Rigidity in higher rank Abelian group actions. Vol.1, Introduction and cocycle problem

321 Pages·2011·1.25 MB·English

by  Katok A.

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185RigidityinHigherRankAbelianGroupActionsI
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Rigidity in Higher Rank
Abelian Group Actions
Volume I.
Introduction and Cocycle Problem
ANATOLEKATOK
PennsylvaniaStateUniversity
VIORELNIT¸ICA˘
WestChesterUniversity,Pennsylvania
CAMBRIDGE UNIVERSITY PRESS
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Firstpublished2011
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RigidityinhigherrankAbeliangroupactions/AnatoleKatok,ViorelNitica.
v. cm.–(Cambridgetractsinmathematics;185–)
Contents:v.1.Introductionandcocycleproblem
ISBN978-0-521-87909-5(hardback)
1. Rigidity(Geometry) 2. Abeliangroups. I. Nitica,Viorel. II. Title.
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Contents
Introduction:anoverview page1
PartI:Preliminariesfromdynamicsandanalysis 9
1 Definitionsandgeneralpropertiesofabeliangroupactions 11
1.1 Groupactions,conjugacy,andrelatednotions 11
1.2 Functorialconstructions 12
1.3 Principalbundles 13
1.4 Cocycles 16
1.5 RootsandWeylchambersforlinearactions 19
1.6 Algebraicactions 20
1.7 Measurableandnon-uniformdifferentiablesetting 25
1.8 Uniformdifferentiablesetting 30
2 Principalclassesofalgebraicactions 39
2.1 Automorphismsoftoriand(infra)nilmanifolds 39
2.2 ActionsofZk, k ≥2ontoriandnilmanifolds 53
2.3 HigherrankRk-actions 77
2.4 Affineactionsbeyondtoriandnilmanifolds 92
3 Preparatoryresultsfromanalysis 100
3.1 Introduction 100
3.2 Preparatorynormestimates 101
3.3 Journe´’stheorem 109
3.4 The Jacobian along the stable leaves of a partially
hyperbolicdiffeomorphism 118
3.5 SmoothregularitybyFouriermethod 122
v
vi Contents
3.6 RealanalyticregularitybyFouriermethod 126
3.7 Smoothregularityviahypoelliptictheory 128
PartII:Cocycles,cohomology,andrigidity 131
4 Firstcohomologyandrigidityforvector-valuedcocycles 133
4.1 Cocyclesovergeneralgroupactions:anoverview 133
4.2 Vector-valuedcocyclesinrank-onehyperboliccase 138
4.3 Cocyclesoverpartiallyhyperbolicsystems 152
4.4 Higherrankresultsforvector-valuedcocycles 163
4.5 CocyclesovergenericAnosovactions 190
4.6 Twistedcocycles 198
5 Firstcohomologyandrigidityforgeneralcocycles 202
5.1 Cocycleswithvaluesincompactabeliangroups 202
5.2 Introduction to rank-one results for non-abelian group-
valuedcocycles 207
5.3 Calculationofcohomologyfornon-abeliancocyclesover
rank-oneAnosovactions 210
5.4 Invariant foliations for Lie group and diffeomorphism
group-valuedextensions 219
5.5 Regularityresultsfornon-abeliancocyclesoverrank-one
Anosovactions 240
5.6 Parry’s general cohomological result for cocycles
withcompactnon-abelianrange 248
5.7 Lift of regularity for the transfer map from
measurabletoHo¨lder 251
5.8 Periodiccyclefunctionals 254
5.9 Non-abeliancocyclesoverTNSactions 259
5.10 Rigidity of non-abelian cocycles over Cartan actions:
K-theoryapproach 269
6 Higherordercohomology 276
6.1 Introductiontohighercohomologyofgroupactions 276
6.2 Cohomology for partially hyperbolic actions by toral
automorphisms 279
6.3 CohomologyforWeylchamberflows 299
References 302
Index 311
Introduction: an overview
1. Rigidityindynamics
In a very general sense, modern theory of smooth dynamical systems deals
with smooth actions of “sufficiently large but not too large” groups or
semigroups (usually locally compact but not compact) on a “sufficiently
small”phasespace(usuallycompact,or,sometimes,finitevolumemanifolds).
Important branches of dynamics specifically consider actions preserving a
geometricstructurewithaninfinite-dimensionalgroupofautomorphisms,two
principal examples being a volume and a symplectic structure. The natural
equivalence relation for actions is differentiable (corr. volume preserving or
symplectic)conjugacy.
Oneversionofthegeneralnotionofrigidityinthiscontextwouldrefertoa
certainclassAofactionsbeingdescribedbyafinitesetofparameters,usually
smoothmoduli.Examplesofsuchclassesareallactionsintheneighborhood
of a given one, or all actions of a continuous group with the same orbits, or
all G-extensions of a given action α to a given principal G-bundle. In some
situationsthisistoostrongand,ratherthanclassifyingallactionsfromA,one
may require that actions equivalent to a given one have a finite codimension
inaproperlydefinedsense,e.g.,appearintypicalorgenericfinite-parametric
familiesofactions.
AperfectextremecaseappearswhenallactionsfromAbelongtoasingle
equivalence class. This may be referred to as rigidity in the narrow sense of
the word. However, one should point out that for abelian group actions even
locallythiscanonlyhappeninthecaseofdiscretegroups,sinceotherwiseone
can always compose the original action with a group automorphism close to
theidentity.
1
2 Introduction:anoverview
2. Limitedextentofrigidityintraditionaldynamics
The material presented in this book relies to a considerable extent on the
classical theory of (uniform) hyperbolic and partially hyperbolic systems,
i.e., the study of rank-one cases of Z+, Z, and R-actions with hyperbolic
or partially hyperbolic behavior. Anosov diffeomorphisms and Anosov flows
(seeSection1.8.1)areprimeexamplesofsuchactions.
Anosov diffeomorphisms and Anosov flows display certain elements of
rigidityofthetopologicalorbitstructure,bothlocally(structuralstability,see
e.g.,[67,Corollary18.2.2])andglobally(topologicalrestrictionsontheambi-
ent manifolds and homotopy invariants [67, Theorem 18.6.1]). Nevertheless,
theirdifferentiablepropertiesarefarfromrigid;atbest,theclassificationwith
respect to differentiable conjugacy is given by infinitely many moduli as in
the case of Anosov diffeomorphisms of T2 ([67, Section 20.4]), or transitive
Anosovflowsonthree-dimensionalmanifolds([105]).Similarly,cohomology
classes of sufficiently regular (Ho¨lder or smooth) rank-one cocycles over an
Anosovsystemaredeterminedbyinfinitelymanyparameters,e.g.,theperiodic
data(seeTheorem4.2.2).
It is also worth pointing out that local differentiable rigidity in the more
looseparametricsensedoestakeplaceforsomenon-hyperbolicsystems,such
as circle rotations with Diophantine rotation number or, more generally, for
Diophantinetranslationsandlinearflowsonatorus.
3. Rigidityforactionsofhigherrankabeliangroups
The goal of this monograph and its projected sequel is to give an up-to-date
and, as much as possible, self-contained presentation of certain rigidity phe-
nomenathatappearforactionsofhigherrankabeliangroupsbysmoothmaps
oncompactdifferentiablemanifolds.Wewillconsiderhyperbolicandpartially
hyperbolicZk-andRk-actions,k ≥2,or,moregenerally,Zk×Rl, k+l ≥2.
CertainresultsforhigherrankabeliansemigroupsZk+,k ≥ 2,arealsodis-
cussed and, by a slight abuse of terminology, we will use the phrase higher
rankabeliangroupsactionsforthemaswell.
The list of known examples of Anosov (normally hyperbolic) and par-
tially hyperbolic actions of higher rank abelian groups which do not arise
from products and other standard constructions is restricted. All such basic
examples are differentiably conjugate to algebraic actions. Basic definitions
appearinSection1.6;principalclassesofexamplesaresurveyedinChapter2.
These actions exhibit a remarkable array of measurable and differentiable