The Ergodic Theory of Lattice Subgroups

136 Pages·2009·0.75 MB·English

by  Alexander Gorodnik

Gorodnik June4,2009
Annals of Mathematics Studies
Number 172
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Gorodnik June4,2009
The Ergodic Theory
of Lattice Subgroups
Alexander Gorodnik
Amos Nevo
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2010
Gorodnik June4,2009
Copyright(cid:2)c 2010byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,Woodstock,Oxfordshire
OX201TW
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Gorodnik,Alexander,1975–
Theergodictheoryoflatticesubgroups/AlexanderGorodnikandAmosNevo.
p.cm.—(Annalsofmathematicsstudies;no.172)
Includesbibliographicalreferencesandindex.
ISBN978-0-691-14184-8(hardcover)—ISBN978-0-691-14185-5(pbk)
1.Ergodictheory.2.Liegroups.3.Latticetheory.4.Harmonicanalysis.5.Dynamics.I.
Nevo,Amos,1960–II.Title.
QA313.G672010
(cid:3)
515.48—dc22
2009003729
BritishLibraryCataloging-in-PublicationDataisavailable
Printedonacid-freepaper.∞(cid:2)
press.princeton.edu
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
Gorodnik June4,2009
Contents
Preface vii
0.1 Mainobjectives vii
0.2 Ergodictheoryandamenablegroups viii
0.3 Ergodictheoryandnonamenablegroups x
Chapter1. Mainresults: SemisimpleLiegroupscase 1
1.1 Admissiblesets 1
1.2 ErgodictheoremsonsemisimpleLiegroups 2
1.3 Thelatticepoint–countingprobleminadmissibledomains 4
1.4 Ergodictheoremsforlatticesubgroups 6
1.5 Scopeofthemethod 8
Chapter2. Examplesandapplications 11
2.1 Hyperboliclatticepointsproblem 11
2.2 Countingintegralunimodularmatrices 12
2.3 Integralequivalenceofgeneralforms 13
2.4 LatticepointsinS-algebraicgroups 15
2.5 Examplesofergodictheoremsforlatticeactions 16
Chapter3. Definitions,preliminaries,andbasictools 19
3.1 Maximalandexponential-maximalinequalities 19
3.2 S-algebraicgroupsandupperlocaldimension 21
3.3 Admissibleandcoarselyadmissiblesets 21
3.4 Absolutecontinuityandexamplesofadmissibleaverages 23
3.5 Balancedandwell-balancedfamiliesonproductgroups 26
3.6 Roughlyradialandquasi-uniformsets 27
3.7 Spectralgapandstrongspectralgap 29
3.8 Finite-dimensionalsubrepresentations 30
Chapter4. Mainresultsandanoverviewoftheproofs 33
4.1 StatementofergodictheoremsforS-algebraicgroups 33
4.2 Ergodictheoremsintheabsenceofaspectralgap:overview 35
4.3 Ergodictheoremsinthepresenceofaspectralgap:overview 38
4.4 Statementofergodictheoremsforlatticesubgroups 40
4.5 Ergodictheoremsforlatticesubgroups:overview 42
4.6 Volumeregularityandvolumeasymptotics:overview 44
Gorodnik June4,2009
vi CONTENTS
Chapter5. ProofofergodictheoremsforS-algebraicgroups 47
5.1 Iwasawagroupsandspectralestimates 47
5.2 Ergodictheoremsinthepresenceofaspectralgap 50
5.3 Ergodictheoremsintheabsenceofaspectralgap,I 56
5.4 Ergodictheoremsintheabsenceofaspectralgap,II 57
5.5 Ergodictheoremsintheabsenceofaspectralgap,III 60
5.6 Theinvarianceprincipleandstabilityofadmissibleaverages 67
Chapter6. Proofofergodictheoremsforlatticesubgroups 71
6.1 Inducedaction 71
6.2 Reductiontheorems 74
6.3 Strongmaximalinequality 75
6.4 Meanergodictheorem 78
6.5 Pointwiseergodictheorem 83
6.6 Exponentialmeanergodictheorem 84
6.7 Exponentialstrongmaximalinequality 87
6.8 Completionoftheproofs 90
6.9 Equidistributioninisometricactions 91
Chapter7. Volumeestimatesandvolumeregularity 93
7.1 Admissibilityofstandardaverages 93
7.2 Convolutionarguments 98
7.3 Admissible,well-balanced,andboundary-regularfamilies 101
7.4 Admissiblesetsonprincipalhomogeneousspaces 105
7.5 TauberianargumentsandHo¨ldercontinuity 107
Chapter8. Commentsandcomplements 113
8.1 Latticepoint–countingwithexpliciterrorterm 113
8.2 Exponentiallyfastconvergenceversusequidistribution 115
8.3 Remarkaboutbalancedsets 116
Bibliography 117
Index 121
Gorodnik June4,2009
Preface
0.1 MAINOBJECTIVES
Let G be a locally compact second countable (lcsc) group and let (cid:2) ⊂ G be a
discrete lattice subgroup. The present volume is devoted to the study of the fol-
lowingfourproblemsinergodictheoryandanalysisthatpresentthemselvesinthis
context,namely:
I. ProveergodictheoremsforgeneralfamiliesofaveragesonG,
II. Solve the lattice point–counting problem (with explicit error term) for gen-
eraldomainsinG,
III. Proveergodictheoremsforarbitrarymeasure-preservingactionsofthelattice
subgroup(cid:2),
IV. Establish equidistribution results for isometric actions of the lattice sub-
group(cid:2).
We will give a complete solution to these problems for fairly general averages
and domains in all noncompact semisimple algebraic groups over arbitrary local
fieldsandanyoftheirdiscretelatticesubgroups. Ourresultsalsoapplytolattices
inproductsofsuchgroupsandthustoallsemisimple S-algebraicgroupsandtheir
lattices.Infact,manyofourargumentsapplyingreatergeneralitystillandserveas
ageneraltemplateforprovingergodictheoremsforactionsofagenerallcscgroup
G andofalatticesubgroup(cid:2). Wewillelaborateonthatfurtherinourdiscussion
below.
Letusproceedtogiveoneconcreteexampleofourresults. Considerthelattices
(cid:2) =SL (Z)⊂SL (R)=G andletσ :SL (R)→SL (R)beanyfaithfullinear
n n n N
representation. Let m be a Haar measure on G. Given any norm on M (R),
G N
letG = {g ∈G; log(cid:8)σ(g)(cid:8)<t}. Wewillestablish,inparticular,thefollowing
t
ergodictheoremsforactionsofthegroup G andthelattice(cid:2) (whichwestatefor
simplicityinlessthantheirfullgenerality).
TheoremA.
1. Inanyergodicmeasure-preservingactionofGonaprobabilityspace(Y,ν),
for f ∈ L2(Y),
(cid:2) (cid:2)
1
f(g−1y)dm (g)−→ fdν ast →∞
mG(Gt) g∈Gt G Y
pointwisealmosteverywhereandinthe L2-norm.
Gorodnik June4,2009
viii PREFACE
2. Furthermore,ifn ≥ 3,orn = 2andtheactionhasaspectralgap,thenfor
almosteverypointtheconvergencetothelimitisexponentiallyfast:
(cid:3) (cid:2) (cid:2) (cid:3)
(cid:3) (cid:3)
(cid:3)(cid:3) 1 f(g−1y)− fdν(cid:3)(cid:3)≤C(y, f)e−θt,
mG(Gt) g∈Gt Y
with θ > 0 an explicit function of the spectral gap and the rate of volume
growthofG .
t
3. ThenumberoflatticepointsinG satisfies
t
(cid:4) (cid:5)
|(cid:2)∩Gt| =(volG/(cid:2))−1+O e−θ(cid:3)t ,
volG
t
withθ(cid:3) >0anexplicitfunctionofθ.
4. Inanyergodicmeasure-preservingactionof(cid:2)onaprobabilityspace(X,μ),
for f ∈ L2(X),setting(cid:2) =G ∩(cid:2):
t t
(cid:2)
(cid:6)
1
f(γ−1x)−→ fdμ ast →∞
|(cid:2) |
t γ∈(cid:2) X
t
pointwisealmosteverywhereandinthe L2-norm.
5. Furthermore,ifn ≥ 3,orn = 2andtheactionhasaspectralgap,thenfor
almosteverypointtheconvergencetothelimitisexponentiallyfast:
(cid:3) (cid:3)
(cid:3) (cid:6) (cid:2) (cid:3)
(cid:3)(cid:3) 1 f(γ−1x)− fdμ(cid:3)(cid:3)≤C(x, f)e−θ(cid:3)(cid:3)t,
(cid:3)|(cid:2) | (cid:3)
t γ∈(cid:2) X
t
withθ(cid:3)(cid:3) > 0anexplicitfunctionofthespectralgapandtherateofvolume
growthofG .
t
6. Iftheactionof(cid:2) isanisometricactiononacompactmetricspace,within-
variantergodicprobabilitymeasureoffullsupport(e.g.,aprofinitecomple-
tionof(cid:2)),thentheconvergenceholdsforallpointsandisuniform,provided
that f iscontinuous.
In the following two sections, we will make some comments aimed at putting
theforegoingresultsinperspectiveandgiveabriefoutlineofourapproachtotheir
proof.
0.2 ERGODICTHEORYANDAMENABLEGROUPS
Classicalergodictheory,developedbyPoincare,vonNeumann,andBirkhoff,stud-
ies a measurable space (X,B) equipped with a probability measure μ and an ac-
tion T of a one-parameter group that preserves the measure. A basic problem is
t
tounderstandthestatisticaldistributionoftheorbits{Ttx}t∈R forμ-genericpoints
Gorodnik June4,2009
PREFACE ix
Figure1 Distributionoforbits
x ∈ X. Givenameasurablefunction f : X → Randaninitialpoint x ∈ X,one
considerstheaveragingoperator
(cid:2)
1 t
(A f)(x)= f(T x)ds,
t s
2t −t
whichsamplesthevaluesof f onpartofthetrajectory{Ttx}t∈Randaveragesthem
withuniformdistributionontheinterval.
Ast → ∞,thequantity(A f)(x)givesanincreasinglyaccuratemeasurement
t
ofthedistributionofthetrajectorystartingatx,sothatcomputinglimt→∞(At f)(x)
is of fundamental importance. This problem was solved by von Neumann and
Birkhoff who showed that under a suitable irreducibility assumption (namely, er-
godicity),
(cid:2)
lim(A f)(x)= f dμ. (0.1)
t→∞ t X
In von Neumann’s formulation, the convergence holds in L2-norm, and in Birk-
hoff’s,μ–almosteverywhere.Whentheactionisbyisometriesofacompactmetric
space,anextensionofWeyl’sequidistributiontheoremassertsthattheconvergence
isuniformandholdsforeveryinitialpointx,provided f iscontinuous.
Classicalexamplesofdynamicalsystemsdescribetheevolutionofphysicalsys-
temsintimeandarenaturallydescribedbyactionsoftheone-parametergroupR.
Inamoregeneralformulation,thebasicgoalofergodictheoryistoanalyzethesta-
tisticaldistributionofgrouporbitsindynamicalsystemsconsistingofaprobability
space(X,B,μ)andalocallycompactgroupGactingonX bymeasure-preserving
transformations. Fixinga(right-invariant)Haarmeasurem onG andanincreas-
G
ing family of compact subsets G of G with m (G ) → ∞, one considers the
t G t
averagingoperators
(cid:2)
1
(A f)(x)= f(g−1x)dm (g),
t m (G ) G
G t Gt
which constitute a sampling of the values of f along the orbit G ·x with respect
to (w.r.t.) the Haar-uniform probability measure on G . Under the assumption of
t
ergodicityoftheG-action(i.e.,thatanymeasurablesetinvariantunderG iseither