequence of a more general one in arbitrary dimension, where we prove the existence of parabolic curves of a tangent to the identity diffeomorphism $F$ at $0\in\mathbb{C}^n$ asymptotic to a given formal invariant curve under some additional conditions, expressed in terms of a reduction of $F$ to a special normal form by means of blow-ups and ramifications along…

Parabolic curves for diffeomorphisms in (C^2,0)

We give a simple proof of existence of parabolic curves for tangent to the identity diffeomorphisms in (C^2,0) with isolated fixed point.

Parabolic implosion for endomorphisms of $\mathbb{C}^2$

We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.

Class of the infinitesimal generator of a diffeomorphism in $(\C^m,0)$

Let $F$ be an analytic diffeomorphism in $(\C^m,0)$ tangent to the identity of order $n$. The infinitesimal generator of $F$ is the formal vector field $X$ such that $\Exp X=F$. In this paper we provide an elementary proof of the fact that $X$ belongs to the Gevrey class of order $1/n$.

复数空间C^n中的花定理

A flower theorem in $\mathbb{C}^n$

本文在复数空间C^n中证明了非退化且减少切到恒等germs在任意维度中固定坐标超空间的花定理类比。

螺旋域与$\mathbb{C}^2$中抛物芽的Lavaurs型重整化

Spiral Domains and Lavaurs-Type Renormalization for Parabolic Germs of $\mathbb{C}^2$

研究$\mathbb{C}^2$中与恒等映射相切的抛物芽的局部动力学，证明螺旋域的存在性，并建立非斜积映射的Lavaurs型重整化定理，得到新的拓扑不变量并构造Fatou分支。