Marco Abate

The residual index and the dynamics of holomorphic maps tangent to the identity

Let f be a (germ of) holomorphic self-map of C such that the origin is an isolated fixed point, and such that dfO = id. Let ν(f) be the degree of the first non-vanishing term in the homogeneous expansion of f − id. We generalize to C the classical Leau-Fatou Flower Theorem proving that there exist ν(f)−1 holomorphic curves f -invariant, with the origin in their boundary, and attracted by O under the action of f . 0. Introduction One of the most famous theorems in one-dimensional holomorphic dynamics is Theorem 0.1: (Leau-Fatou Flower Theorem [L, F]) Let g(ζ) = ζ+akζ+O(ζ), with k ≥ 2 and ak 6= 0, be a holomorphic function fixing the origin. Then there are k − 1 disjoint domains D1, . . . , Dk−1 with the origin in their boundary, invariant under g (that is, g(Dj) ⊂ Dj) and such that (g|Dj ) → 0 as n → ∞, for j = 1, . . . , k − 1, where g denotes the composition of g with itself n times. Any such domain is called a parabolic domain for f at the origin, and they are (together with attracting basins, Siegel disks and Hermann rings) among the building blocks of Fatou sets of rational functions (see, e.g., [CG] for a modern exposition). A natural problem in higher dimensional holomorphic dynamics is to find a generalization of this result, where the function g is replaced by a germ f of self-map of C fixing the origin and tangent to the identity, that is such that dfO = id. After preliminary results in C obtained by Ueda [U] and Weickert [W], a very important step in this direction has been made by Hakim [H1, 2] (inspired by previous works by Ecalle [E]). To describe her results, we need a couple of definitions. Let f be a germ of holomorphic self-map of C fixing the origin and tangent to the identity. A parabolic curve for f at the origin is a injective holomorphic map φ: ∆→ C satisfying the following properties: (i) ∆ is a simply connected domain in C with 0 ∈ ∂∆; (ii) φ is continuous at the origin, and φ(0) = O; (iii) φ(∆) is invariant under f , and (f |φ(∆)) → O as n→∞. Furthermore, if [φ(ζ)] → [v] ∈ Pn−1 as ζ → 0 (where [·] denotes the canonical projection of C \ {O} onto Pn−1) we say that φ is tangent to [v] at the origin. Writing f = (f1, . . . , fn), let fj = zj +Pj,νj +Pj,νj+1 + · · · be the homogeneous expansion of f in series of homogeneous polynomial, where degPj,k = k (or Pj,k ≡ 0), and Pj,νj 6≡ 0. The order ν(f) is defined by ν(f) = min{ν1, . . . , νn}. A characteristic direction for f is a vector [v] = [v1 : · · · : vn] ∈ Pn−1 such that there is λ ∈ C so that Pj,ν(f)(v1, . . . , vn) = λvj for j = 1, . . . , n. If λ 6= 0 we shall say that [v] is non-degenerate; otherwise it is degenerate. Then Hakim’s result is: Theorem 0.2: (Hakim [H1, 2]) Let f be a (germ of) holomorphic self-map of C fixing the origin and tangent to the identity. Then for every non-degenerate characteristic direction [v] of f there are ν(f) − 1 parabolic curves tangent to [v] at the origin. This is a very good generalization of Theorem 0.1, but applies only to generic maps: if f has no non-degenerate characteristic directions, this theorem gives no informations about the dynamics of f . 1 Partially supported by Progetto MURST di Rilevante Interesse Nazionale Proprietà geometriche delle varietà reali e complesse. 1991 Mathematical Subjects Classification: Primary 32H50, 32H02, 58F23

Discrete Holomorphic Local Dynamical Systems

This chapter is a survey on local dynamics of holomorphic maps in one and several complex variables, discussing in particular normal forms and the structure of local stable sets in the non-hyperbolic case, and including several proofs and a large bibliography.

The Infinitesimal Generators of Semigroups of Holomorphic Maps (

Vs, teR § ~sO~t=~s+t . This notation is a continuous analogue of the concept of sequence of iterates of a map fe Hol(X, X); indeed, a sequence of iterates can be characterized as a map (P: N-~ --~ Hol (X, X) such that ~0 = idx and satisfying (1) for every s, t ~ N. The first paper concerning one-parameter semigroups of holomorphic maps seems to be [T], where problems somehow regarding the asymptotic behavior of one-parameter semigroups on 4, the unit disk in C, are studied. Later on, the typical approach used to be via the idea of fractional iteration; loosely stated, one wants to find a sensible way of defining, at least locally, the r-th iterate of a holomorphic function for any positive real number r. For a recent work on this subject, see [C]. The real break-through in the study of one-parameter semigroups in one complex variable is due to BERKSON and PORTA [BP] and HEINS [H]. Following [W2], they related semigroups and the theory of ordinary differential equations, being able to classify all one-parameter semigroups on Riemann surfaces (for a unified account of their results see [A3]). Strangely, there seems to be almost no papers on semigroups in several complex variables; as far as we know, they have been studied only in [A1, 2] and [V]. In this paper we want to generalize to arbitrary complex manifolds some of the results of [BP]; in particular, we want to describe at some extent the relationships between semigroups and ODE in several complex variables. First of all, we fix some notations. Let X and Y be two complex manifolds. A sequence {f, } c Hol (X, Y) is said compactly divergent if for every pair of compact sets

The Julia-Wolff-Caratheodory theorem in polydisks

The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls inCn and by the author to holomorphic maps between strongly (pseudo)convex domains. Here we describe Julia-Wolff-Carathéodory theorems for holomorphic maps defined in a polydisk and with image either in the unit disk, or in another polydisk, or in a strongly convex domain. One of the main tools for the proof is a general version of the Lindelöf principle valid for not necessarily bounded holomorphic functions.

Angular Derivatives in Several Complex Variables

1. Introduction 2. One Complex Variable 3. Julia’s Lemma 4. Lindelof Principles 5. The Julia-Wolff-Caratheodory Theorem References

Parabolic curves in ℂ3

We discuss a family of holomorphic self-maps of ℂ3 tangent to the identity at the origin presenting dynamical phenomena not appearing for lower-dimensional maps.

Holomorphic curvature of Finsler metrics and complex geodesics

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov iff the Finsler metric is Kähler, has constant holomorphic sectional curvature −4, and its curvature tensor satisfies a specific simmetry condition—which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature −4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.

Common fixed points in hyperbolic Riemann surfaces and convex domains

In this paper we prove that a commuting family of continuous selfmaps of a bounded convex domain in Cn which are holomorphic in the interior has a common fixed point. The proof makes use of three basic ingredients: iteration theory of holomorphic maps, a precise description of the structure of the boundary of a convex domain, and a similar result for commuting families of self-maps of a hyperbolic domain of a compact Riemann surface.

Holomorphic Dynamical Systems

This chapter is a survey on local dynamics of holomorphic maps in one and several complex variables, discussing in particular normal forms and the structure of local stable sets in the non-hyperbolic case, and including several proofs and a large bibliography.

A characterization of hyperbolic manifolds

In this note we prove that a complex manifold X is Kobayashi hyperbolic if and only if the space Hol(A, X) of holomorphic maps of the unit disk A into X is relatively compact (with respect to the compact-open topology) in the space C(A, X*) of continuous maps from A into the onepoint compactification X* of X.