Lecture Notes ni Mathematics Edited by .A Dold dna .B Eckmann 902 ydderF reitromuD oluaP .R Rodrigues treboR eirassuoR Germs of smsihpromoeffiD ni eht enalP galreV-regnirpS Berlin Heidelberg New York 1891 Authors Freddy Dumortier Limburgs Universitair Centrum, Universitaire Campus B-3610 Diepenbeek, Belgium Paulo .R Rodrigues Departemento de Geometria, Instituto de Matematica Universidade Federal Ftuminense 24000 Niteroi, Brazil Robert Roussarie Departement de Math6matique, Universite de Dijon - UER MIPC Laboratoire de Topologie ERA No.945 du CNRS, 21000 Dijon, France AMS Subject Classifications (1980): 34C25, 34 D10, 34 D30, 58 F10, 58F14, 58F22, 58F30 ISBN 3-540-11177-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11177-8 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. 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(cid:14)9 yb Springer-Verlag Berlin Heidelberg 1891 Printed ni Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210 TABLE FO STNETNOC Summary, some motivation and acknowledgments . . . . . . . . . . 1 - Chapter I : Introduction, definitions, formal, study and - statement of the results . . . . . . . . . . . . . 7 w I Introduction . . . . . . . . . . . . . . . . . . . . . . . 8 w 2 The blowing-up method . . . . . . . . . . . . . . . . . . 14 w 3 Statement of the fundamental theorem . . . . . . . . . . . 20 w 4 Decomposition in sectors for singularities ........ of vector fields in IR 2 and characteristic lines ..... 22 w 5 Statement of the results concerning characteristic lines and decomposition in sectors for certain germs of planar diffeomorphisms; reduction to the fundamental theorem . . . . . . . . . . . . . . . . . . . . 26 w 6 Statement of the principal results 34 w 7 Statement of the topological results . . . . . . . . . . . 39 w 8 Some applications and examples . . . . . . . . . . . . . . 43 - Chapter II : Stability of type I- and type II- singularities.. 48 w I Singularities of the "hyperbolic contraction"-type .... 48 w 2 Singularities of the "quasi-hyperbolic contraction"-type 15 w 3 Singularities which are quasi-hyperbolic contractions with respect to a degenerate Finsler-metric ....... 64 w q The "attracting corner"-singularities . . . . . . . . . . . 74 w 5 Attracting arcs . . . . . . . . . . . . . . . . . . . . . . 79 6 "Saddle-type"-corners . . . . . . . . . . . . . . . . . . . 80 IV Chapter III : Stability of type lll-singularities . . . . . . . 83 w I Simplified form of the "type IIl-singularities" . .... 84 2 -Existence of a C ~ center manifold . . . . . . . . . . . . 86 3 Reduction of the C ~ problem to a formal problem ..... 105 4 Reduction of the formal problem 30) to a difference equation . . . . . . . . . . . . . . . . . . . 108 5 Resolution of the difference equations 39 end {40) . . . . . . . . . . . . . . . . . . . . . . . . . 113 Chapter IV : Proof of the C ~ results . . . . . . . . . . . . . 126 - I On the unicity of germs of flat C ~ diffeomorphisms commuting with attracting diffeomorphisms of type I and II . . . . . . . . . . . . . . . . . . . . . . . . 127 2 Characterization of germs of flat C ~ diffeomorphisms commuting with saddle-type diffeomorphisms of type I and IIJ . . . . . . . . . . . . . . . . . . . . . . . . . 139 w 3 Construction of C ~ conjugacies in hyperbolic and parabolic sectors with only type I singularities .... 153 4 COnstruction of differentiable invariants in elliptic sectors . . . . . . . . . . . . . . . . . . . . . . . . . 160 5 Final touch to the proofs of theorem B and C ....... 170 Chapter V : Proof of the topological results . . . . . . . . . 173 - i Preliminary results . . . . . . . . . . . . . . . . . . . 173 w 2 Final touch to the proof of the theorems D and E ..... 190 3 Final touch to the proof of the theorems F and G ..... 192 References . . . . . . . . . . . . . . . . . . . . . . . . . . 193 - Subject index . . . . . . . . . . . . . . . . . . . . . . . . . 196 - ,yrammuS emos motivation dna stnemgdelwonkca The aim of this lecture note is to study germs of C diffeomorphisms in IR 2 from a topological and a C point of view (C means smooth or infinitely differentiable). Although our methods could also be used C r for a study we do not pay attention to this here. We especially emphasize the following problems : 1 (cid:12)9 When can such a germ or a power of it be C ~ or C embedded in the germ of a flow ? 2. When are such germs C determined by their ~ -jet ? 3. When are such germs C O determined by some finite jet ? We restrict our attention to the germs oocuring in generic n-parameter families of diffeomorphisms and having a characteristic line. The possibility of embedding a diffeomorphism in a flow in a C o or C ~ way {i.e. to show that the diffeomorphism is C ~ or C~-conjugated to the time I mapping of the flow of a vector field has at least a twofold advantage. Firstly the study of the diffeomorphism is reduced to the study of a vector field which in most cases reveals to be an easier tasK. Secondly up to a homeomorphism the orbits of the vector field are Kept invariant under the diffeomorphism, so that we find an invariant singular foliation {C ~ or C ~ restricting the topological complexity of the diffeo- morphism in essentially the same way as a first integral does. A perhaps mere important aspect can be seen in the study of periodic solutions for periodic time-dependent differential equations(cid:12)9 In this context we would like to refer to the Floquet-Liapunov theory for a linear periodic system of differential equations stating among other things that the system can be transformed into an autonomous linear system by means of a coordinate change given by a periodic matrix func- tion. Let us now take X to be a more general T-periodic system of differential equations on ~n which we want to study in the neighbourhood of some T-periodic solution y. As usual we associate to X an autonomous system of differential equations or vector field Y = X + ~tdefined on n+1 ~ n Because of the T-periodicity of X we can consider Y to be a vector field n in ~ x S where S = R/TZ. For simplicity in exposition let us suppose that y is the zero solution, i.e. Y = {O}x S. ~n We take f : x {0}~ to be the first return mapping Poincar6 mapping) associated to ,Y which in this case is x ~Tx{X,T where YX denotes the global solution of X. In analogy with the Floquet-Liapunov theory we can state that f cr-embeds C r ~ n in a flow if and only if there exists a diffeomorphism H : x S -~, ,x tJ ~ HtxJ,tJ with the property that this coordinate change H trans- forms the vector field Y into an expression Z + ~-~ with Z autonomous. In that way the study of the diffeomorphism or the study of a system of differential equations to which can be associated a diffeomorphism in casu the Poincar@ mapping) is then reduced to a further investigation of an autonomous vector field in a space of the same dimension. In this lecture note we deal with germs of diffeomorphisms f in the plane satisfying a so called ~ojasiewicz inequality, exhibiting a characteristic line and having a l-jet which can be expressed as R+N with N nilpotent and R p = I for some p E ~. For exact definitions we refer to the first chapter. Roughly spoken the first condition means that the diffeomorphism is not too degenerate, although the condition is rather weak since all germs of diffeomorphisms showing up in generic n-parameter-families of diffeomorphisms, for whatsoever ,n are of tojasiewicz-type. The second condition is one of good sense; as a matter of fact in the other case the orbits indefinitely spiral around the fixed point and the study of this phenomenum is already fairly complicated and not completely understood in the vector field case. The third condition means that we do not pay attention, except in the introductory remarks in chapter ,I to the already well Known diffeo- morphisms like the hyperbolic and partially hyperbolic ones, as well as to diffeomorphlsms whose associated R semi simple part of the 1-jetJ is an irrational rotation. In all the cases treated here we find for the diffeomorphism a same Kind of decomposition in parabolic, elliptic and hyperbolic sectors as for an R-equivariant vector field X. This X has the property that up to a C ~ change of coordinates the ~-jet of f is the same as the ~-jet of RoX I where X I is the time l-mapping of the flow X t of X. Moreover the union of the boundaries of these sectors is a C ~ image of the union of the boundaries for the X-decomposition. Let us remark that in case R = Identity these sectors for f are "invariant" sectors while for general R we have for each sector S P that f leaves "invariant"iU 0 fics) with fPs = "S Knowing that in the s type-case vector fields only have but one topological model of attracting, expanding, hyperbolic and o elliptic sector (up to C conjugacyJ we in this work prove the same for the diffeomorphism taKe the case R = Id), except for the hyper- bolic sector. We however show that in the interior of a hyperbolic sector orbits only stay a finite number of iterates. We use all this to prove that the diffeomorphism f (case R = Id) is weakly-C~ to the time 1- mapping X I. Such f as we deal with is hence weaKly-C o -embeddable in a flow and is up to weaK-C o -conjugacy determined by some finite jet. These results can be ameliorated if we do not allow certain partially hyperbolic singularities in a desingularisation of X obtained after successive blowing up. Then as a matter of fact we find that f is C -conjugated to X I on the union of parabolic and hyperbolic sectors. Hence under the just mentioned extra assumption which we only need inside the hyperbolic sectors) f is C o -conjugated to X 1 and is up to C o -conjugacy determined by some finite jet. The elliptic sectors give C problems, even under these extra conditions on the desingularisation. Under these extra conditions we are able to describe a complete infinite dimensional) C ~ modulus for flat C ~ con- jugacy conjugacy by means of C ~ diffeomorphisms which are infinitely near the identity). 5 The reason essentially is that a flat C ~ conjugacy between two elliptic sectors is uniquely determined in a conic neighbourhood of each of the two boundary lines. These uniquely defined diffeomorphisms do not need to match together in the middle of the sector and this obstruction can be fully described. At least for a large class of germs of diffeomorphisms in ~2 we so prove that the whole C = structure only depends on the =-jet. In other cases we get that this definitely is not the case, In many cases o we show the diffeomorphism to be C determined by some finite jet so that the investigation of the topological structure of the diffeomorphism becomes a problem concerning polynomial vector fields. In order to make the lecture note accessible for non-specialists we added an extensive introduction in chapter I. It contains besides the definition of most notions, a list of well Known facts related to our study and a description of the main technique, namely the blowing-up method. Moreover in chapter I we enumerate all our results in a rather self-con- tained way with a guide for travelling through the proofs; at the end we present some nice applications. The rest of this note is then completely devoted to the proof of the theorems. Some of the results in this paper have first been announced and proved in limited cases by Rodrigues~and Roussarie during a stay of the first at the university of Dijon. The method of proof has been adapted and completed by Dumortier and Roussarie during a sejourn of both authors at the "Institut des Hautes Etudes Scientifiques" in Bures-s-Yvette. The writing has essentially been finished while Dumortier remained at the university of Dijon. We want to thanK the mentioned institutions for their hospitality. 6ranted by the CNPq of Brazil