Holomorphic Diffeomorphisms Research Articles

The Maslov triple index on the Shilov boundary of a classical domain

Let D be an irreducible Hermitian symmetric space of tube-type, S its Shilov boundary, G its group of holomorphic diffeomorphisms. For a generic triple of points ( σ 1, σ 2, σ 3)∈ S× S× S, a characteristic G-invariant ι( σ 1, σ 2, σ 3), called the Maslov index was introduced in [Transform. Groups 6 (2001) 303]. For D of classical type (i.e. for all cases except for the exceptional domain associated to Albert’s algebra), the definition of the Maslov index is extended to all triples, by using a holomorphic embedding of D into a Siegel disc, which corresponds to an embedding of S into a Lagrangian manifold. When D is the Lie ball, the extension of the definition is obtained through a realization of S in the Lagrangian manifold of a spinor space.

Stable manifolds of holomorphic diffeomorphisms

We consider stable manifolds of a holomorphic diffeomorphism of a complex manifold. Using a conjugation of the dynamics to a (non-stationary) polynomial normal form, we show that typical stable manifolds are biholomorphic to complex Euclidean space.

Limit at resonances of linearizations of some complex analytic dynamical systems

We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of $({\Bbb C},0)$ and of the semi-standard map.We prove that for each resonance there exists a suitable blow-up of the Taylor series of the linearization under which it converges uniformly to an analytic function as the multiplier, or rotation number, tends non-tangentially to the resonance. This limit function is explicitly computed and related to questions of formal classification, both for the case of germs and for the case of the semi-standard map.

Holomorphic diffeomorphisms of complex semisimple Lie groups

Holomorphic diffeomorphisms of complex semisimple Lie groups

On the classification of nilpotent singularities

We deal in this paper with the analytic classification of singular holomorphic foliations defined in a neighborhood of the origin of C2 and having nonzero linear part. In the “semi simple” case this classification which is closely related to the ratio of the eigenvalues of the linear part, is quasi complete. It appears in particular that the moduli are essentially the ones of a holonomy diffeomorphism (see [2,4,6,7,10,11], . . .). Our aim here is to study the analytic classification in the “nilpotent case”, i.e., when both eigenvalues are zero. Let Λ denote the set of germs at 0 ∈ C2 of holomorphic 1-forms for which the origin is an isolated singularity. Let also Diff(C,0) (respectively Diff(C,0)) be the group of germs at 0 ∈ C of holomorphic (respectively formal) diffeomorphisms which fix the origin. Two elements ω1 and ω2 of Λ are said to be holomorphically (respectively formally) conjugated if there exists an element φ of Diff(C2,0) (respectively Diff(C2,0)) such that φ∗ω1 ∧ ω2 = 0. We will write ω1 hol ∼ ω2 (respectively ω for ∼ ω2). The moduli space ωfor of ω is the set

Analyticity, scaling and renormalization for some complex analytic dynamical systems

We review some results about the analytic structure of Lindstedt series for some complex analytic dynamical systems: in particular, we consider Hamiltonian maps (like the standard map and its generalizations), the semi-standard map and Siegel's problem of the linearization of germs of holomorphic diffeomorphisms of ( C ,0). The analytic structure of those series can be studied numerically using Padé approximants, and one can show the existence of natural boundaries for real, diophantine values of the rotation number; by complexifying the rotation number, we show how these natural boundaries arise from the accumulation of singularities due to resonances, providing a new intuitive insight into the mechanism of the break-down of invariant KAM curves. Moreover, we study the Lindstedt series at resonances, i.e. for rational values of the rotation number, by suitably rescaling to 0 the value of the perturbative parameter, and a simple analytic structure emerges. Finally, we present some proofs for the simplest models and relate these results to renormalization ideas.

Quasi-conformal mapping theorem and bifurcations

LetH be a germ of holomorphic diffeomorphism at 0 ∈ ℂ. Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze 2πi )=H○S(z) (1). IfH λ is an unfolding of diffeomorphisms depending on λ ∈ (ℂ,0), withH 0=Id, one introduces its ideal $$\mathcal{I}_H$$ . It is the ideal generated by the germs of coefficients (a i (λ), 0) at 0 ∈ ℂ k , whereH λ(z)−z=Σa i (λ)z i . Then one can find a parameter solutionS λ (z) of (1) which has at each pointz 0 belonging to the domain of definition ofS 0, an expansion in seriesS λ(z)=z+Σb i (λ)(z−z 0) i with $$(b_i ,0) \in \mathcal{I}_H$$ , for alli. This result may be applied to the bifurcation theory of vector fields of the plane. LetX λ be an unfolding of analytic vector fields at 0 ∈ ℝ2 such that this point is a hyperbolic saddle point for each λ. LetH λ(z) be the holonomy map ofX λ at the saddle point and $$\mathcal{I}_H$$ its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals σ, τ, transversal to the separatrices of the saddle point, such that the difference between the transition mapD λ(z) and the identity is divisible in the ideal $$\mathcal{I}_H$$ . Finally, suppose thatX λ is an unfolding of a saddle connection for a vector fieldX 0, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz∈σ, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX λ, is finite and can be computed explicity in terms of the Bautin ideal.

The classification of curvilinear angles in the complex plane and the groups of $\pm $ holomorphic diffeomorphisms

La classification des configurations des germes des courbe lisses et analytiques reelles dans le plan complexe est connue comme probleme local de geometrie conforme de Poincare ([5], [6], [10], [11]). Ce probleme a ete classiquement etudie en terme de groupe des germes de diffeomorphismes a holomorphes du plan complexe engendre par les reflexions antiholomorphes de Schwarz par rapport aux composants lisses. Dans cette note, nous etudions la structure du groupe en utilisant la methode du cylindre sectoriel de Ecalle-Voronin, et nous determinons l'espace de modules des couples de courbes lisses et tangentes ainsi que des fronces.

Non linearizable holomorphic dynamics having an uncountable number of symmetries

We construct germs of holomorphic diffeomorphisms of (C, 0),f(z) =e 2πiα z +O(z 2), α∈R−Q, non linearizable and having an uncountable centralizer, that is an uncountable number of symmetries. This gives a positive answer to a question of M.R. Herman. We construct such examples having no periodic orbit (except 0), and also with a sequence of periodic orbits tending to 0 (distinct from 0). Moreover the examples constructed have an infinite number of finite order symmetries, and the dynamics is quite explicit. We also construct non linearizable analytic diffeomorphisms of the circle having an irrational rotation number and uncountable centralizer with similar properties.

Sur un théorème de Dulac

We shall consider holomorphic vector fields of (ℂ n ,0) with a non zero diagonal linear part and which eigenvalues λ i ∈ℂ satisfy to some resonnance’s relations all generated by one relation (r,λ)=0 for a non zero vector r∈ℕ n . We shall show that such vector fields can be transformed, by a local holomorphic diffeomorphism near 0∈ℂ n , into a preliminary normal forms while exhibiting invariant varieties, if an hypothesis of diophantine small divisors is made. Our results generalize, to any dimension, those done by H. Dulac in dimension 2.

Distribution of periodic points of polynomial diffeomorphisms of C2

This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of $\C^2$: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibrium measure $\mu$ of the set $K$ of points with bounded orbits. In [BLS] $\mu$ is also characterized dynamically as the unique measure of maximal entropy. Thus $\mu$ is also an equilibrium measure from the point of view of the thermodynamical formalism. In the present paper we give another dynamical interpretation of $\mu$ as the limit distribution of the periodic points of $f$.

Normal forms of invariant vector fields under a finite group action

Let G be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of (Cn, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.

Square Integrable Representations of Semisimple Lie Groups

Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component ${A_0}(D)$ of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle $E \to D$ admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle $E \to D$ must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of $E \to D$ by defining a map $\Phi :G \to {\text {GL}}(V)$ (V being the fiber of the vector bundle) which extends the classical “universal factor of automorphy” for the action of ${A_0}(D)$ on D. Then, we study the space H of all square integrable holomorphic sections of $E \to D$. The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over $G/Z$ (Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in ${L_2}(G/Z)$.

Square integrable representations of semisimple Lie groups

Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component A 0 ( D ) {A_0}(D) of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle E → D E \to D admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle E → D E \to D must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of E → D E \to D by defining a map Φ : G → GL ( V ) \Phi :G \to {\text {GL}}(V) (V being the fiber of the vector bundle) which extends the classical “universal factor of automorphy” for the action of A 0 ( D ) {A_0}(D) on D. Then, we study the space H of all square integrable holomorphic sections of E → D E \to D . The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over G / Z G/Z (Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in L 2 ( G / Z ) {L_2}(G/Z) .

Holomorphic Lefschetz fixed point formula

We are concerned with the problem of computing L((p). REMARK. Let G be a compact Lie group acting on X as a group of holomorphic diffeomorphisms and cp e G. The problem in this case has been solved by Atiyah and Singer, see [2]. Also in the case cp has isolated fixed points, the problem was solved in the nondegenerate case (see §2 for definition) by Atiyah and Bott in [1] and by Toledo and Tong ii>[6] and [7] in the degenerate case.