Topological classification of families of diffeomorphisms without small divisors

183 Pages·2010·1.516 MB·English

by  Javier Ribon

MEMOIRS
of the
American Mathematical Society
Number 975
Topological Classification
of Families of Diffeomorphisms
without Small Divisors
Javier Ribo´n
September 2010 • Volume 207 • Number 975 (end of volume) • ISSN 0065-9266
American Mathematical Society
Number 975
Topological Classification
of Families of Diffeomorphisms
without Small Divisors
Javier Ribo´n
September2010 • Volume207 • Number975(endofvolume) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Rib´on,Javier,1974-
Topologicalclassificationoffamiliesofdiffeomorphismswithoutsmalldivisors/JavierRib´on.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 975)
“September2010,Volume207,number975(endofvolume).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4748-0(alk. paper)
1.Topologicaldynamics. 2.Diffeomorphisms. 3.Differentiabledynamicalsystems. I.Title.
QA611.R53 2010
515(cid:2).39—dc22 2010022791
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Contents
Preface vii
Chapter 1. Outline of the Monograph 1
Chapter 2. Flower Type Vector Fields 11
2.1. Definition and basic properties 11
2.2. Families of vector fields without small divisors 23
Chapter 3. A Clockwork Orange 27
3.1. Exterior dynamics 27
3.2. The magnifying glass 43
Chapter 4. The T-sets 49
4.1. Unstable set and bi-tangent cords 49
4.2. Dynamical instability 57
4.3. Disassembling the graph 60
Chapter 5. The Long Limits 65
5.1. Setup and non-oscillation properties 66
5.2. Definition of the Long Limits 68
5.3. Structure of the Long Limits 70
5.4. Evolution of the Long Limits 73
Chapter 6. Topological Conjugation of (NSD) Vector Fields 79
6.1. Orientation 80
6.2. Comparing residues 81
6.3. Topological invariants 87
Chapter 7. Families of Diffeomorphisms without Small Divisors 97
7.1. Normal form and residues 98
7.2. Comparing a diffeomorphism and its normal form 100
7.3. Long orbits 107
Chapter 8. Topological Invariants of (NSD) Diffeomorphisms 111
8.1. Topological invariants 111
8.2. Theorem of topological conjugation 116
Chapter 9. Tangential Conjugations 123
9.1. Plan of the chapter 124
9.2. Preparation of ϕ and ϕ 124
1 2
9.3. Shaping the domains 126
9.4. Base transversals 131
iii
iv CONTENTS
9.5. The M-interpolation process 135
9.6. Regions and their limiting curves 139
9.7. Conjugating a diffeomorphism and its normal form 145
9.8. Comparing tg-conjugations 148
List of Notations 161
Bibliography 163
Index 165
Abstract
We give a complete topological classification for germs of one-parameter fam-
ilies of one-dimensional complex analytic diffeomorphisms without small divisors.
In the non-trivial cases the topological invariants are given by some functions at-
tached to the fixed points set plus the analytic class of the element of the family
corresponding to the special parameter. The proof is based on the structure of the
limits of orbits when we approach the special parameter.
ReceivedbytheeditorJune28,2005,and,inrevisedform,December29,2007.
ArticleelectronicallypublishedonApril30,2010;S0065-9266(10)00590-9.
2000 Mathematics Subject Classification. Primary37C15,37F45,37G10;Secondary37F75,
37G05.
Key words and phrases. Diffeomorphisms,topologicalclassification,bifurcationtheory,nor-
malform,structuralstability,tangenttotheidentitygermsofdiffeomorphism.
ThismonographistheresultofaworkwhichtookplaceinUniversit´edeRennesI,Univer-
sidaddeValladolid,UCLA,IMPAandUFF..
TheauthorwassupportedinpartbyMinisteriodeEducaci´onCulturayDeporte. Programa
de BecasPostdoctoralesen Espan˜ay en el extranjero. Conv:2002. Programa: EX. Fecha Resol.
18/04/02.
TheauthorwassupportedinpartbyMinisteriodeEducaci´onCulturayDeporte. Referencia:
HBE2004-0002. FechaResol. 23/11/04.
TheauthorwassupportedinpartbyCNPq. Referˆencia: PDJ151377/2005-7.
(cid:2)c2010 American Mathematical Society
v

Preface
In this paper we give a complete topological classification for germs of one-
parameter families of one-dimensional complex analytic diffeomorphisms without
small divisors. More precisely, we study germs of diffeomorphism in (C2,0) of the
form
ϕ(x,y)=(h(x,y),y)
The curve Fix(ϕ) ⊂ C2 of fixed points of ϕ is given by h(x,y)−x = 0. We asso-
ciate ϕ ∈ Diff(C,0) to every point (x ,y ) ∈ Fix(ϕ); it is the germ defined
(x0,y0) 0 0
by ϕ|y=y0 in a neighborhood of x = x0. The diffeomorphism ϕ is the unfolding of
ϕ . Thereare twokindofphenomena whichcanproduceacomplicateddynam-
(0,0)
ical behavior for a diffeomorphism ϕ.
Presence of small divisors. Wesaythatϕhassmalldivisorsifthereexistj ∈Z
and P ∈ Fix(ϕj) such that (∂ϕj /∂x)(P) ∈ S1 and (∂ϕj /∂x)(P) is not a Bruno
P P
number [Brj71]. Then the dynamics of ϕj is very chaotic if ϕj is not linearizable
P P
[Yoc95], [PM97].
Evolution of the dynamics. In absence of small divisors the dynamics of ϕ|y=s
admits a simple description. In some sense it depends continuously on s for s(cid:4)=0,
but it can change dramatically for different values of the parameter s.
There are some works identifying regular zones in the parameter space, i.e.
zones where the dynamics of ϕ|y=s converges regularly to the dynamics of ϕ|y=0
when s → 0 (see [Ris99] for the case where j1ϕ is an irrational rotation or
(0,0)
[DES] for the case j1ϕ ≡Id). But so far there was no description of the zones
(0,0)
in the parameter space where the dynamical behavior does not commute with the
limit. There was also no information about the dependence of the dynamics of
ϕ|y=s with respect to s (s (cid:4)= 0) except in the topologically trivial case. Here we
provide a description of these phenomena in the absence of small divisors.
A diffeomorphism ϕ without small divisors will be called (NSD) diffeomor-
phism. The (NSD) character implies that we are in one of the following cases:
• ϕ is analytically conjugated to (λ(y)x,y) for some λ∈C{y}.
• j1ϕ=(λx,y) for a root λ∈S1 of the unit.
• j1ϕ=(x+μy,y) for some μ∈C.
Thefirstsituationistrivial. Forj1ϕ=(λx,y)andλp =1wecanrelatethedynam-
ics of ϕ with the dynamics of ϕp. Thus, we can suppose j1ϕ=(x+μy,y) for some
μ ∈ C up to replace ϕ with an iterate. Naturally, from now on we assume that
(NSD) means (NSD)+(j1ϕ = (x +μy,y)) or in other words (NSD)+unipotent.
In the one-variable case the topological [Lea97], [Cam78], [Shc82] formal and
vii
viii PREFACE
analytical classifications [E´ca78], [Vor81], [MR83], [Mal82] of unipotent diffeo-
morphisms are well-known (see [Lor99] for an excellent survey on these topics).
We are interested on giving a complete characterization of whether or not two
(NSD)diffeomorphismshavethesamedynamicalbehavior, orinotherwordswhen
they are conjugated by a homeomorphism defined in a neighborhood of 0 in C2.
Suchaconjugating homeomorphism canbe wild; for instance ingeneral it is notof
the form (σ (x,y),σ (y)). Since we want to describe the evolution of the dynamics
1 2
of ϕ|y=s we impose two natural conditions. Let ϕ1, ϕ2 be (NSD) diffeomorphisms
conjugated by a germ of homeomorphism σ; we say that σ is normalized if
• y◦σ ≡y.
• σ|Fix(ϕ1)\(y=0) ≡Id.
Ifsuchaconjugationexistswedenoteϕ ∼ ϕ ,wesaythatσisastrongtopological
1 st 2
conjugation. We denote the topological and the analytic conjugations by ∼ and
top
∼ respectively.
ana
Ifwehaveϕ ∼ ϕ for(NSD)diffeomorphismsϕ ,ϕ thenFix(ϕ )=Fix(ϕ ).
1 st 2 1 2 1 2
This equation has to be understood as a relation between analytic sets with not
necessarilyreducedstructure; forinstancewehaveFix(x+x2,y)(cid:4)=Fix(x+x3,y).
Letϕbea(NSD)diffeomorphism. Wedenotebym(ϕ)theuniquenon-negative
numbersuchthatym dividesx◦ϕ−xbutym+1 doesnotdividex◦ϕ−x. Consider
the decom(cid:2)position x ◦ ϕ − x = ymf1n1...fpnp in irreducible factors. We define
N(ϕ) = p ν(f (x,0)) where ν(a(x)) stands for the order of a(x) ∈ C[[x]] at
j=1 j
x = 0. Then for every sufficiently small neighborhood U of (0,0) and y (cid:4)= 0 in
0
a neighborhood of 0 we obtain N(ϕ) = (cid:7)(Fix(ϕ)∩U ∩{y = y }). The couple
0
(N(ϕ),m(ϕ)) is a topological invariant.
Let ϕ be a (NSD) diffeomorphism. Consider γ (cid:4)= {y = 0} an irreducible
componentofFix(ϕ). WedefineResγ :γ\{(0,0)}→Casthefunctionassociating
ϕ
to P the residue of the diffeomorphism ϕ (see def. 7.3). The value Resγ(P) is a
P ϕ
formal invariant of ϕ . The function Resγ is holomorphic. We prove:
P ϕ
Main Theorem. Let ϕ , ϕ be two (NSD) diffeomorphisms with same invari-
1 2
ant (N,m). We have
• If N =0 or (N,m)=(1,0) then ϕ ∼ ϕ ⇔Fix(ϕ )=Fix(ϕ ).
1 st 2 1 2
• For the remaining cases ϕ ∼ ϕ if and only if
1 st 2
– Fix(ϕ )=Fix(ϕ ).
1 2
– ym(Resγ −Resγ ) extends continuously by 0 to (0,0) for all irre-
ducible cϕo1mponenϕt2γ (cid:4)={y =0} of Fix(ϕ ).
1
– ϕ ∼ ϕ .
1,(0,0) ana 2,(0,0)
Moreover if (N,m) (cid:4)= (1,0) then σ|y=0 is complex analytic for every normalized
germ of homeomorphism σ conjugating ϕ and ϕ .
1 2
Suppose m = 0 throughout this paragraph. The condition ϕ ∼ ϕ
1,(0,0) ana 2,(0,0)
is much stronger than ϕ ∼ ϕ for N >1 since the analytic classes con-
1,(0,0) top 2,(0,0)
tained in a topological class are parameterized by a functional invariant. Suppose
ϕ ∼ ϕ ; we have
1 st 2
(N,m) situation in y =0 existence of irregular zones
N =1, m=0 ϕ ∼ ϕ NO
1,(0,0) top 2,(0,0)
N >1, m=0 ϕ ∼ ϕ YES
1,(0,0) ana 2,(0,0)
PREFACE ix
Roughly speaking, an irregular zone Z attached to a (NSD) diffeomorphism ϕ is a
subset of the parameter space such that the limit of the dynamics of ϕ|y=s when
s→0ands∈Z isricherthanthedynamicsofϕ|y=0. Therigidityprovidedbythe
maintheoremisattachedtotheexistenceofirregularzonesintheparameterspace.
Our work unveils a new phenomenon whose existence is based on the structure of
the limits of orbits in the irregular zones.
Let us say a word about the proof of the main theorem. We study at first
the real flow of a vector field X = f∂/∂x such that exp(X) is a normal form of a
(NSD)diffeomorphism ϕ. Weuse techniquesanalogous tothose in[DES] tostudy
the real flow (cid:11)(X) of the vector field X. In fact we classify topologically all the
vector fields (cid:11)(X) where X ∈ H(C2,0) and exp(X) is a (NSD) diffeomorphism.
The same techniques can be used to classify the real flows of all the vector fields of
the form X = f∂/∂x for any f ∈ C{x,y}. Anyway, we do not do it for simplicity
and because it is of no utility to study the (NSD) diffeomorphisms.