Quasi-conformal deformations of nonlinearizable germs

0.1 MB·English

by  Kingshook Biswas

QUASI-CONFORMAL DEFORMATIONS OF NONLINEARIZABLE
GERMS
0
KINGSHOOKBISWAS
1
0
2 Abstract. Letf(z)=e2πiαz+O(z2),α∈Rbeagermofholomorphicdiffeo-
n morphisminC. Forαrationalandf ofinfiniteorder,thespaceofconformal
a conjugacyclassesofgermstopologicallyconjugatetof isparametrizedbythe
J Ecalle-Voronin invariants(and in particular is infinite-dimensional). When α
isirrationalandf isnonlinearizableitisnotknownwhetherf admitsquasi-
2
conformaldeformations. Weshowthatiff hasasequenceofrepellingperiodic

orbitsconvergingtothefixedpointthenf embedsintoaninfinite-dimensional
]
V familyofquasi-conformallyconjugategermsnotwoofwhichareconformally
conjugate.
C
AMS Subject Classification: 37F50
.
h
t
a
m
Contents
[

1 1. Introduction 1
v 2. Deformations. 2
0 2.1. Deforming repelling periodic orbits. 3
9
2.2. Deforming germs with small cycles. 4
2
References 4
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.
1
0
0
1. Introduction
1
:
v Let f(z) = e2πiαz+O(z2),α ∈ R/Z be a germ of holomorphic diffeomorphism
i
X fixing the origin in C. We consider the question of when f admits quasi-conformal
deformations, i.e. when do there exist germs g which are quasi-conformally but
r
a not conformally conjugate to f? If f is linearizable (i.e. analytically conjugate
to the rigid rotation R (z) = e2πiαz) then any germ topologically conjugate to f
α
is linearizable. In the nondegenerate parabolic case (i.e. α = p/q ∈ Q,fq 6= id),
the quasi-conformal conjugacy class of f contains an infinite dimensional family of
conformalconjugacyclassesparametrizedbytheEcalle-Voronininvariants([Eca75],
[Vor81]). In the irrationally indifferent nonlinearizable case (α irrational, f not
linearizable), it seems to be unknown whether quasi-conformal deformations are
possible. We show the following:
Theorem 1.1. Let f be an irrationally indifferent nonlinearizable germ with a
sequence of repelling periodic orbits accumulating the origin. Then there is a family
of quasi-conformal maps {h }Λ ∈ M parametrized by sequences M = {(λ ) :
Λ n n≥0
Research partly supported by Department of Science and Technology research project grant
DyNo. 100/IFD/8347/2008-2009.
1
2 KINGSHOOKBISWAS
λ ∈ C,|λ | > 1} such that all conjugates of g = h ◦f ◦h−1 are holomorphic,
n n Λ Λ Λ
and g ,g are conformally conjugate if and only if all but finitely many terms of
Λ1 Λ2
Λ and Λ agree. For fixed z, h (z) depends holomorphically on each λ .
1 2 Λ n
Theproofproceedsasfollows: givenΛ=(λ )thegermg isobtainedbyquasi-
n Λ
conformally deforming f near its periodic orbits. Each periodic orbit is attracting
for f−1, and it is possible to construct an f-invariant Beltrami differential on each
basin of attraction such that the quasi-conformal map h rectifying the Beltrami
Λ
differential conjugates f to a holomorphic germ g having multipliers λ at the
Λ n
periodic orbits. The multipliers at periodic orbits being invariant under conformal
conjugacies, the conclusion of the theorem follows.
Examplesofgermssatisfyingthehypothesisofthetheoremaregermsofrational
mapsofdegreedwithαsatisfyingtheCremerconditionofdegreed(seeforexample
Milnor [Mil99], Ch. 8)
loglogq
n+1
limsup >logd
q
n
(where(p /q )arethecontinued fraction convergents ofα)which ensuresthat the
n n
fixed point is accumulated by periodic orbits (only finitely many of which can be
repelling for a rational map).
Perez-Marcohasshown([PM97])foranynonlinearizablegermf theexistenceof
auniquemonotone one-parameter family (K ) offull, totally invariant continua
t t>0
called hedgehogs containing the fixed point. In [Bis09] it is proved that any confor-
mal mapping in a neighbourhood of a hedgehog K of a germ f mapping K to a
1
hedgehog of a germ f necessarily conjugates f to f . As a corollary of Theorem
2 1 2
1.1 we have
Corollary 1.2. There exists a holomorphic motion φ : D∗ ×Cˆ → Cˆ of Cˆ over
D∗ and a hedgehog K such that all the sets φ(t,K) are hedgehogs, all of which are
quasi-conformal images of K, but for s6=t, φ(s,K) cannot be conformally mapped
to φ(t,K) .
Acknowledgements. The author thanks Ricardo Perez-Marco for many helpful
discussions and comments. This article was written in part during a visit to the
Chennai Mathematical Institute. The author is very grateful to the organizers for
their warmth and hospitality.
2. Deformations.
We fix a germ f(z)=e2πiαz+O(z2),α∈R−Q and a neighbourhood U of the
origin such that f and f−1 are univalent on a neighbourhood of U. By a periodic
orbit or cycle of f of order q ≥ 1 we mean a finite set O = {z ,...,z } ⊂ U such
1 q
that f(z ) = z ,1 ≤ i ≤ q −1 and f(z ) = z . The multiplier at the periodic
i i+1 q 1
orbit is defined to be λ = f′(z )f′(z )...f′(z ) = (fq)′(z ). The periodic orbit is
1 2 q i
called attracting, indifferent and repelling according as |λ| < 1,|λ| = 1 and |λ| > 1
respectively. A periodic orbit for f of multiplier λ is a periodic orbit for f−1 of
multiplierλ−1. Thebasinofattractionofanattractingperiodiccycleisdefinedby
A(O,f):={z ∈U :fn(z)→O as n→+∞}.
QUASI-CONFORMAL DEFORMATIONS OF NONLINEARIZABLE GERMS 3
We observe that f can have only finitely many cycles of a given order q in U
(by the uniqueness principle). Since f is asymptotic to an irrational rotation, i.e.
f(z)/z →e2πiα asz →0, it follows that iff hassmall cyclesthentheordersof the
cycles must go to infinity.
2.1. Deforming repelling periodic orbits. Given a repelling periodic orbit O
of f with multiplier λ and |λ′|>1 we deform the multiplier of f quasi-conformally
from λ to λ′ by constructing a corresponding f-invariant Beltrami differential µ=
µ(O,λ,λ′) on the basin of attraction A(O,f−1) as follows:
By Ko¨enigs linearization theorem (see [Mil99], Ch. 6) for any repelling periodic
orbit O of f with multiplier λ there exists a unique holomorphic map φ defined on
a neighbourhood of O such that φ(z ) = 0,φ′(z ) = 1,i = 1,...,n and φ(fq(z)) =
i i
λφ (z). The conformal isomorphism L : C∗ → C/Z,w 7→ ξ = 1 logw conjugates
λ 2πi
thelinearmapw 7→λw onC∗ tothetranslationξ 7→ξ+τ onC/Zwhereτ =L(λ)
and Imτ <0.
Letτ′ =L(λ′)andletKbethereallinearmaponCdefinedbyK(1)=1,K(τ)=
τ′. ThenKcommuteswiththetranslationbyoneandhencegivesaquasi-conformal
orientation preserving (since Imτ,Imτ′ < 0) homeomorphism K˜ : C/Z → C/Z.
The Beltrami differential of K˜ is constant and invariant undertranslations of C/Z,
and K˜ conjugates the translation ξ 7→ξ+τ on C/Z to ξ 7→ξ+τ′.
We let µ be the Beltrami differential of K˜ ◦L◦φ restricted to a small neigh-
λ
bourhoodDofz ∈O. ThemapK˜◦φ conjugatesfq tothetranslationξ 7→ξ+τ′.
1 λ
So at points z,z′ =fq(z) in D, µ satisfies the invariance condition
µ(z′)(fq)′(z)=µ(z)(fq)′(z)
SoµextendedtotheneighbourhoodV =D∪f(D)∪···∪fq−1(D)ofO byputting
(fj)′(z)
µ(fj(z))=µ(z)
(fj)′(z)
forz ∈D,j =1,...,q−1isanf-invariantBeltramidifferential. Similarlytheabove
equationallowsustoextendµtoanf-invariantBeltramidifferentialµ(O,λ,λ′)on
A(O,f−1)=∪ fn(V).
n≥1
Lemma 2.1. Any quasi-conformal homeomorphism h with Beltrami coefficient
equal to µ = µ(O,λ,λ′) on a neighbourhood of O conjugates f to a map g =
h◦f◦h−1 with a periodic orbit h(O) of multiplier λ′. The dependence of µ(O,λ,λ′)
on λ′ is holomorphic.
Proof: Fori=1,...,q weletψ bethebranchofφ−1 sending0toz . Byconstruc-
i i
tion the map k defined by k =ψ ◦L−1◦K˜ ◦L◦φ(z) for z in a neighbourhood of
i
z has Beltrami coefficient equal to µ and conjugates f on a neighbourhood of O
i
to a holomorphic map f =k◦f ◦k−1 with periodic orbit k(O) and multiplier λ′.
1
Since h and k have the same Beltrami coefficient the map h◦k−1 is holomorphic,
and conjugates f to g, so the multipliers of f and g are equal. The Beltrami dif-
1 1
ferential of K˜ is constant equal to −log(λ′/λ) which depends holomorphically
2log|λ|+log(λ′/λ)
onλ′,sotheBeltramidifferentialµ(O,λ,λ′)ofK˜◦L◦φ dependsholomorphically
λ
on λ′. ⋄
4 KINGSHOOKBISWAS
2.2. Deforming germs with small cycles. We use the Beltrami differentials
µ(O,λ,λ′) to deform a germ with small cycles as follows:
Proof of Theorem 1.1: Let f be a germ with a sequence of repelling small cycles
of orders (q ) (which we may assume to be strictly increasing). The basins of
n
attraction A(O,f−1) of distinct repelling cycles of f are disjoint, so for any Λ =
(λ ) ∈ M we can define an f-invariant Beltrami differential µ on U by putting
n Λ
µ (z) = µ(O,λ,λ )(z) when z belongs to A(O,f−1) for a cycle O of order q
Λ n n
and multiplier λ, and µ (z) = 0 otherwise. Let h be the unique quasi-conformal
Λ Λ
homeomorphismwithBeltramicoefficientµ fixing0,1,∞givenbytheMeasurable
Λ
Riemann Mapping Theorem. Then g =h ◦f ◦h−1 is a holomorphic germ fixing
Λ Λ Λ
the origin with small cycles. By Naishul’s theorem [Nai83] the multiplier at an
indifferent fixed point is a topological conjugacy invariant so g′ (0) = e2πiα. By
Λ
Lemma2.1, themultiplierofg atallitsrepellingcyclesoforderq isequaltoλ .
Λ n n
If φ is a holomorphic germ conjugating two such germs g ,g , φ must take
Λ1 Λ2
all repelling cycles of g of order q (in its domain) to repelling cycles of g
Λ1 n Λ2
of order q and preserve multipliers, so the sequences of multipliers Λ ,Λ must
n 1 2
agree for all but finitely many terms. It follows from Lemma 2.1 that µ depends
Λ
holomorphically on each λ , hence by the Measurable Riemann Mapping Theorem
n
so does h (z) for fixed z. ⋄
Λ
ProofofCorollary1.2: Letf beasaboveandK ⊂U beahedgehogoff. Fort∈D∗
we let Λ be the constant sequence (λ =1/t). Then µ depends holomorphically
t n Λt
on t, and φ:(t,z)7→h (z) gives the required holomorphic motion. ⋄
Λt
References
[Bis09] K. Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. To
appear in DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2009.
[Eca75] J. Ecalle. Th´eorie it´erative: introduction `a la th´eorie des invariantes holomorphes. J.
Math pures et appliqu´es 54,pages183–258,1975.
[Mil99] J. Milnor. Dynamics in one complex variable. Friedr. Vieweg and Sohn, Braunschweig,
1999.
[Nai83] V.I.Naishul.Topological invariantsofanalyticand area-preservingmappingsandtheir
applicationstoanalyticdifferentialequationsinC2 andCP2.Trans. Moscow Math. Soc.
42,pages239–250,1983.
[PM97] R.Perez-Marco.Fixedpointsandcirclemaps.Acta Mathematica 179:2,pages243–294,
1997.
[Vor81] S. M. Voronin. Analytic classification of germs of conformal mappings (C,0) → (C,0)
withidentitylinearpart.Funktsional. Anal. i Prilozhen 15,pages1–17,1981.
Ramakrishna Mission Vivekananda University, Belur Math, WB-711202, India
email: [email protected]