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.Comment: 48 page

Christopher, Colin

Rousseau, Christiane

English

arXiv

Colin Christopher

Christiane Rousseau

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 345 (2007) 695–698http://france.elsevier.com/direct/CRASS1/Dynamical SystemsThe moduli space of germs of generic families of analyticdiffeomorphisms unfolding a parabolic fixed point ✩Colin Christopher a, Christiane Rousseau ba School of Mathematics, University of Plymouth, Plymouth PL4 8AA, Devon, UKb DMS and CRM, Université de Montréal, Montréal H3C 3J7, CanadaReceived 27 June 2007; accepted after revision 13 October 2007Available online 26 November 2007Presented by Étienne GhysAbstractWe describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point ofcodimension 1. A complete modulus is given by unfolding the Écalle–Voronin modulus over a sector of opening greater than 2π inthe canonical parameter ε. In the region of overlap (Glutsyuk sector of parameter space) where the two fixed points are connectedby orbits, we identify the necessary compatibility between the two representatives of the modulus. The compatibility conditionimplies the existence of a normalization for which the modulus is 12 -summable in ε, non-summability occurring in the directionof real multipliers of the fixed points. We show that the compatibility condition together with the summability is sufficient forrealization of the modulus. To cite this article: C. Christopher, C. Rousseau, C. R. Acad. Sci. Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméL’espace des modules des germes de familles génériques de difféomorphismes analytiques déployant un point fixe para-bolique. On donne l’espace des modules des germes de familles génériques de difféomorphismes analytiques déployant un pointfixe parabolique de codimension 1. Un module complet est donné par le déploiement du module d’Écalle–Voronin sur un secteurd’ouverture plus grande que 2π du paramètre canonique. Dans le sous-secteur recouvert deux fois (sous-secteur Glutsyuk), là oùles deux points fixes sont connectés par des orbites, on identifie une condition de compatibilité nécessaire satisfaite par les deuxreprésentants du module. Cette condition implique l’existence d’une normalisation sous laquelle le module est 12 -sommable enε, la non-sommabilité se produisant dans la direction des multiplicateurs réels aux points fixes. On montre que la condition decompatibilité, jointe à cette propriété de sommabilité, est suffisante pour réaliser le module. Pour citer cet article : C. Christopher,C. Rousseau, C. R. Acad. Sci. Paris, Ser. I 345 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.✩ This work is supported by NSERC in Canada.E-mail addresses: C.Christopher@plymouth.ac.uk (C. Christopher), rousseac@dms.umontreal.ca (C. Rousseau).URL: http://www.dms.umontreal.ca/~rousseac (C. Rousseau).1631-073X/$ – see front matter © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2007.10.033696 C. Christopher, C. Rousseau / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 695–6981. Statement of the resultsWe consider germs of generic analytic 1-parameter families of diffeomorphisms of (C,0) unfolding a parabolicpoint. Such families can be ‘prepared’ so that the parameter becomes an analytic invariant [2]. A prepared family withcanonical (complex) parameter ε has the formfε(z) = z + (z2 − ε)(1 + b(ε) + c(ε)z + O(z2 − ε)), (1)so that 1/√ε = 1/ln(f ′ε(√ε )) − 1/ln(f ′ε(−√ε )). The formal invariant, a(ε), is the analytic function in ε given bya(ε) = 1/ln(f ′ε(√ε )) + 1/ln(f ′ε(−√ε )).In [2] (and [5]) it is shown that a modulus for such unfoldings can be obtained from the formal parameter, a(ε),and two analytic functions, Ψ 0ε̂and Ψ ∞ε̂, which describe the shift between two choices of Fatou coordinates on theirregion of overlap. The Fatou coordinates are chosen so thatlimImW→−∞Ψ0ε̂= id, limImW→+∞Ψ∞ε̂= T−2πia(ε), (2)where Tb denotes translation by b. Here, ε̂ represents the parameter ε when lifted to the universal cover of thepunctured disc. These functions are always assumed to satisfy T1 ◦ Ψ 0,∞ε̂ = Ψ 0,∞ε̂ ◦ T1. That is, they could also beconsidered as lifts of maps ψ0,∞ε̂between neighborhoods of the poles of two spheres. We demonstrate the followingtheorems, which identify the extra conditions on the Ψ 0,∞ε̂to form a moduli space for analytic unfoldings, and provetheir sufficiency:Theorem 1.1. Consider a prepared germ of the form (1). Then for δ ∈ (0,π) there exists ρ > 0 and a representativeof the modulus, (Ψ 0ε̂,Ψ ∞ε̂), defined for ε̂ ∈ Vδ,ρ = {ε̂; |ε̂| < ρ, arg ε̂ ∈ (−δ,2π + δ)} such that.(i) there exists Y0 > 0 such that Ψ 0ε̂ (resp. Ψ∞ε̂) is analytic on ImW < −Y0 (resp. ImW > Y0) for all ε̂ ∈ Vδ,ρ ;(ii) Ψ 0,∞ε̂are 12 -summable in ε (i.e. 1-summable in√ε̂) with direction of non-summability given by R+ (see [3] fork-summability);(iii) Ψ 0,∞ε̂give rise to the following compatibility condition.We cover the region of overlap arg(ε) ∈ (−δ, δ) (Glutsyuk sector) by the two subsectorsV = {ε̂;0 < |ε̂| < ρ, arg ε̂ ∈ (−δ,+δ)}, Ṽ = {ε̂;0 < |ε̂| < ρ, arg ε̂ ∈ (2π − δ,2π + δ)}, (3)on which we denote ε̂ by ε̄ and ε̃ respectively (and Ψ 0,∞ε by Ψ̃0,∞ε̃and Ψ 0,∞ε ). We also defineα0 = −2πi(1 − a(ε)√ε̂ )/2√ε̂, α∞ = −2πi(1 + a(ε)√ε̂ )/2√ε̂.On Ṽ (resp. V ), α0,∞ takes values α̃0,∞ (resp. ᾱ0,∞). Then there exist maps H 0,∞ε̄ and H̃ 0,∞ε̃ holomorphic in W andε̂ and commuting with T1 which satisfyon Ṽ{H̃ 0ε̃◦ Tα̃0 ◦ Ψ̃ 0ε̃ = Tα̃0 ◦ H̃ 0ε̃ ,H̃∞ε̃◦ Tα̃0 ◦ Ψ̃ ∞ε̃ = Tα̃∞ ◦ H̃∞ε̃ ,on V{ H 0ε̄ ◦ Ψ 0ε ◦ Tᾱ0 = Tᾱ0 ◦ H 0ε̄ ,H ∞̄ε ◦ Ψ ∞̄ε ◦ Tᾱ0 = Tᾱ∞ ◦ H ∞̄ε .(4)The compatibility condition isH̃∞ε̃ ◦(H̃ 0ε̃)−1 = T2πia(ε) ◦ H 0ε̄ ◦ ( H ∞̄ε )−1 ◦ TD(ε) (5)for some constant D(ε) = −2πia(ε) + O(exp(− A|√ε| )) with A > 0.Conversely, we have the following:Theorem 1.2. Consider a germ of a function a(ε) analytic in ε and a germ of a family (Ψ 0ε̂,Ψ ∞ε̂)ε̂∈Vδ,ρ for someδ ∈ (0,π) and ρ > 0, which satisfies (i), (ii) and (iii) of Theorem 1.1. Then there exists a germ of family of analyticdiffeomorphismsfε = z + (z2 − ε)(1 + O(ε) + O(z)) (6)C. Christopher, C. Rousseau / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 695–698 697whose modulus over a fixed neighborhood in (z, ε)-space is given by (Ψ 0ε̂,Ψ ∞ε̂)ε̂∈Vδ,ρ and a(ε).Remark 1. In the unfolded modulus, the map Ψ 0ε̂(resp. Ψ ∞ε̂) refers to the dynamics near −√ε̂ (resp. √ε̂). Whenε̂ makes a full turn, −√ε̂ and √ε̂ are exchanged. We therefore have two different ways of describing the dynamicsnear each singular point in the region of overlap. The functions H̃ 0,∞ε̃and H 0,∞ε̄ are lifts of normalizations of fε̂ atthe fixed points. Such maps must exist due to the hyperbolicity of the fixed points in the region of overlap [1]. Thecompatibility condition guarantees that H̃ 0,∞ε̃in Ṽ and H 0,∞ε̄ in V glue to the same dynamics. It is an interesting factthat though the functions H̃ 0,∞ε̃and H 0,∞ε̄ have no geometrical meaning at ε = 0, yet their limits can be calculatedand are well-behaved. This fact is of some importance in step (2) of the proof below.Remark 2. The correspondence established in Theorems 1.1 and 1.2 can be extended in a natural way to the casewhen fε depends on a number of auxiliary analytic parameters.2. The proofsThe proof of Theorems 1.1 and 1.2 is composed of four parts:(1) the construction of (Ψ 0,∞ε̂)ε̂∈Vδ,ρ : this is done in [2];(2) the derivation of the compatibility condition from which the 12 -summability of Ψ0,∞ε̂follows;(3) the ‘local realization’ over sectorial neighborhoods in ε̂ of small opening: this yields the realization by a familygε̂ ramified in ε with uniform limit g0 when ε̂ → 0 along any ray arg ε̂ = Const. The local realization does notuse (2);(4) the global realization: from gε̂ , using the compatibility condition we construct a uniform family fε over an abstract2-dimensional manifold. The 12 -summability of Ψ0,∞ε̂allows application of the Newlander–Nirenberg theorem toshow that this manifold is an open set of C2.In more detail:(2) To decide when two diffeomorphisms fε unfolding a parabolic point are conjugate, we embed them in the flow ofthe vector field vε = z2−ε1+a(ε)z ddz on adequate sectorial domains and measure the obstruction to a global embedding. Forthis it is easier to work in a coordinate W which is the time of the vector field vε . The four maps H̃0,∞ε̃and H 0,∞ε̄ canbe seen as the lifting of a change of coordinate embedding the map in a flow near the hyperbolic fixed points ±√ε. Inthe W -coordinate this flow is that of ∂∂W. The comparison of these embeddings is an invariant: this is exactly what isexpressed by the compatibility condition, modulo a scaling given by the constant D(ε). To derive the 12 -summabilityof Ψ 0ε̂we explicitly calculate the maps H̃ 0,∞ε̃and H 0,∞ε̄ in some region inside ImW > Y0. There exists A,C > 0 suchthat { | H 0ε̄ − id | < C exp(−A/|√ε̂|),|H̃ 0ε̃− id | < C exp(−A/|√ε̂|), and{ |( H ∞̄ε )−1 − Ψ ∞̄ε − 2πia| < C exp(−A/|√ε̂|),|H̃∞ε̃− Ψ̃ ∞ε̃− 2πia| < C exp(−A/|√ε̂|),so the compatibility condition yields the 12 -summability of Ψ∞ε̂by the theorem of Ramis–Sibuya [4]. A similar studyin some region inside ImW < −Y0 allows to prove the 12 -summability of Ψ 0ε̂ .(3) The local realization for ε̂ in a sector of small opening is first done for fixed ε̂ by gluing two sectors U±ε̂as inFig. 1 with maps Ξ0,∞ε̂constructed from Ψ 0,∞ε̂(Ξ0,∞ε̂is the conjugate of Ψ 0,∞ε̂under the change z 	→ W ). TheU±ε̂are images of modified strips in the W domain. This yields a complex manifold Mε̂ which we endow with adiffeomorphism given on each U±ε̂by the time-one map of the vector field vε . The theorem of Ahlfors–Bers allowsrecognition of the abstract manifold Mε̂ as a neighborhood of ±√ε in C and to fill the holes at the fixed points. Theconstruction can be made to depend analytically on ε̂ with a continuous fixed limit M0 as ε̂ → 0 along a ray. In thisway we realize the modulus in a ramified family gε̂ for ε̂ in some Vδ,ρ .698 C. Christopher, C. Rousseau / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 695–698Fig. 1. The sectors U±ε̂for ε̂ ∈ Vδ,ρ : U−ε̂ is light grey, U+ε̂ is middle grey and their intersection is dark grey.(4) The compatibility condition ensures that gε̄ and gε̃ are conjugate by some Jε̂ for arg ε̄ ∈ (−δ, δ) and ε̃ = e2πi ε̄.Moreover gε̂ is12 -summable in ε̂ with directions of non-summability given by arg ε̂ ∈ {0,2π}. An indirect con-sequence of this is that |Jε̂ | = O(exp(− A|√ε̂| )) for some positive A. We construct an abstract manifold by gluingB(0, r) × {arg ε̂ ∈ (−δ, δ)} with B(0, r) × {arg ε̂ ∈ (2π − δ,2π + δ)}, by means of (z, ε̄) 	→ (Jε̂(z), ε̃) and pasting inB(0, r) × {ε = 0} to fill the hole. This gives us a C∞ manifold M with an almost complex structure. We apply theNewlander–Nirenberg theorem to realize M as a neighborhood of the origin in C2 on which gε̂ induces a family ofanalytic diffeomorphisms fε realizing the modulus.AcknowledgementsWe are grateful to Reinhard Schäfke, Loïc Teyssier and the referee for helpful discussions or remarks.References[1] A.A. Glutsyuk, Confluence of singular points and nonlinear Stokes phenomenon, Trans. Moscow Math. Soc. 62 (2001) 49–95.[2] P. Mardešić, R. Roussarie, C. Rousseau, Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms, MoscowMath. J. 4 (2004) 455–502.[3] J.-P. Ramis, Les séries k-sommables et leurs applications, in: Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Proc.Internat. Colloq., Centre Phys., Les Houches, 1979, in: Lecture Notes in Phys., vol. 126, Springer, Berlin, New York, 1980, pp. 178–199 (inFrench).[4] J.-P. Ramis, Y. Sibuya, Hukuhara’s domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type,Asymptotic Anal. 2 (1989) 39–94.[5] X. Ribon, Modulus of analytic classification for unfoldings of resonant diffeomorphisms, Moscow Math. J., in press.

The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point

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The moduli space of germs of generic families of analytic
  diffeomorphisms unfolding a parabolic fixed point

The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point

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