In this paper we describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1.
In [MRR] (and also [R]), it was shown that the Ecalle-Voronin modulus can be unfolded to give a complete modulus for such germs. The modulus is defined on a ramified sector in the canonical perturbation parameter $\eps$. As in the case of the Ecalle-Voronin modulus, the modulus is defined up to a linear scaling depending only on $\eps$.
Here, we characterize the moduli space for such unfoldings by finding the compatibility conditions on the modulus which are necessary and sufficient for realization as the modulus of an unfolding.
The compatibility condition is obtained by considering the region of sectorial overlap in $\eps$-space. This lies in the Glutsyuk sector where the two fixed points are hyperbolic and connected by the orbits of the diffeomorphism. In this region we have two representatives of the modulus which describe the same dynamics. We identify the necessary compatibility condition between these two representatives by comparing them both with their common Glutsyuk modulus.
The compatibility condition implies the existence of a linear scaling for which the modulus is 1/2-summable in $\eps$, whose direction of non-summability coincides with the direction of real multipliers at the fixed points. Conversely, we show that the compatibility condition (which implies the summability property) is sufficient to realize the modulus as coming from an analytic unfolding, thus giving a complete description of the space of moduli.
We prove the multisummability of the infinitesimal generator of unfoldings of finite codimension tangent to the identity 1-dimensional local complex analytic diffeomorphisms. We also prove the multisummability of Fatou coordinates and extensions of the Ecalle-Voronin invariants associated to these unfoldings. The quasi-analytic nature is related to the parameter variable. As an application we prove an isolated zeros theorem for the analytic conjugacy problem. The proof is based on good asymptotics of Fatou coordinates and the introduction of a new auxiliary tool, the so called multi-transversa…
We prove that a topological homeomorphism conjugating two generic 1-parameter unfoldings of 1-variable complex analytic resonant diffeomorphisms is holomorphic or anti-holomorphic by restriction to the unperturbed parameter. We provide examples that show that the genericity hypothesis is necessary. Moreover we characterize the possible behavior of conjugacies for the unperturbed parameter in the general case. In particular they are always real analytic outside of the origin. We describe the structure of the limits of orbits when we approach the unperturbed parameter. The proof of the rigidity …
We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension 1 (i.e. a double fixed point) under conjugacy. These generic unfolding depend on one real parameter. The classification is done by assigning to each such germ a weak and a strong modulus, which are unfoldings of the modulus assigned to the antiholomorphic parabolic point. The weak and the strong moduli are unfoldings of the Écalle-Voronin modulus of the second iterate of the germ which is a real unfolding of a holomorphic parabolic point. A preparation of the unfolding allows to ide…
Let G a group of germs of analytic diffeomorphisms in (C^2,0). We find some remarkable properties supposing that G is finite, linearizable, abelian nilpotent and solvable. In particular, if the group is abelian and has a generic dicritic diffeomorphisms, then the group is a subgroup of a 1-parametric group. In addition, we study the topological behaviour of the orbits of a dicritic diffeomorphisms. Last, we find some invariants in order to know when two diffeomorphisms are formally conjugates.
We investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curve, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, commuting holomorphic and antiholomorphic parabolic germs.
We provide a complete system of analytic invariants for unfoldings of non-linearizable resonant complex analytic diffeomorphisms as well as its geometrical interpretation. In order to fulfill this goal we develop an extension of the Fatou coordinates with controlled asymptotic behavior in the neighborhood of the fixed points. The classical constructions are based on finding regions where the dynamics of the unfolding is topologically stable. We introduce a concept of infinitesimal stability leading to Fatou coordinates reflecting more faithfully the analytic nature of the unfolding. These impr…
We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension~$k$ (i.e.~a fixed point of multiplicity $k+1$) under conjugacy. Such generic unfoldings depend real analytically on $k$ real parameters. A preparation of the unfolding allows to identify real analytic \emph{canonical parameters}, which are preserved by any conjugacy between two prepared generic unfoldings. A modulus of analytic classification is defined, which is an unfolding of the modulus assigned to the antiholomorphic parabolic point. Since the second iterate of such a germ is a…
Let f_1,...,f_N be commuting germs of holomorphic diffeomorphisms in C fixing the origin with irrational rationally independent rotation numbers alpha_1,...,alpha_N. We adapt Yoccoz' renormalization of germs to this setting to show that a Brjuno-type condition on simultaneous Diophantine approximability of the rotation numbers is sufficient for simultaneous linearizability of f_1,...,f_N. This generalizes a result of Moser's. In the absence of periodic orbits we show that a weaker arithmetic condition analogous to that of Perez-Marco's for the case of a single germ is sufficent for linearizabi…
Generic three-parameter unfolding of a completely degenerate germ of a vector field on $\mathbb{R}^3$
University of Kinshasa（金沙萨大学）；Center for Research on Mathematics Education in the Democratic Republic of the Congo（刚果民主共和国数学教育研究中心）；University Institute of Technology, University of Ngaoundere（恩冈代雷大学理工学院）；Institut des Hautes Études Scientifiques, Université Paris-Saclay（法国高等科学研究所，巴黎萨克雷大学）
We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group can be linearized holomorphically near the fixed point. It rests on a "small divisors condition" of the family of linear parts. The second part of this article is devoted to the study families of totally real intersecting n-submanifolds of (C n , 0). We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it…
We deal with the problem of the validity of Livsic's theorem for cocycles of diffeomorphisms satisfying the orbit periodic obstruction over an hyperbolic dynamics. We give a result in the positive direction for cocycles of germs of analytic diffeomorphisms at the origin.
The formal class of a germ of diffeomorphism $ϕ$ is embeddable in a flow if $ϕ$ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at $({\mathbb C}^{n},0)$ ($n>1$) whose formal class is non-embeddable. The examples are inside a family in which the non-embeddability is of geometrical type. The proof relies on the properties of some linear functional operators that we obtain through the study of polynomial families of diffeomorphisms via potential theory.
We prove the existence of hedgehogs for germs of complex analytic diffeomorphisms of $(\mathbb{C}^{2},0)$ with a semi-neutral fixed point at the origin, using topological techniques. This approach also provides an alternative proof of a theorem of Pérez-Marco on the existence of hedgehogs for germs of univalent holomorphic maps of $(\mathbb{C},0)$ with a neutral fixed point.
The aim of this article is twofold: First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants. Their restriction to an irreductible component of the zero locus of the first integral reduces to the Birkhoff--Écalle--Voronin modulus of the 1-dimensional restricted parabolic germ.…
In this paper we give a geometric description of the foliation of a generic real analytic family unfolding a real analytic vector field with a weak focus at the origin, and show that two such families are orbitally analytically equivalent if and only if the families of diffeomorphisms unfolding the complexified Poincaré map of the singularities are conjugate. Moreover, by shifting the leaves of the formal normal form in the blow-up (quasiconformal surgery) by means of a fibered transformation along a convenient complex cross-section, one constructs an abstract manifold of complex dimension 2 e…
We study Siegel's center problem on the linearization of germs of diffeomorphisms in one variable. In addition of the classical problems of formal and analytic linearization, we give sufficient conditions for the linearization to belong to some algebras of ultradifferentiable germs closed under composition and derivation, including Gevrey classes. In the analytic case we give a positive answer to a question of J.-C. Yoccoz on the optimality of the estimates obtained by the classical majorant series method. In the ultradifferentiable case we prove that the Brjuno condition is sufficient for the…
In this paper we give complete analytic invariants for germs of holomorphic foliations in $(\mathbb{C}^2,0)$ that become regular after a single blow-up. Some of them describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in finite dimensional complex vector space. Such singularities admit separatrices tangent to any direction at the origin. When enough separatrices coincide with their tangent directions (a condition that can always be attained if the mutiplicity of the germ at the origin is at most four) we are able to describe and realize …
We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ${\mathbb C}^{n}$. More precisely, we are interested on the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms $ϕ, η$ such that every formal conjugation $\hatσ$ (i.e. $η\circ \hatσ = \hatσ \circ ϕ$) does not extend to the fixed points set $Fix (ϕ)$ of $ϕ$, meaning that it is not transversally formal (or semi-convergent) along $Fix (ϕ)$. We focus on unfoldings of 1-dimensional tangent to the id…
We study the Siegel--Schröder center problem on the linearization of analytic germs of diffeomorphisms in several complex variables, in the Gevrey--$s$, $s>0$ category. We introduce a new arithmetical condition of Bruno type on the linear part of the given germ, which ensures the existence of a Gevrey--$s$ formal linearization. We use this fact to prove the effective stability, i.e. stability for finite but long time, of neighborhoods of the origin for the analytic germ.
Let $f$ be a germ of holomorphic diffeomorphism of $\C^n$ fixing the origin $O$, with $df_O$ diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of $df_O$ and some restrictions on the resonances, $f$ is locally holomorphically linearizable if and only if there exists a particular $f$-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.
We show that at the vicinity of a generic dissipative homoclinic unfolding of a surface diffeomorphism, the Hausdorff dimension of the set of parameters for which the diffeomorphism admits infinitely many periodic sinks is at least 1/2.
In this paper we consider germs of k-parameter generic families of analytic 2-dimensional vector fields unfolding a saddle-node of codimension k and we give a complete modulus of analytic classification under orbital equivalence and a complete modulus of analytic classification under conjugacy. The modulus is an unfolding of the corresponding modulus for the vector field with the saddle-node. The point of view is to compare the family with a "model family" via an equivalence (conjugacy) over canonical sectors. This is done by studying the asymptotic homology of the leaves and its consequences …
We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\mathbb{C}P(2)$. Let $\{\mathcal{F}_t\}_{t \in \mathbb{D}_ε}$ be topological trivial (in $\mathbb{C}^2$) analytic deformation of a foliation $\mathcal{F}_0$ on $\mathbb{C}^2$. We show that under some dynamical restriction on $\mathcal{F}_0$, we have two possibilities: $\mathcal{F}_0$ is a Darboux (logarithmic) foliation, or $\{\mathcal{F}_t\}_{t \in \mathbb{D}_ε}$ is an unfolding. We obtain in this way a link between the analytical classification of the unfolding and the one of its germs at the…
In this paper we give moduli of analytic classification for parabolic Dulac i.e. almost regular germs. Dulac germs appear as first return maps of hyperbolic polycycles. Their moduli are given by a sequence of Ecalle-Voronin-like germs of analytic diffeomorphisms. We state the result in a bigger class of parabolic generalized Dulac germs having power-logarithmic asymptotic expansions.
In this paper we describe the bifurcation diagram of the$2$-parameter family of vector fields $\dot z = z(z^k+ε_1z+ε_0)$ over $\mathbb C\mathbb P^1$ for $(ε_1,ε_0)\in \mathbb C^2$. There are two kinds of bifurcations: bifurcations of parabolic points and bifurcations of homoclinic loops through infinity. The latter are studied using the tool of the periodgon introduced in a particular case in \cite{CR}, and then generalized in \cite{KR}. We apply the results to the bifurcation diagram of a generic germ of 2-parameter analytic unfolding preserving the origin of the vector field $\dot z = z^{k+1…
In this paper, we consider the normal form problem of a commutative family of germs of diffeomorphisms at a fixed point, say the origin, of $\mathbb{K}^n$ ($\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$). We define a notion of integrability of such a family. We give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic (resp. a smooth) transformation if the initial diffeomorphisms are analytic (resp. smooth).
We give normal forms for generic k-dimensional parametric families $(Z_\varepsilon)_\varepsilon$ of germs of holomorphic vector fields near $0\in\mathbb{C}^2$ unfolding a saddle-node singularity $Z_0$, under the condition that there exists a family of invariant analytic curves unfolding the weak separatrix of $Z_0$. These normal forms provide a moduli space for these parametric families. In our former 2008 paper, a modulus of a family was given as the unfolding of the Martinet-Ramis modulus, but the realization part was missing. We solve the realization problem in that partial case and show th…
Perez-Marco proved the existence of non-trivial totally invariant connected compacts called hedgehogs near the fixed point of a nonlinearizable germ of holomorphic diffeomorphism. We show that if two nonlinearisable holomorphic germs with a common indifferent fixed point have a common hedgehog then they must commute. This allows us to establish a correspondence between hedgehogs and nonlinearizable maximal abelian subgroups of Diff$(\bf C,0)$. We also show that two nonlinearizable germs are conjugate if and only if their rotation numbers are equal and a hedgehog of one can be mapped conformall…
We study germs of singular holomorphic vector fields at the origin of $\Bbb C^n$ of which the linear part is 1-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some "sectorial domains" which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey ordrer of the latter is linked to the diophantine type of th…
We give a complete classification of analytic equivalence of germs of parametric families of systems of complex linear differential equations unfolding a generic resonant singularity of Poincare rank 1 in dimension $n = 2$ whose leading matrix is a Jordan bloc. The moduli space of analytic equivalence classes is described in terms of a tuple of formal invariants and a single analytic invariant obtained from the trace of monodromy, and analytic normal forms are given. We also explain the underlying phenomena of confluence of two simple singularities and of a turning point, the associated Stokes…
In this article we study the dynamics generated by germs of parabolic diffeomorphisms f : (C; 0)->(C; 0) tangent to the identity. We show how formal classification of a given parabolic diffeomorphism can be deduced from the asymptotic development of what we call directed area of the epsilon-neighborhood of any orbit near the origin. Relevant coefficients and constants in the development have a geometric meaning. They present fractal properties of the orbit, namely its box dimension, Minkowski content and what we call residual content.
We give geometric and algorithmic criterions in order to have of a proper Galois closure for a codimension one germ of quasi-homogeneous foliation. We recall this notion recently introduced by B. Malgrange, and describe the Galois envelope of a group of germs of analytic diffeomorphisms. The geometric criterions are obtained from transverse analytic invariants, whereas the algorithmic ones make use of formal normal forms.
Gambaudo and Pécou introduced the ``linking property'' to study the dynamics of germs of planar homeomorphims and provide a new proof of Naishul theorem in their paper "A topological invariant for volume preserving diffeomorphisms" (Ergodic Theory Dynam. Systems 15 (1995), no. 3, 535--541). In this paper we prove that the negation of Gambaudo-Pécou property characterises the topological dynamics of holomorphic parabolic germs. As a consequence, a rotation set for germs of surface homeomorphisms around a fixed point can be defined, and it will turn out to be non trivial except for countably man…
In [{\it American J. Mathematics}, 124(2002), 107--127] we proved that for a germ of $C^\infty$ hyperbolic diffeomorphisms $F(x)=Ax+f(x)$ in $(\mathbb R^n,0)$, if $A$ has a real logarithm with its eigenvalues weakly nonresonant, then $F(x)$ can be embedded in a $C^\infty$ autonomous differential system. Its proof was very complicated, which involved the existence of embedding periodic vector field of $F(x)$ and the extension of the Floquet's theory to nonlinear $C^\infty$ periodic differential systems. In this paper we shall provide a simple and direct proof to this last result. Next we shall …
We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf of a foliation. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, ...). Such system of equivalences also characterizes the existence of suitable {\it integrating factors}, i.e., invariant vector fields and one-forms associated…
In this article, we study analyticity properties of (directed) areas of epsilon-neighborhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using epsilon-neighborhoods of orbits in the simplest formal class. We show that the coefficient in front of epsilon^2 term in the asymptotic expansion in epsilon, which we call the principal part of the area, is a sectorially analytic function of initial point of the orbit. It satisfies a cohomological equation similar to the standard trivialization equation for parabolic diffeomorphisms. We give necessa…
We study the dynamics of generic unfoldings of saddle-node circle local diffeomorphisms from the measure theoretical point of view, obtaining statistical stability results for deterministic and random perturbations in these kind of one-parameter families.
In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.
We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, whe…
When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in the neighborhood of a singular point? A way to answer is to use normal forms. But there are large classes of dynamical systems for which the change of coordinates to a normal form diverges. In this paper we discuss the case of singularities for which the normalizing transformation is $k$-summable, thus allowing to provide moduli spaces. We explain the common geometric features of these singularities, and show that the study of their unfoldings allows understanding both the singul…
In this paper, we identify the moduli space for germs of generic unfoldings of nonresonant linear differential systems with an irregular singularity of Poincaré rank $k$ at the origin, under analytic equivalence. The modulus of a given family was determined in \cite{HLR}: it comprises a formal part depending analytically on the parameters, and an analytic part given by unfoldings of the Stokes matrices. These unfoldings are given on "Douady-Sentenac" (DS) domains in the parameter space covering the generic values of the parameters corresponding to Fuchsian singular points. Here we identify exa…
We describe the equivalence classes of germs of generic $2$-parameter families of complex vector fields $\dot z = ω_ε(z)$ on $\mathbb{C}$ unfolding a singular parabolic point of multiplicity $k+1$: $ω_0= z^{k+1} +o(z^{k+1})$. The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fields $\dot z = z^{k+1} + ε_1 z + ε_0$ over $\mathbb{CP}^1$. This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) uniq…
Let f be a germ of holomorphic diffeomorphism with an irra- tionally indifferent fixed point at the origin in C (i.e. f(0) = 0, f'(0) = e 2pi i alpha, alpha in R - Q). Perez-Marco showed the existence of a unique family of nontrivial invariant full continua containing the fixed point called Siegel compacta. When f is non-linearizable (i.e. not holomorphically conjugate to the rigid rotation R_{alpha}(z) = e 2pi i z) the invariant compacts obtained are called hedgehogs. Perez-Marco developed techniques for the construction of examples of non-linearizable germs; these were used by the author to …
Considering random noise in finite dimensional parameterized families of diffeomorphisms of a compact finite dimensional boundaryless manifold M, we show the existence of time averages for almost every orbit of each point of M, imposing mild conditions on the families. Moreover these averages are given by a finite number of physical absolutely continuous stationary probability measures. We use this result to deduce that situations with infinitely many sinks and Henon-like attractors are not stable under random perturbations, e.g., Newhouse's and Colli's phenomena in the generic unfolding of a …
We give essentially unique ``normal forms'' for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity which is a p:q resonant-saddle. Hence each vector field of that type is conjugate, by a germ of a biholomorphic map at the singularity, to a preferred element of an explicit family of vector fields. These model vector fields are polynomial in the resonant monomial.Abstract. This work is a followup of a similar result obtained for parabolic diffeomorphisms which are tangent to the identity, and solves the long standing problem of finding explici…
The first part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $(\Bbb C^n,0)$. We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which interesection with the family can be holomorphically staightened. The second part is devoted to the study real analytic $(n+r)$-submanifolds of $(\Bbb C^n,0)$ having a CR-singularity at the origin ($r$ is a nonnegative integer). We consider deformations of quadrics and we define genera…
The classical Lagrange inversion formula is extended to analytic and non--analytic inversion problems on non--Archimedean fields. We give some applications to the field of formal Laurent series in $n$ variables, where the non--analytic inversion formula gives explicit formal solutions of general semilinear differential and $q$--difference equations. We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferenti…
We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$, $(F-id)$ is a $κ$-smoothing map ($κ\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps $F_j^\prime$ such that $F_j-F_j^\prime=O(|u|^2)$ and each $\…
Dynamics of generic 1-parameter perturbations of a vector field with a singular point of codimension k
Institut de Mathématiques de Toulouse, Université Paul Sabatier（图卢兹数学研究所，图卢兹第三大学）；Département de mathématiques et de statistique Université de Montréal（蒙特利尔大学数学与统计系）
本文通过构造几何模型（星形域）和分类通用族，详细描述了复向量场族在奇点展开时的局部分岔图，并与简单模型$\dot z = z^{k+1}-ε$进行比较。
