Annals of Mathematics
The Metric Entropy of Diffeomorphisms: Part II: Relations between Entropy, Exponents and
Dimension
Author(s): F. Ledrappier and L.-S. Young
Source: Annals of Mathematics, Second Series, Vol. 122, No. 3 (Nov., 1985), pp. 540-574
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1971329 .
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122 (1985), 540-574
AnnalsofMathematics,
The metricentropyof diffeomorphisms
Part II: Relationsbetween entropy,
exponentsand dimension
By F. LEDRAPPIER
AND L.-S. YOUNG
We continueto considerf: (M, m) -> (M, m), a C2-diffeomorphism
f of a
a Borelprobability
measurem. As
compactRiemannianmanifoldM preserving
before,let hm(f) denotethemetricentropyof f,let Xl(x) > ... > r(x)(x)be
the distinctLyapunovexponentsat x and let Di < r(x)Ei(x) be thecorresponding
decompositionof TxM. Both partsof this paper concernthe relationbetween
to
entropyand Lyapunovexponents.As we have indicatedin the introduction
Part I, muchofthisworkis motivatedby thetheoremsofMargulis,Ruelle[Ru 1]
conditionon the
and Pesin [Pe]. Part I is about a necessaryand sufficient
measureunderwhichPesin'sentropyformulaholds.The maingoal herein Part
and all
II is to prove an analogousformulathatworksforall diffeomorphisms
invariantmeasures.
on m, any equationrelating
of absolutecontinuity
Withoutany hypotheses
entropyand exponentsmustinvolvesome notionof fractionaldimension.This
fact has been observedfor certainspecificexamplesfor some time (see for
instance[Bi]). Generalresultsrelatingthesequantitieshave so farbeen confined
in nature(e.g. [L], [Y]). Our main theorem
to maps that are one-dimensional
(TheoremC) extendstheseresults.We show thatforany invariantprobability
measurem,
(* *)
hm(f) =fXtYidm
where yi denotes,roughlyspeaking,the dimensionof m "in the directionof the
in (7.3) and (7.4).
are givenin ? 7, in particular
subspace Ei". (Precisedefinitions
it takessomeworkto see thatthesenumbersas definedin Section
Unfortunately
mentionedabove.)
7 have the intuitiveinterpretation
Researchsupportedin partby NSF GrantMCS 8120790 and AFOSR GrantAFOSR-83-0265.
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METRIC ENTROPY OF DIFFEOMORPHISMS,
II
541
Since yi lies between0 and dimEi, our formulaimpliesthe inequalityof
Margulisand Ruelle [Ru 1] (althoughthisis a ratherindirectway of arrivingat
theirresult).Pesin'sformulacorresponds
to the case where yi= dimEi.
We give an indicationof the proofof (* *). When two or more distinct
positiveexponentsare present,it is difficult
to estimatedimensionas is done in
[Y]. In these situations,however,there is a hierarchyof unstable manifolds
correspondingto the largest k exponents,k = 1,... , u(x) (see (1.1)). These
unstablemanifoldsgiveriseto a nestedfamilyof invariantfoliations.
Conditioning our estimateson the leaves of these foliationsand workingour way up
successivelayers,we are able to focuson one exponentat a time.
In our argumentwe use thenotionof "entropyalongan invariantfoliation"
or "partial entropies".This allowsus to distinguish
betweenrandomnessoccurin muchthe same way thatone studiesthe growthof
ringin different
directions
the derivativeby decomposingTxM into subspaces corresponding
to distinct
exponents.These ideas mayhave further
ramifications.
We have expressedentropyin termsof exponentsand dimension.One may
wishalso to expressdimensionin termsoftheotherinvariants.
We do nothave a
completesolutionto thisproblem.We showhoweverthatthe dimensionof m is
bounded above by Eiyi (TheoremF). A corollaryto this(CorollaryI) partially
confirms
a conjectureofYorke'son the dimensionof attractors
[FKYY]. Incidentally,Pesin's formulacan also be deduced fromtheoremF and (* *) in the case
when m is equivalentto Lebesgue.
This paper beginswithSection7 in whichwe give precisestatementsof
resultsand relateddefinitions.
Section8 containslocal unstablemanifoldsestimates.The notionofentropyalongan unstablefoliationis introducedin Section
9. Theorem C is provedin Sections10 and 11 and TheoremF is proved in
Section 12.
Some of the proofsin Section8 and especiallyin Section11 are similarto
thosegivenin PartI. We tryto makethe statements
oflemmasas self-contained
as possiblehere but willreferthe readerto PartI forcertainproofs.
We remarkalso thatwhile our theoremscontainno ergodicassumptions,
mostofthemcan be provedby reductionto theergodiccase. For thisreason-and
forsimplicityof exposition-wehave chosen to presentmost of our ideas and
proofsassumingthatm is ergodic.
Finally we should mentionthat the theoryof dimensions,as well as its
relationto entropyand exponents,seemsto be of someinterestto mathematical
was underpreparation,
physicists.In fact,whilethismanuscript
two physicists,
Grassbergerand Procaccia(see [G]), derivedheuristically
of
(and independently
this work) a relationverysimilarto ( * * ). For a discussionof these invariants
that is more orientedtowardsapplicationswe referthe readerto the survey
article[ER].
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542
F. LEDRAPPIER AND L.-S. YOUNG
StandingHypotheses
M is a C' compactRiemannianmanifoldwithoutboundary;
f: M *' is a C2 diffeomorphism;
m is an finvariantBorelprobability
measureon M;
and exceptin (7.5), (7.6) and Section12,
m is ergodic.
of results
and statements
7. Definitions
(7.1) Unstablemanifolds.As in Section1, let F-' be the set of pointsthat
are regularin the senseof Oseledec [0]. For x c F%,let
XA(x) >
2) >
...
> Xr(x)(x)
denoteits distinctLyapunovexponentsand let
D
TxM= El(x) ED.. *Er(x)(x)
of its tangentspace. Since m is assumedto
be the corresponding
decomposition
be ergodic,the functionsx r(x), Xi(x) and dimEi(x) are constantm-a.e.
Let u = max{i: Xi > 0}. For 1 < i < u, define
Wi(x)=
y EM: limsup n1log
d(f -nx, f -ny) <-A i
o
Then Wi(x) is a C2 (-d? dimE)dimensional immersedsubmanifoldof M
tangentat x to EDj < iE (x). (See forinstance[Ru2].) It is called the ith unstable
manifoldof f at x. We sometimesreferto { W'(x): x c F- } as the Wi-foliation
on M. Using thislanguagethen,we have a nestedfamilyof a.e. foliations
W1c
W2c
...
c
Wu
to the distinctpositiveexponentsof f.
corresponding
each W'(x) inheritsa Riemannianstructure
immersed
submanifold,
As an
fromM. This givesriseto a Riemannianmetricon each leaf of Wi. We denote
thesemetricsby d&.
The measurem also definesconditionalmeasureson theleavesof Wi. More
precisely,a measurablepartition( of M is said to be subordinateto the
of x
iffora.e. x, ((x) C W'(x) and containsan open neighborhood
Wi-foliation
in W'(x). Associatedwitheach measurablepartitionis a systemof conditional
measures.(See (1.3) ofPartI or [Ro].) In thenexttwosubsectionswe shalldefine
subordinateto Wi.
some invariantsusingthe conditionalmeasuresforpartitions
(7.2) Entropyalong unstablemanifolds.Fix 1 < i < u. We now definea
notion of entropyalong the Wi-foliation.In the ergodic case this notionis
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METRIC ENTROPY OF DIFFEOMORPHISMS,
II
543
describedby a numberhi whichmeasuresthe amountof randomnessalong the
Following[BK] we takea
leaves of W'. Thereare severalequivalentdefinitions.
pointwiseapproach:
Let E > 0. For x E F' and n E Z+, define
V?(x, n, c) = {y e W?(x): di(fkx ,fky) < EforO < k < n}.
to Wi and let { mx} be a systemof
Let ( be a measurablepartitionsubordinate
conditionalmeasuresassociatedwith(. Define
hi (x E
=
n, E)
lim
inf- Ilog myVi(x,
n
n--
=
n, E)
limsup- -log
n mXVi(x,
and
hi(X,
00
1
fl --p 00
o
are indeedmeasurathatthesefunctions
(The cautiousreadermaywishto verify
ble.)
PROPOSITION
7.2.1. At m-a.e.x,
limhi(x,E,)
=
limhi(x, E,).
These limitsexistbecause hi(x,E, ) and hi(x,E, () increaseas E I 0. The
occupiesSection9.
proofof thisproposition
Let hi(x, {) denote the limit in Proposition7.2.1. Using the essential
uniquenessof conditionalmeasuresand the invarianceof m it is easy to verify
that hi(fx,() = hi(x,() m-a.e.and hence hi(x,() is constanta.e. The reader
shouldcheck also thatthisconstantis independentof the choice of ( or { mx}.
of hi.
This completesthe definition
has in factbeen exploitedin PartI
The conceptofentropyalonga foliation
of
The mainproposition
introducethisterminology.
thoughwe did notexplicitly
Part I (see Section5) combinedwiththe discussionin Section9 gives:
COROLLARY
7.2.2. hu = hm(f).
(7.3) Dimensionof conditionalmeasureson unstablemanifolds.Again fix
1 < i < u. Let B'(x, E) denotethe di-ballin W'(x) centeredat x of radius E.
Let { be a measurablepartitionsubordinateto WZ withconditionalmeasures
{mO. For x e ' define
bi( x, ()=lim
inf log
and
Si(X,)
=
mXBi(x,
lim sup
log
E)
)
E--+log
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544
F. LEDRAPPIER AND L.-S. YOUNG
As in the last sectionone verifiesthat Si(x, () and Si(x, () are constantalong
orbitsand thatthe two numbersSi and Si are well definedindependentof (.
Propositions7.3.1 and 7.3.2 are consequencesof our resultsin Sections10
and 11.
PROPOSITION
7.3.1.
8
=
8
We denote this commonvalue by Si and call it the dimensionof m on
knownfact
let us recallthe following
Wi-manifolds.To justifythisterminology
(see forinstance[Y]):
Suppose 1& is a finiteBorel measureon M and X c M is a Borel set with
LX > 0. If foreveryx E X,
lim inflog tB(X'
loge
E2J'o
E)
> 8
and
l
lim sup
ogLB(x, F)
logE
< 8,
thenthe Hausdorff
dimensionof X, writtenHD(X), satisfies
8
< HD(X) < 8.
Suppose now that2 < i < u. RememberthatWi is a (Ej 2
idimEj)-dimenM
in
turn foliated by the
sional foliation on
each leaf of which is
(Xj i- dimEj)-dimensionalfoliationW2-1. We shallsee in Section11 thatthe
number Si - i-1 can be interpretedas the "transverse dimension" of m on
W2/W'- 1. In particular,we shall prove:
PROPOSITION
7.3.2. 0 <
Si - Si-1 < dimEi for 2 < i < u.
(7.4) The main result:ergodiccase. In (7.2) and (7.3) we describedsome
naturalinvariants
associatedwiththedynamicalsystemf: (M, m) +- . Our main
result establishesthe connectionbetween these numbersand the Lyapunov
exponentsof f.
THEOREM C'. Let f: M < be a C2 diffeomorphism
of a compactRiemannian manifoldand let m be an ergodicBorel probabilitymeasureon M. Let
.. *> Xu denote the distinctpositive Lyapunov exponentsoff, and let Si
i >
Thenfor 1 < i < u thereare numbers
be the dimensionof m on Wi-manifolds.
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METRIC ENTROPY OF DIFFEOMORPHISMS,
II
545
yi with 0 < yi< dimEi such that
i=
j<i
Yj
fori =1,. ..,u and
=
hm(f)
i<U
XiYi.
threepartialresults:
of the following
TheoremC' is the amalgamation
(i)
h1 =
X 18 ,
(ii) hi-hi_
= Xi(8i -
(iii) hu = hm(f).
Settingy1= 81 and yi =
l) for i
=
2,. .., u, and
i- Si-L for i = 2,. .., u, one obtains the conclusion
(i), (ii) and (iii).
by addingformulas
of TheoremC' immediately
The dimensionof a
We explaintheseformulasbeforeproceedingfurther.
suchas entropyand Lyapunov
measureis directlyrelatedto dynamicalinvariants
generatesets thatare essentiallyround balls.
exponentsif one can dynamically
This can be done whenall theexponentsare equal. In thiscase equation(i) is the
basic principle relatingentropy,exponentsand dimension.In general,one
attemptsto decomposethe dynamicalstructuresassociatedwith a map into
ratesof expansion.Equation(ii) is then
to the different
directionscorresponding
dynamicsand dimensionof Wi- 1
of (i) concerningthe transverse
a restatement
insideWi-leaves.This is theidea of partof theproofof (ii). (See Section11.) To
complete the proofof (ii) we need anotherargument,which is explainedin
and
Section 10. Finallyequation(iii) says thatwhathappensin the contracting
to theentropyof f. That was basicallythe
neutraldirectionsdoes notcontribute
concernof Part I. (See Section9.)
fromTheoremC'.
The nexttwo corollariesfollowimmediately
COROLLARY
ho
=
D'. With the same hypothesesas in TheoremC', and with
?5
j~i
h
h=
fori=1,...,u.
i
In (7.3) we definedSi = Si(f) and in TheoremC' we definedyi= yi(f) for
i = 1, ... ., u. Rememberthatthese yi's carrygeometricmeaningas the transverse dimension of m between successive Wi-foliations.Now consider
f
and define analogously 'i = yi(f- 1) corresponding to the
positiveexponentsof f -L (or equivalentlythe negativeexponentsof f). Let
Yi=
1: (M, m)
Yr-i
+
when Xi < 0 and yi= dimEi if Xi = 0.
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546
F. LEDRAPPIER AND L.-S. YOUNG
COROLLARY
definedabove,
as in TheoremC' and withyi as
E'. Withthesame hypotheses
ExiYi
=
0.
i
This followsfromTheoremC' and thefactthathm(f) = hm(f'). We state
two morecorollariesofTheoremC' thatare knownresults:
COROLLARY
7.4.1. hm(f) < Li<uXidimEi.
COROLLARY 7.4.2. If m has absolutelycontinuousconditionalmeasureson
unstablemanifolds(see Definition1.4.2), then
hm(f) = E Xidim Ei.
i<u
Corollary7.4.1 followsreadilyfromTheoremC' and Proposition7.3.2, as
does Corollary7.4.2 since Su = Li , udimEi when m has absolutelycontinuous
conditionalmeasureson Wu.
The nonergodicversionof these resultsis statedin the next section.We
mentionhere that Corollary7.4.1 (withoutthe assumptionof ergodicity)was
by Margulisand Ruelle [Rul]. Corollary7.4.2 (again not
provedindependently
necessarilyergodic)is the "if" partof TheoremA and was provedin [LS]. As
indicatedin Part I, thisgivesan alternateapproach,thoughthe proofsin [Rul]
and are muchmoredirect.
assumptions
and [LS] requireweakerdifferentiability
2 case by Manning[Mg].
A
in
Axiom
dimension
TheoremC' was first
proved the
(7.5) Definitionsand the main results:nonergodiccase. Unlikethe situationwhen m is ergodic,thefunctionsx p-> r(x), Xi(x) and dimEi(x) (see (7.1))
are now no longer constanta.e. Let u(x) = max{i: Xi(x) > 0). Then for
i = 1,...,u(x), we can defineWZ(x) as in (7.1) exceptthat Xi shouldnow be
relacedby Xi(x).
Let Fi = {x e F': i < u(x)}. A measurablepartition( of M is said to be
subordinateto Wi on hi if for m-a.e. x e Fi, {(x) c W'(x) and containsan
open neighborhoodof x in W'(x). We extendthe notionof entropyalong Wi
to the nonergodiccase usingconditional
and dimensionof m on Wi-manifolds
of thistype.These extensionsare ratherformal.To make
measuresforpartitions
it the readershould considersets of the formri k =
of
out
sense
geometric
{ E rI Ej <idim E,(x) = k} one at a time or simplydisintegratem into its
ergodiccomponents.Note thatthe entireleaf Wt(x) is containedin the ergodic
componentof x (see e.g. (6.2)).
First we definethe notionof entropyalong Wi, which is now a class of
definedon Fi. Let ( be a measurablepartitionsubordinate
measurablefunctions
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METRIC ENTROPY OF DIFFEOMORPHISMS,
II
547
to Wi withconditionalmeasures{ m,}. Definehi: Fi -* R by
hi(x) = hi(x
, f mx1)
= lim liminf - -log mxVi(xn, E)
E- 0
n
n-0o
= lim lim sup
E? *O
-
n-0
mxV'( x nE
-log
n
We knowthatthe liminfand limsup definitions
coincide m-a.e.on Fi because
Proposition7.2.1 saystheydo fortheergodiccomponentsof m. To verifythathi
is indeed a well definedclassofmeasurablefunctions
one mustshowthatif (' is
a measurablepartition
ofthesame typeas ( and { m' } is a systemofconditional
m-a.e. on Fi.
measures for (' then hi(x,5 , { mj) = hi(x, ',{m})
Corollary7.2.2 now reads
hm(f).
fhU(x)(x)dm(x)
Similarlythereare two measurablefunctions
(or morepreciselytwo classes
of measurablefunctions)Si, i: Fi -* R such thatif ( is a measurablepartition
subordinateto Wi on Fi and { mx} is a systemofconditionalmeasuresfor(, then
at m-a.e. x E ro
li inf
0
and
logm B'
Lg
limsup
(x)
E)
loge
x5 E)
xlog
)
=
i( X)
Propositions7.3.1 and 7.3.2 are valid (so we can write 3i = 3. = 3i) again
because theyare validfortheergodiccomponentsof m. The functionSi is called
the dimensionof m on Wi-manifolds.
We are finallyreadyto statethe mainresultof thispaper:
and let m be an
be a C2 diffeomorphism
finvariant Borel probabilitymeasure on M. Let Xi(x) > ... > Xu(x)(x) be the
distinctpositiveexponents
at x and let Si: Fi -> R be thedimensionof m on Wi.
Then thereexistmeasurablefunctionsyi: Fi -- R with 0 < yi(x) < dimEi(x)
such thatat m-a.e.x,
THEOREM
C. Let f: M
Si W
fori = 1,...
, u(x)
=
-
E yjx)
j<i
and
hm(f) =
(x)
| ii(x)
E
dmu(x)x
i < u(x)
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548
F. LEDRAPPIER AND L.-S. YOUNG
fromTheoremC' by decompositionof m
TheoremC followsimmediately
into ergodiccomponents.
The nonergodicversionsof CorollariesD', E', 7.4.1 and 7.4.2 are obvious.
(7.6) A volumelemmaand some consequences.TheoremC and Corollary
D give some information
on the dimensionof the conditionalmeasuresof m on
unstablemanifolds.We wishnow to discussthedimensionof m itself.Since this
to Wu(x)
we shallreferexclusively
discussiondoes notinvolvetheotherfoliations,
when we speak of unstablemanifolds-whichwe denote simplyby Wu. Note
is assumedin thissectionunlessotherwisestated.
also thatno ergodicity
F. Let f: M <' be a C2 diffeomorphism
of a compactRiemannian manifoldand let m be an ftinvariant
Borelprobabilitymeasureon M. Let
ofm on Wu, Ss be thedimensionofm on W', and AC(x)be
Su be thedimension
the multiplicity
of 0 as an exponentat x. Thenat m-a.e.x,
THEOREM
E)
limsuplogmB(x,
-
limsup
< Su(x)
+ Ss(x)
+ Sc(x)
log
Note that Su + Ss + Sc = EYjy.The influenceof the neutralexponenton
and withoutfurther
one cannotexpecta
dimensionis unpredictable
assumptions
sharperestimate.TheoremF is provedin Section12.
Let us say thata measureon M has full dimensionifat m-a.e.x,
limsuplogmB(x,
logE? --0
=
dimM.
G. Let f: M *- be a C2 diffeomorphisrn
of a compactRiemannian manifoldand let m be an f-invariant
Borelprobabilitymeasurewithfull
m
dimension. Then has absolutelycontinuousconditionalmeasureson both
stable and unstablemanifoldsand thefollowingformulashold:
COROLLARY
hmff) = JEX+dimEjdm,
hm(f) = fEZ-
dimEjdm,
and
JlogjDet
Df Idm = 0.
relatively
easy to
Whethera measurem has fulldimensionis experimentally
is a weakerassumptionthan absolute
verify.Note also thatfulldimensionality
to Lebesgue.Hence CorollaryG can be regardedas a slightextension
continuity
of some of the resultsin [Pe].
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METRIC ENTROPY OF DIFFEOMORPHISMS,
II
549
of a compactRiemannian
H. Let f be a C2 diffeomorphism
manifold. An invariantBorel probabilitymeasurewithfull dimensionis absolutely continuouswith respectto Riemannianmeasure if any one of the
followingconditionshold:
(1) Thereis no zeroexponent,i.e. SC(x) = 0 m-a.e.;
map of a flowgeneratedby a C2
(2) 8 C(x) = 1 m-a.e.and f is thetime-one
vectorfield;
and f is an ergodicalgebraicautomorphism.
(3) M is an n-torus
COROLLARY
Assertions(1) and (2) followfromabsolute continuitypropertiesof the
stablefoliation.See [Be] for(3).
When the measureis not of fulldimension,our resultsare relatedto a
conjectureof Yorke's (see [FKYY]). Suppose m is ergodic. We definethe
AidimE>i 0.
Lyapdim(m),to be dimM if Xi
Lyapunovdimensionof m,written
Otherwisedefine
Lyapdim(m)=
j<i
ji .X dimE.
dimE+
li
Xdim E 2 0.
where i < r is thelargestintegerwithY < iX
of a compactRiemanI. Let f: M +- be a C2 diffeomorphisin
ergodicBorelprobabilitymeasureon
nian manifoldand let m be an f-invariant
M. Then
COROLLARY
limsup
E+
logmB(x, ) < Lyapdim(m)
loge
onlyif m has absolutelycontinuat m-a.e.x. Equality holdsalmosteverywhere
ous conditionalmeasureson unstablemanifolds.In fact, if equalityholds,then
thereis some i with yj = dimEj for j < i and yj = O for j > i, y, being the
numbersin CorollaryE'.
A weakerinequalityinvolvingliminfcan be obtainedby a simplerproof.
(See [L].)
To proveCorollaryI, observethatby 7.3.2 we have
(*)
(xi-
i
X3yj <
j<i
(Xj
-
Xi)dimE1.
Dividingby - Xi and applyingCorollaryE', we get
Eyj < Lyapdim(m)
I
whichtogetherwithTheoremF givestheinequalityin CorollaryI. Supposenow
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550
F. LEDRAPPIER AND L.-S. YOUNG
that equalityis attained.This forcesequalityin (*), fromwhichfollowsthe last
assertionin CorollaryI. That m has absolutelycontinuousconditionalmeasures
in thiscase is a consequenceof TheoremA and the factthat i > u.
8. Moreon local unstablemanifolds
In the firsthalfof Section8 we recordsome estimatesin Lyapunovcharts.
(8.1) and (8.2) parallel(2.1), (2.2), (2.3) and (4.2). For the convenienceof the
to PartI (or [Ru2]) forproofs.
thenotationhere,referring
readerwe reintroduce
The second halfof Section8 containsa descriptionof some special coordinates
These coordinatesare used mainlyin Section11.
on Wu-manifolds.
(8.1) Lyapunovcharts.As usual, d denotesthe Riemannianmetricon M.
We writeRdimM = RdimEl x * .. xR dimE- and forx E RdimM, let (x1,.. ., Xr) be
Define
its coordinateswithrespectto thissplitting.
jxj = maxlxili
is the Euclideannormon Rdim E Let
where
=
Ri(p)
( xi e
RdimEj: IXili < p}
{x E RdimM. IXI < p).
and R(p)
We now describe some changes of coordinates.Given e < (1/100) X
mina*besla - bI where S = {0} U {X p, ., Xr}, there exists a measurable
function1:17 --*(1, co) with1(f+x) < ei1(x) and an embeddingFix:R(l(x)
M foreach x E F' suchthatthe following
fiveconditionshold.
(i) xO = x, and DFJ(0) takesRdimEi to Ei(x) fori = 1,..., r.
(ii) exp[' o coincideswithDF>J(0)on R(l(x) ').
(iii) Let fx= OjxF1? f o AX be the connectingmap betweenthe chartat x
and the chart at fx, defined whereverit makes sense, and let fx- =
-X f 0 (DXbe definedanalogously.Then forall v E RdimE
-
e
<
i-'|V|
lDfx(O)vl
<eI+?Il
(iv) If L(g) denotesthe Lipschitzconstantof the functiong, then
L(fx
L
fX-
Dfx(0))
?
E,
Dfjx-(0))
?
E
-
and
L( Dfx),L(Dfjx-) < 1(x).
(v) For all z, z' EJR(l(x) - 1),
K7-d((4'xzA'Dxz') < Iz
-
z'l
?
l(x)d()xzJADzZ')
forsome universalconstantKr.
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METRIC ENTROPY OF DIFFEOMORPHISMS,
551
II
x E IF )
ThroughoutPartII we shallreferto anysystemoflocal charts{ FDX:
of (e, l)-chartsis more
satisfying(i)-(v) above as (e, l)-charts.This definition
than the one givenin Part I, whereno attemptwas made to control
restrictive
of Lyapunov
the behaviorin individualOseledec subspaces.The construction
chartsgiven in the appendixto Part I extendseasilyto meet these additional
requirements.
(8.2) Local unstablemanifoldestimates.Let { (DX:x e )} be a systemof
(e, l)-charts.For x E IF and 1 < i < u, let WxV(x)be the subsetof R(l(x)-')
the lyimage of whichis thecomponentof WiZ(x)n 4DX
R(l(x) -1) containingx.
We write R(z)(p) = R1(p) x
LEMMA
that
x Rl(p) and Rr(i)(p)
= R+ 1(p) x ...
8.2.1. Thereis a functiong': R(z)(l(x)- )
=0
(1) g'(O)
Dg'(O)
=
xlRr(p).
Rr-(i)(l(x)-l) such
0,
<3 and
(3) graph(g ) = Wx(x).
(2) ?Dg
Clearly,fx-'Wx(x) c W X(f x).
8.2.2. Let x e r. Thenthereexists , 0 < T < 2, such thatforall
and 1 < i < u, there is a function
y E .Fx(Wx(x) n R(Tl(x)'))
LEMMA
- Rr-(i)(l(X)-l)
R(i)(l(x)'-)
(1) 4y' y e graph gXy
(2) IIDg, ? 4, and
gi
such that
(3)
raph
'
=
e WU(x): limsup
n-oo
frt.tx
yj ?
-
Xi + )2,}
=
by WU(x). Note that WXUT(X)
We denote Wxu(x)n
R(Tl(x<')
call
and
Also we denotethe graphof g' by Wxi(y)
graph(gxuR(U)(Tl(x)-l)).
is an easy
corollary
a local ith unstablemanifoldat y. The following
its 4(X-image
consequenceof 8.2.2(3).
COROLLARY 8.2.3. Let x E I" and let T be as in 8.2.2. Then for all
y ( 'DXWX
T()
(1)
and
W(y)
C Wxz+?(y)
theneitherWxz(y)= Wx(z) or Wxf(y)fl Wxf(z)= 0,
(2) If z E FDWXU7(x)
(3) If y c F, then4y-lWi(y)
n
Wxu(x)= Wxi(y).
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552
F. LEDRAPPIER AND L.-S. YOUNG
Assertion(2) above saysthat{ W(( y): y E 4>Wx 7(x)4is a foliationon some
i's thesefoliations
set containingWxu(x). (1) saysthatfordifferent
are "nested",
and (3) says thatforeach i the global and local Wi-foliations
induce the same
partitionon Wxu,(x),a factwe willneed in Section11.
LEMMA8.2.4. Let x E IF.
(1) If z z' E R(e-XI-3'1(x)
E R(l(fx)
l) then fxzfxz'
lfxz -fxz'l
< Iz
and
- z-'Ie~
(2) If y E4'-WX T(x) and z z' E Wxy), then
L1 z --fx-
1z - Z',
< e-x2
?Zl
inside local
Finally we state a lemma that tells us that the Wi-foliation
Wu-leavesis at least Lipschitz.Let L(Rn,Rk) denotethe space of linearmaps
from Rn to Rk. For x E F' and 1 < i < u . let Gi: UYoWU(X)W(y)
L(Ryj s idim Ej Ryi < ?rdimEj) be definedby
G'(z)
z
= Dg"
Y(Z, ..
)
and z E WxA(y).In particular,if Oxz E F', then
where y E 4DxWxu,(x)
DO
x(z)(graph G'( z)) = ED3 E j(oxZ)
LEMMA8.2.5. Thereis a constantD > 0 such thatfor all x E F' and for
every1 < i < u, themap G' is LipschitzwithLip constant< Dl(x).
The proofis identicalto thatin (4.2).
(8.3) Special coordinateson local unstablemanifolds.For each x E F, we
now definea coordinatesystemon thelocal Wu-manifold
at x. These coordinates
into parallelplanes and at the same time
"straightenout" local Wi-manifolds
preservegood dynamicalestimates.
We firststatea lemmaaboutLipschitzfoliations
thathas nothingto do with
and will be omitted.Fix positive
dynamics.The proofis quite straightforward
integers n1,... ., nk and a number0 < p < 1. Let Bi(p) be a closed disk
centered at 0 of radius p in Rn'i.Consider B(p) = Bi(p) x ... xBk(p) as a
subsetof R' I +nk
LEMMA 8.3.1 (straighteningout lemma). For i = 1,..., k - 1, let F1 be a
Ink containing
Lipschitz foliation with C' leaves on some subset of R n+?
B( p). Assumethateach leaf of Fj is thegraphof a function
g: B (2p)
x
...
x Bz(2p)
-
Rni+l+
nk
with IIDg < ' and thatthefunctionx -- TxFj has Lipschitzconstant< some
numberC. Assumealso thattheF1's are nested;i.e., if Fi(x) denotestheleafof
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METRIC ENTROPY OF DIFFEOMORPHISMS,
553
II
Fi containingthe pointx, thenFi(x) C Fi 1(x) for all x E B(p) and for all i.
Define (9=
(1,...A
k): B(p)
-
Rnl+'
+nk
as follows: Forx
= (xl,...,xk)
B(p), let C91(x)= x1, and let Ci(x) be theith coordinateof theunique pointof
intersection
of Fi 1(x) and { O} x ... x {O} x Rni+ + nkfori ? 2. Then
(1) (9 is a homeomorphism
betweenB(p) and its image;
(2) For everyx, y E B(p), C1x= C9jyfor j = i + 1,... . u if and only if
y E Fi(x) and
(3) Both (9 and (- 1 are LipschitzwithLip constantdependingonlyon C
(assumingk is fixed).
Consider now a system of Lyapunov charts { Ix 4. For x E F' let
be the naturalprojection.Then pxIWxu(x)is a
R(u)(l(xfl)
px: R(l(xf')
lipeomorphism
and the pr-images
of Wxi(y),y E=I4xWxu(x), formtheleaves of a
LipschitzfoliationFi. It is easyto checkthattheseFi's (i = 1,... , u - 1) satisfy
the hypothesesof 8.3.1 with B(p) = R(u)(Tl(x)-l)
and C = Dl(x) (see 8.2.5).
Ri__dim'i
in
be
the
Let Cx:J(u)(rl(xl)
map 8.3.1.
We are now ready to defineour "special coordinates"on Wxu,(x).Let
'7T: Wxu(x)
R?iudimEi be given by
17x =
Ox ? Px.
From 8.3.1 we can concludethat7rxis a lipeomorphism
betweenWxu,(x)and its
image with Lip(7Trx),Lip(7T[l)
< Nx where Nx depends only on 1(x) (and other
lies on a (E j i dimEj)-dimenconstantsdeterminedby f). Moreover,vxWxi(y)
sional plane parallel to REj<idimEj x {O} x ... x{O} and if Wx(y) = Wx(y')
lie on distinctplanes.
then 7rxWxW(y)
and 7rxWx(y')
LEMMA
(
7Tx
8.3.2. Let z E WxU,(x)and supposeIZi
<
Te-X3?.
When zx =
. . . 5
7Tx )
IT
fxzl< ex?3 IZI
for i = 1, ... ., .
Proof Make theestimateone i at a time.For i ? 2 theproofis identicalto
thatof Lemma 2.3.2.
(8.4) Special coordinateson a set of positivemeasure.In (8.3) we describe
some coordinateson local unstablemanifoldsthathave desirablegeometricas
well as dynamicalproperties.
These coordinates,
however,are not compatiblein
the sensethatforD C eFxWxu7(x)
flI)D
1) need notbe equal
to (7x,o x11) D. The aim of this subsection is to construct a map
udimEi, whereS is a set of positivemeasure,suchthat7rrestricted
to
7J: S --I R
out Wi-leavesas in (8.3).
each local Wu-leafin S straightens
For x Ec F,
Let lo be chosenso thatA = { 1 < lo4 has positivem-measure.
let E'(x)
= Ej < iEE(x). As mentioned in (8.2), Ei(y), i = 1, ... .,u, is actually
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554
F. LEDRAPPIER AND L.-S. YOUNG
In fact,thereexistsa constantlo
A4XWXT(X).
definedfor every y E A =
such thatforeveryy E=A, IIDfy-nvI? -e (1x3)nhlvIl forall n ? 0 and forall
v E EZ(y). This impliesthaty 4 E3(y) is continuouson A.
Let w E A be a densitypoint of XA *m. Then using the continuityof
=
y -+Ez(y) on A, we can choose To, 0 < To <X, so that for all x EA
if y E F>WxU(x) n FDwR(Tol-'),then foreach i = 1,...., 4,
A l IR(Colo'),
n R(2T0l-1) is the graph of a functionfrom Rtfm(2T01-1)
wlxW~xwi(y)
Rr-(i)(2Tol
1) the normof the derivativeof whichis < 3. For each x E A
then,Lemma 8.3.1 givesa map (% : R(u)(Tslo') --*Ri'"udim Ei thatstraightens
out the foliationwhoseleavesare subsetsof pwfF'xWx(y).
Let S = UXEA (4,xWxu(x)n (FwR(T01-1)). Define
0~~
~o~'y
To
7JY
ow,X?
7T:
S
-
Ri<udimEi
by
?4ly
W
where y E xWU(x), x E ATO The readershouldverifythat thismap is wellof 7r(y)is independentof the choiceof x.
defined;i.e. the definition
We note also that
(1) S is the disjointunionof sets Doawhereeach Da fl { 1 < lo} # 0, and
also,m(S n {l < lo}) > 0;
if x E Daf,,l(1< 101thenDoa= S n (DxWxu(x);
=
.
continuous
foliation,it is easy to see
1,.
,
u,
is
a
(2) Because WZIS, i
of ( in Lemma8.3.1 that gris in facta continuousmap; and
fromour definition
in (1), foreach a, 7TID,,and (7rID,),) are Lipschitz
With
the
notation
(3)
maps withLip constant< someconstantNo.
9. Entropyalongunstablefoliations
subordinateto Wi. Let q1 and 'q2 be measurable
(9.1) Increasingpartitions
partitionson M. Recall the followingdefinitions:q1 refines'q2( 1 > q2) if
for m-a.e.x, and q1 is increasingif q1> f'q1. In (3.1) we
'ql(x) c 'q2(x)
described certain increasingpartitionsthat are subordinateto Wu. In this
subordinateto W', i = 1,..., U.
subsectionwe constructsimilarpartitions
LEMMA 9.1.1.
partitions ,.u.. on M suchthat
Thereexistmeasurable
foreach i,
(1) (i is an increasingpartitionsubordinateto W3,
(2) i > i+l and
(3)f -fnli generatesas n -x oo.
Outlineof proof.We takea systemof( E, 1)-localcharts{ 4D) and choose 10
and S =UDa asin(8.4). Fori = 1,...,ulet
Oi~x)
= (Wi(x) n Da
ifx E/D
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METRIC ENTROPY OF DIFFEOMORPHISMS,
555
II
of S and 8.2.3(3) thatif x e Da fl { l < 10), then
It followsfromour definition
every (i-element contained in Da is equal to S n <VxWx(y)for some
> du obviouslywe have
y E FxWx'u(x).Let (i =
Of, i. Since 4j >
>u
41>>
Some care has to be takento ensurethat (i(x) containsan open neighborhood of x in W'(x) for m-a.e.x. (In fact,Lebesgue-a.e.To small enough for
detailsbecause thisargumentis identical
purposes(8.4) willdo.) We omitfurther
to thatin [LS].
Let x E F'. Then y E (i(x)
characterization.
Note that (i has thefollowing
if and onlyif forall n ? 0,
(1) ffny e S ifand onlyif f =x e S,
(2) f-nx, ffny lie in thesameDa whenever
f-nx E S, and
(3) y e W(x).
In (9.2) and (9.3) we shallprove
im hi(x, E, i) = H(tijfti) = limhi(x,E, i).
fromthis.Remarkalso thatif is another
Proposition7.2.1 followsimmediately
increasingpartitionsubordinateto Wi constructedin the proofof 9.1.1, then
H(tIfJ) = hi as well.
In (9.2) and (9.3) let ( (i = 1,. . ., u) be as above and foreach i let { mx}
be a systemof conditionalmeasuresassociatedwith(.
t
(9.2) Proof that hi ? H(tiIfti). Let E > 0 be given. For 8 > 0, we let
Then mA8Tl as 8 0. Let g(x) =
A8 = {x: B'(x,8) C (i(x)).
- log m'(f -'
)(x) and choose 8' smallenoughthat
fg 2 H(f
-
-Ij?
Define
=
H(tilf~i) -
n (f-i)(x)
U'(x, n, 8) =
{j: O<j<n
and fixeA8}
Then by definitionV'(x, n, 8) c U'(x, n, 8) and - log m'U'(x, n, 8) >
*g)(fix). Thus whenever8 < 8' we have form-a.e.x,
jn=O(X
hi(=x,8,
liminfn logm vi(x, n, 8)
n -oo
?fg
? H(iAfO)
-
(9.3) Proofthathi < H((iIf~i),
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556
F. LEDRAPPIER AND L.-S. YOUNG
LEMMA9.3.1. Let 9 be a partitionwithHm(jA) < Doo.Thenform-a.e.x,
lim
-
v
n log Mi(g
)(x)
=
(i
H(tjfti)
n--*
oo
wherean( x) denotestheelementof thepartitionV . - jf-a thatcontainsx.
Proof Define the informationfunctionof a partition9 given ) by
(1 )(x) =-log mX2(x) where { mj is a familyof conditionalmeasures
associatedwith q. Then
v (i)njtiJ(X)
nI((gz
-
-I(9It)(x)
1n-1
+ n-
k=i
I(( iv ii
v
5
k)(fx).
the secondtermabove
Since supnI(g V (ijftj V ??n) is an integrablefunction,
thelimitfunction
is constant
convergesm-a.e.and in L' (see [Pa]). By ergodicity
and is therefore
almosteverywhere
equal to
lim
1
n- oo
H((-q v (i)onjti
whichcan be writtenas
lim
1H((tj)nti)
+ liM 1H(_qonif-n+I
0
n- oo n
The firsttermis equal to H(jIjft3). The second termgoes to 0 since f -n
E
generates.
n--oo n
We now constructa partition O where Y&0n(x)can be compared to
V'(x,n,E). Let {?x), S and 10 be as in (9.1) and let S' = S nl {l < 10). Then
mS' > 0. Firstwe definen+ and n_: S' -- N by
n+(x) = Inf{n > 0: fnx E S.'
and
n- (x) = Inffn > 0: f -nx e S').
Then we let 41: M
E)
--
R be given by
K
-K112e-max(XI+3e
Xr+3e} max(n+(x), n(x)}
S'
otherwise
if x E
where E' is a preassignedsmall number.Since f - log 4 dm < xo, thereis a
partition92 with H(,O) < co such that O(x) c B(x, 4(x)) foralmosteveryx.
(See [Me] or (2.4) fora similarconstruction.)
Finallywe define4A+by replacing
n in the definition
max( +, ,)
of 41by n+.
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METRIC ENTROPY OF DIFFEOMORPHISMS,
Let x E S'.
(1) If y E x~Wx(x)
satisfies4y'-lyj < l04'(x),
II
557
LEMMA 9.3.2.
then
d'(fix, fjy) < E
for 0 ? j < n+(x) and
fn+(x)y
(2) If yEcon(x)
E
Ix).
f+(X)xWf+(X)x(f
l FxWxi(x)
forsomen > 0, then
d'(fjX, fly) < ?'
for 0 < j < n.
Proof It follows from our assumptions on y and 8.2.4(1) that
fx(Ox-y) e Wfjx(fix)and
1~x((D1y)l < jD- lyle(A?+2e)i
forj > 0
provided that I4D[lyIe( 1?2e)k ? l(fk)leA13e
for all 0 < k < j. This is
is a graphover R(')(l(fix)-') with
guaranteedforj < n+(x). Since Wf/,x(fix)
slope < 1 and
|
is the maxnorm,
d'(fix,fjy) < KrI4iX(f'y)I <
forO < j < n+(x).
To prove (2), firstobserve that if y E 9q(x), then I -ly[I < lo(x) <
lob4?
+( x) so thatwe have thedesiredconclusionfor0 < j < n +(x). Furthermore,
if n > n+(x) and y E ?gon(x),then
fn(x)y
GE 9(f
n+(x)x )
n
and we can apply (1) with fn+(x)x and
f+(X)x W+(X)Ix(fn+(x)
fn+(x)y
argumentcompletesthe proofof (2).
in place of x and y. An inductive
3q withH(9 ') < xo
LEMMA 9.3.3. Forevery
a' > 0, there
existsa partition
and a functionno: M -- N such thatform-a.e.x,
gon(x) n
(x)
c V'(x, n, E')
for all n ? no(x).
Proof: Let S be as above and let
no(x)
= Inf{n
? 0:
fnx E
S'}.
Consider an arbitrary point x with the property that fnx E S' infinitelyoften as
n -*+ ?o.
Let y E gono(x)(x) n
We claim that d'(fix, fiy) < ?' for 0 <
(i(x).
j < no(x) and that ffo(X)y C 4(xXWO(X)V
fno(x)X
It suffices to prove this claim, for if y E 3on(x) n j(ix), then the second
assertion above allows us to apply 9.3.2(2) to ffo(X)y, which then tells us that
dz(fix, fiy)
< ?' for no(x) < j < n.
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558
F. LEDRAPPIER AND L.-S. YOUNG
To provethe claim,let us assumethatx 4 S' and forsimplicity
of notation
writero=- n_(fno(x)x)+ no(x). That is, rois thelargestinteger< 0 suchthat
frox E S'. Since (i is increasing,
we have froy E .i(frox) whichby ourchoiceof
S lies in FroxWjox(
fox). Also, 4 is chosenin sucha way thatf -ij(fno(x)x)
lies
+
j
no(x)x forj = 1,2,..., n - (fno(x)x). Thus
well insidethe chartsat f
I4fo
<o
<
?Ilx+
fno(x)fYoI
10.
< ?fIfo(x)X?
<
whichis < 10?+(frox). Lemma9.2.2(1) givesexactlywhatwe need.
LI
Finally,combiningLemmas 9.3.1 and 9.3.3, we have at m-a.e.x and for
everysmallE',
hi(x,E', ()
=
1
limsup - -log mixV(x,
E'
n -oo
< lim sup - 1log mi gon(X)
- n-.
nn
oo
< lim - I'logm~iX iV g) 0n(X)
n
n
-
H(~Ilft ).
Proofof Corollary7.2.2. It followsfrom(9.2), (9.3) and Corollary5.3 that
hU= H(tujftu) = hm(f).
10. Relatingentropy,
exponentsand dimension:ProofofTheoremC'
We now begin to prove TheoremC'. The fivesets of inequalitieslisted
below are dealt withseparately:
-h
(1) a.
<
h
b. 81 2 A
2 < i < u,
(2) hi- hi_1 2 Xi(8iSi
- ),
2 < i < u,
(3) hi- hi1 < Xi(8i - 8i0),
2 < i < u,
(4) hi-hi1? <XidimEi,
(5) hu = hm(f).
Inequalities(1)a. and b. tell us that 81 = 81 and that hi = ix1. This
togetherwith (2) and (3) proveinductively
thatfori = 2,.. ., u, Si = i (Proposition 7.3.1) and hi - hi-_ = Xi(i-b i-,) Adding these equations and
setting yi = -Si -i-1
(80 = 0), we obtain hu = Ei uyiXi. That yi < dimE.
(Proposition7.3.2) is a consequenceof (4).
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METRIC ENTROPY OF DIFFEOMORPHISMS,
II
559
The fifthequationwas statedas Corollary7.2.2 and was provedin the last
section. In Section 10 we prove (1) and (2). The remainingtwo inequalities
involveexplicitestimateson "transversedimension",a notionwe shallelucidate
in Section 11.
(10.1) Proofthathi = Six,. Since thereis onlyone rateof expansionalong
1
in [Y]. Let - > 0 be given.Choose a system
W thisproofmimicsthearguments
of (E, l)-charts{ (Dj, an increasingpartition4l subordinateto W1 and a system
of conditionalmeasures{ ml) for 4. From Section 9, (9.2) in particular,we
knowthatthereexists8 > 0 such thatform-a.e.x,
hl(x,
KA8
E.
2 hi
01
and let z = <>xy. Then z E Wx'(x) and
By 8.2.4(3), IkzI ? lze(Ai+2e)k for k > 0 provided all
e-n(Al+3e)81(X)-l.
iteratesup throughk staywellinsidecharts.This is guaranteedfork = 1,. . . , n.
< 8 for0 < k < n and since fkZ Ee Wfkx(fkx), we have shownthat
Thus IfXkzI
C V'(x, n, KrS). This gives
B'(x, e-n(X1+3e)81(x)-2)
Considery E Bl(x, e-n(l+3e)8l(x)-2)
Izl <
S 1(x) = liminf logm'B'(x,p)
iminf
.log
= lim inf
n
n-oo
>
mlB 1(x,
- n(xl
I
+
A1 + 3liminf3En
o
e
n(A?+3e)S1( X)-2)
+ 3E)
+
log1l(x)2
Ilog m1V1(x,n,Kr8)
r
n
- E
A1 + 3e
hi
we have proved81 ? (h/Al).
form-a.e.x.The choiceof E beingarbitrary,
let 9 be the partitionin (9.3) withi = 1. It
To provethe otherinequality,
was shown in the proof of Lemma 9.3.3 that for almost every x, if fnX E S',
n > 0, then fn((1(x) n gon(x)) c ! W nx(fnX) Fix x and n ? no(x). Con-
sider y E (1(x)
Since
fky
n gon(x).Let k be the largestinteger <
E !I?(fkX),
since k ? no, fky C
<
1VD-lyI ?IDi(fky)le-(cX-2e)k
n such thatfkx E S'.
< l04(ffkx) < e-K-11-le-(1+?3e)(n-k).
Also,
10_kj(fky)
fkXWk(fkx). It then follows from Lemma 8.2.4(2) that
< K-le-(A-2e)n.
Thus
(l(x)
nr3on(x) c Bl(x,
e-n(Ai-2e))
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560
F. LEDRAPPIER AND L.-S. YOUNG
forall n 2 no(x). Proceedingas before,we have
Sj(X
) =
<
logm'B(x, e-n(Al-2e))
lim sup
--n(A
n(x1 -E-
n--oo
A
?
2e-)
*lim sup-
nlogmlgon(x)
n -oo
l
hi
2?
We knowfromCorollary4.1.4 that81 < dimE1.
? i(iSi
(10.2) Proofthathi - hi_ X2
-i1). Let ? > 0 be given.Choose
(? 1)-charts{?x) and measurablepartitions(i, 1 < i < u, as in (9.1). In
particularthen, each (i is an increasingpartitionsubordinateto Wi and
4l > * > (*. For each i, choose a systemof conditionalmeasuresassociated
among the
with (. We now fix i, 2 < i < u, and focus on the relationships
invariantsdefinedon Wi and Wi1.
Thereexistsa partition9 withH(9) < xo and a measurable
are
functionno: M -- N such thatfor m-a.e. x, thefollowingfour properties
satisfiedforall n > no( x):
LEMMA
10.2.
(1)
logm- lB'-l(x,e
- hnnlog Mi
(2)
(3)
-2
n(i2e))
)-1
lgn(X)
>
l gon(x) c Bi(x,
~i(x)
(4)
-
nlogm~b2In(x)
< hi+
e-n(Xi- 2E))
e.
Proof: By definitionof Si -,
there exists a function n 1: M -* N such that
property(1) is satisfiedform-a.e.x and n ? nl(x). By (9.2), we can choose d
and n2: M -- N suchthatform-a.e.x and
n?
n2(x), - 1/nlogm7- 'V'- (x,
n, ')
By 9.3.2 thereare a partition9 and a functionn': M
> hi -1 -*
N suchthatform-a.e.x
and n ? n'(x),
So
n(X)
n (ij1x)
c Vi
(x,
n. ?l).
By 10.1.1 thereis a partitionR and a functionn3: M -* N suchthatform-a.e.x
and n > n3(x), (i(x) n _Wn(x) c Bi(x, e- n(Xi 2E)). (Our argumentin 10.1 is
given for i = 1 but this is clearly valid for all i = 1,.. ., u.) Let be= 2 V R.
Then properties(2) and (3) are satisfied
forn > n2(x), n'(x) and n3(x). Finally,
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METRIC ENTROPY OF DIFFEOMORPHISMS,
561
II
since 910'(x) D ( V )0(x), by 9.3.1 thereis an n4: M - R such that for
m-a.e.x and n ? n4(x) property (4) is satisfied. We put no =
E
max(n1,n2,n2,n3,n4) and thelemmais proved.
For the remainderofthissectionlet 91 and no be as in the lastlemma.Fix
No large enoughthatif F = { no < No) then mIF > 0. Considerthe following
fouradditionalconditions:
def
(i-2L = B i 1(x e n(2e))
(5)
(6)
i_(X)
1
mx'-(LfnF)
log m'Bi(x,
C
-
2eXi2e)n
1
nlog2
(8)
We claim that fora.e.x E F, thereexistsn(x) > No such thatproperties
(1)-(8) are satisfiedwithn = n(x). First,(1)-(4) are satisfiedforall x e F and
n ? No. For a.e.x e F, condition(5) holdsforall n largeenough(dependingon
of Si, fora.e.x, (7) holdsfor
x), as does condition(6). (See (4.1.2).) By definition
numberof n's.
an infinite
To completethe argumentwe now pick (and fix)x and n = n(x) > No for
hold. By (6) and (1), m'-'(L n IF) > 2mx-'(L)
which (1)-(8)
> le- (+i2E)(1-?E). But for y E L n F, we have by (2), miY-Al n(y) <
L n IFis
e- (hi-I-)n.
It followsthatthenumberofdistinctgon-atomsintersecting
bigger than m-j'(L n F)e(hi-I-)n. By (5), L c (i_l(x) C (i(x); so distinct
L n F.
L n F definedistinct(hi V 9'0n)-atoms
intersecting
Yr-atomsintersecting
F
c
Now (i(y) n glon(y) BZ(y,e-n(X1-2E)) for y E n) L, which guaranteesthat
L n F are containedin BZ(x,2en(X12E)).
all the ( i V 9U0n)-atoms
intersecting
(Here we used the fact that d'(x, y) < d- 1(x, y) for y E L.) Puttingthese
togetherwe can concludethat
m'Bz(x,
2e-n(2e))
> #{distinct(son v )-atomsintersecting
L nl F }
x minimalmeasureofsuchan atom
> IC n(Xi-2F)(8j_j+,-)en(hil-e)- . e- n(hi+0
(7), we have
Comparingthiswithproperty
-
e)(X,
2E) < (8i-1 + e)(i
-
2)
+ nlog 2 + hi - hi 1 + 2e
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562
F. LEDRAPPIER AND L.-S. YOUNG
and by (8),
hi
hi-1 + 3
of ?.
The desiredinequalityfollowsfromthe arbitrariness
11. Transversedimension:ProofofTheoremC' (continued)
In thissectionwe provethetwo remaining
relationsleadingto TheoremC'.
We have to estimatefromabove an increasein entropy,and thisis similarto
whatwe did in PartI. In fact,theprooffollowsthesame schemeas in Sections3
and 5, and we shallnot givedetailswhentheycan be obtainedby a straightforstatements
in PartI.
ward adaptationof the corresponding
of partitionsand relatednotions.We consideragain
(11.1) Construction
the familyof increasingpartitions(i (i = 1,... , u) subordinateto WZ givenby
are definedusinga set S withthe
Lemma 9.1. Rememberthatthesepartitions
Ye
be
a
finite
entropypartitionadapted to
in
(8.4).
Let
propertiesdiscussed
({J, ,re-X1-3-) (see Definition2.4.1). We requirethat b refine{S, M - S}
and anotherpartitionto be specifiedlater.Define
Qi =
Viv g+.
11.1.1. The familyof measurablepartitionsqli, i = 1,... , u, has
thefollowingproperties:
(1) Each qi is increasing,
(2) qj < qj1i5l2 < i < a,
fori = 1,... , u and m-a.e.x,
(3) qi(x) c Dx(Wxi(x) fl R(Te-X1-3Il(x)-l))
(4) hm(f,qji)= H(tiIfti) = hi, i = 1, ... .
LEMMA
of S'.
Property3) is a consequenceof Lemma 2.2.3 B ii) and our definition
Property4) is Lemma 3.2.1.
LEMMA
11.1.2. For m-a.e.xand everyy E
F. nl
qi(x),
1(y) n ni(x) = j -1(y).
Dxffxi
of (
That kDXWxl(y) fl mi(x)c q i-1(y) followsfromour characterization
in (9.1).
LEMMA
11.1.3. Form-a.e.xand everyy Ec F'I n i(x),
f - (i
- (y))
=
ji -(f1y)
nflf'(2 j(x)).
The last two lemmasare analogsof Lemmas3.3.1 and 3.3.2.
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METRIC ENTROPY OF DIFFEOMORPHISMS,
563
II
For each i > 2, we now definea metricon qi(x)/,qi-l form-a.e.x.As in
choiceofa positivemeasureset E.
(3.4), thismetricwilldependon thearbitrary
Let E c S n { 1 < 10) be a measurableset with mE > 0. We assumethatthe
{ E, M - E} and defineS U _0fn E -- RdimEi as fol9 refines
partition
E
lows: If x E, 9'(x) = '7(x) where '7: S -- Ryi<udim Eg is as in (8.4), and in
general 'T(x) = '7(ff(X)X) where n(x) is the smallest nonnegative integer such
that f nx e E. Since m> {E, M - E}, for each x E Un>ofnE, either
f- n( i(X)) C E or f- n(7i(x)) n E = 0. Thus foreach x, ' T 7i(x) is a lipeomorphism between q i(x) and a subset of REj<idimEjcarryingdistinct qi --elements
(See (8.4).)
to subsetsof distinctRi <idimEi-planes.
>
Forx EUn>0fnE. i 2and y,y' E= qi(x),define
=
d'(yy')
I-
~U This induces a metricon qi(x)/i_1 makingit isometric to a subset of (RdimE1, .*). For each i, i > 2, let Mihbe a systemof
where
= ......
, EU).
conditionalmeasuresassociatedwith qi and let
B(xp)
= {y
E qi(X): di(x, y) < P}.
Define:
MrXBT(XP)
.. flog
p-+
log p
P
We call the functionY, the transversedimension of qi(x)/,qi
LEMMA 11.1.4. ji
<
.
dim E,, i = 2,..., U.
This is a consequenceof the above discussionand (4.1.4).
LEMMA11.1.5. Thereexistsa numberN > 0 such thatforall i > 2, forall
x E E and forall y,y' E qi(x),
NI'x' o
(y)
-
?(y")I < dJ'(yy')
< NJ oID-[(y)
-
z
(- 1(
"
where rxiis as in 8.3.2.
This followsfromremark(3) at the end of (8.4) and the uniformLipschitz
propertyof rx? Ox-1 on S n {l < lo}. Note that N does not depend on E as
long as E c S n { ? lo}.
(11.2) Proposition5.1 revisited.
11.2. Supposef: M < is a C2 diffeomorphism
of a compact
Riemannianmanifoldand m is an ergodicBorelprobabilitymeasureon M. Let
f > 0 be given. Then thereexistsa familyof increasingpartitions%5, i =
PROPOSITION
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564
F. LEDRAPPIER AND L.-S. YOUNG
1,..., u, satisfyingLemma 11.1.1 such thatform-a.e.xand fori = 2,. .., u,
(Xi + l)jYi(x) > (1 - f3)(hi - hi-, - /A),
dimensionof qi/qinwhere Y is the transverse
i.
The proofof Proposition11.2 is parallelto thatof Proposition5.1, except
if x E E,
directionis now Xi. Moreprecisely,
thatthe exponentin thetransverse
y
then
E
fE
E
and
f'x
n(7i(fx)),
y)
dJ(fnx fy) < N2e(Ai?3E)nd'(x,
(apply Lemmas 8.3.2 and 11.1.5).
and Lemma 11.1.4 that
It followsfromthisproposition
XidimEi > hi - hi- 1
whichis the fourthinequalitystatedat the beginningof Section10.
To completethe proofof inequality(3), we need to relateY to Si - Si-,
dimensionthatdoes notdependon the ij's.
We do thisvia a notionoftransverse
(11.3) Generallemmasin dimensiontheory.We statetwo lemmasrelating
the dimensionof a measureand thoseof its decompositions.
measureon RP x Re, X theprojecLEMMA11.3.1. Let m be a probability
of m withrespectto iT. Define
tiononto RP, mt a disintegration
BP
m z g (pt
y(t) = liminf log
p)
and let 8 ? 0 be such thatat m-a.e.(s, t)
8
logmtBq(s, p)
logp
o
P----
< liminf
Then, at m-a.e.(s, t):
8
+ y(t) < liminf -ogmBP? ((st),p)
P--+0log
p
Proof.Fix a > 0; we can findN1 and a set A1 withmA1 ? 1 - a suchthat
forall (s, t) E A1 and n ? N1,
mtBq(s,2e n) < e
e
By the Lebesgue densitytheorem(see 4.1.2) we can findN2 and a set A2 with
mA2 >I - 2a such thatforall (s, t) E A2 and n ? N2,
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If (so,
to)
565
II
METRIC ENTROPY OF DIFFEOMORPHISMS,
E A2 and n > N2,we have
mBP +(sto)5
21
e-)
e-n
BP(to,
<
2ee-nS enmo
mt(Al n Bq(So e -n))m
Z- 'BP(to
0
z - l(dt)
e-n)
because foreach t, thereexistssome u(t) with(t, u(t)) E A1 n Bq(S, e-n) and
f g-f1{t}. The lemma folthus A1 n Bq(s0, e-n) fl - 1{t) C Bq(u(t),2e-n)
lows when n -x oo and a -O 0.
If instead of RP, we have no structureat all, a similarproofgives the
followinglemma:
LEMMA
11.3.2. Let (Q, v) be an abstract probabilityspace and m a
probabilityon S x R9 writtenm(dco,ds) = fm,(ds)v(dco). Let y > 0 be such
that at m-a.e. (co, s)
?
Then,
liminflog m.,(Bq(s, p))
logp
o
P--+
< liminflogm(B q(s5 P)) m-a
logp
Plo
Transverse dimension. Let ,B > 0 be given. We choose
E 10 ST:
S - RisudimEi, E and 5, i = 1,...,u, as above and let Y be the
of in/in For each i, let { m) 4be a systemofconditional
dimension
transverse
measuresassociatedwith4j (see 9.1).
Fix x E S and i, 2 < i ? u. Let 'T = 7TIji(x): ((x) - R j idimEj
Define:
R 'dimEj X R i'E and write mX= mXo07T
(11.4)
1.
y(y)
=
liminflog mX{Izi YiI
--+
p
Po 0log
Note thatby the Lipschitzpropertyof 7Tand
~~^-1
<
PI
'7T,
1ogm-ly-IY0T- fizi YJ < PI
log
--+
logpp
liminf
Yi(T1y)=
P 0
formi-a.e.y,ifwe assumeM'(Un fnE) = 1. Alsowe mayassumethat
l
(;
(1
ly) >
-
3)[hi - hi-, -
for m'-a.e.y. Applying (11.3.2) with 52 = (i(x)/lq
1]
and Rq
= RdimEt,
concludethat
-2 (1-'
)[h, -
h
-
-
]
mi-almost
everywhere.
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we can
566
F. LEDRAPPIER
AND L.-S. YOUNG
Consider now the partition of ' ( i(x)) into planes of the form
{ zi = constant).We mayassumethatat mX-a.e.y:
-8
liminflog m{z-YI<p}
--+
logp
pro
1Uo T-1({zi
logM,'^-
liminf -o
= yi, 1Z -y I
~logp
P--+0
Lemma 11.3.1 thentellsus thatform'-a.e.y:
(*)
8-
-2Y,(y)2
(1
-
-
< PI}
_
and
= ad
hi-l _ /]
whichcompletesthe proofof inequality(3) and hence ofTheoremC'.
Remark.Observethat yi as definedabove has geometricmeaningas the
dimensionof m on W'/W -1. It is not hardto see thatthisquantityis in fact
independentofourchoiceof S and 'i. It thenfollowsfrom(*) and inequality(2)
that
Yi =
-a i-1
12. Volumelemmaand dimensionofinvariantmeasures:
ProofofTheoremF
(12.1) Somereductions.We do not assumein thissectionthatthe measure
m is ergodic.Instead,we divideM intoa countablenumberof invariantsetson
are moreor less constant.By (4.1.2) it
each one of whichall relevantfunctions
suffices
to considerthesesetsone at a time.
For x E F', set
Eu(x) =
eD Ei(x),
Xj(x)> 0
Es(x)
=
e
Xj(x)< 0
Ei(x),
Ec(x) = Eio(x)(x)
where Xi,(X)(x) = 0; Wu and Ws have thesame meaningas in (7.6), and Su and
Ss are the dimensionof m on Wu and Ws manifoldsas definedin (7.5). In the
proofof TheoremF we do not distinguish
betweenexponentsof the same sign.
Thus the chartsand estimatesin PartI are moresuitableforthissection.
For E > 0, integersu0, so and numbersXA, XA, S+, 8- and h, with
A - A-> 100e, let F(e, u0, s0, X+, X- , 8+, S- , h) =
{ x E F' (i) dim Eu(x) = u0, dim Es(x) = s0,
(ii) min Xi(x) 2 X+, max Xi(x) < XA,
(iii)
XA(x)> 0
E <
8+-
XA(x)< 0
-(- E < as(X)
U(X) < 8+'
(iv) h < hn(f) < h + E}.
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<
-
METRIC ENTROPY OF DIFFEOMORPHISMS,
567
II
where{ 11,} is a decomposition
of m intoergodiccomponents.Clearly,thereis a
countable numberof invariantsets of this type,the union of whichhas measure 1.
Firstreduction.We mayassumethatforsome E, u0,
and h,
mIF(E5 U05,0sX
SO,
XA+,X-,
8+,
s-
+, A- , 8+5 8- , h) =1
Let {?(} be a systemof(E, l)-charts(see Remark(3) in the appendixat the
end of PartI). We construct
an increasingpartition{u subordinateto Wu and a
decreasing
partition
{S (i.e. fis > AS) subordinate
to Ws as in (3.1). We may
assumethatthesame lo is used in bothconstructions,
but since m is notergodic,
we can onlyensurethatLemma3.1.1 holdsforboth f and f'- 1 on a set B with
mB > 1 - E'. But again by (4.1.2) it sufficesto prove the theoremfor the
inducedmeasureon setsof measurearbitrarily
close to 1.
Secondreduction.We mayassumethat mB = 1.
More specifically
we provethe following
proposition:
12.1. Suppose (i), (ii), (iii), (iv) are satisfiedform-a.e.x, and
PROPOSITION
Muand (S are respectively
increasingand decreasingpartitionssubordinateto
Wu and Ws on M. Then,givenE' > 0, thereexistsa set A withmA > 1 - 5e'
such thatforx E A,
mB
limsup log (xP
?
< 8++
-
+x)
+8c + 5( 1 +l!
where &c = dimM - u0 - so is dimEC(x) form-a.e.x.
We now pick once and for all a systemof conditionalmeasures muX
associatedwith{U. a systemofconditionalmeasuresms associatedwith{S and a
decompositionFnXof m intoergodicmeasures.Discardinga set ofmeasure0, we
thateach M- is an ergodicinvariantmeasure,forwhichmu
may assumefurther
and ms are conditionalmeasuresassociatedwith(u and As.
Let us reviewsome of the factsat our disposal.Firstrecallthatat m-a.e.x,
we have:
(1)
Hj(tujIf{u)
(2)
(3)
limsup logxp)
(3)
X
lmSup <onBSxp
log
lim+
lis 0
log
=
hn(f)
8 ,
p
(Corollary5.3),
and
?8
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568
F. LEDRAPPIER
AND L.-S. YOUNG
If S is a finiteentropypartitionand a, b > 0, then by the Shannontheorem,
Breiman-McMillan
urn - '10gm (9
(4)
n
+ b)(h + ?)
nb(x)) <(a
containingx. Morewhere pq(x) denotesthe atomof the partitionVW-f1
over,if = UU v go 0 and { mx} is a systemofconditionalmeasuresassociated
witha, thenwe claimthatat m-a.e.x,
lim - - logm
(5)
n
n
logrn
x (x) = h(f,- m)
Bu(x p)
and
.
<
_
limsup
P--+O logp
These two assertionsare consequencesof(1) and (2) and twoothergeneralfacts,
the proofsof whichwe postponeuntil(12.4).
From our resultsin Section 9 (it is not hard to adapt them to the
non-ergodiccase), it followsthatthereexista finiteentropypartitionY'1 and a
set A1 withmA 1> 1 c-' such thatat everyx in A1,
(6)
(7)
limnsup-
n
'log msx(gAl)?n(x) 2 H(Csjf')
-
2 h -E
We may assume also that hw(f.
,1)
we have
(8)
limn
? h - E on A1. Combiningthiswith(5),
mx(gl)n(x)
-log
n
? h
on A1.
In the case when thereare no zero exponents,using argumentsthat are
standardby now,we can choosea partitionS > 91 suchthatforsome suitable
choice of a, b > 0, the atomsof gnab have diameter < e-n. Proposition12.1
then followsfrom(3), (4), (6), (7), (8) and the countingargumentwe give in
partitionthe atoms
we have to further
(12.3). In the presenceof zero exponents,
theneutraldirection.This accountsfor
of Y2-nb-moreor less arbitrarily-along
the factor8 c whichappearsin the statementof Proposition12.1. This partition
procedureis describedin (12.2).
(12.2) Local picture.
X -2e).
There exist a set
a
C
and
a
constant
>
finiteentropy
an
1
E',
integerN0,
A2 with m(A2)
partitiong? refining91, withthefollowingproperty:For everyn ? N0, there
S9 > g-na suchthatif x E A2 and qn(x) is the atom of9 n
existsa partition
LEMMA 12.2.1. Let a = (1/X+- 2e), b = (1/-
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containing x, then:
(i) diamqn(x) < 2e-
569
II
METRIC ENTROPY OF DIFFEOMORPHISMS,
and
C-'e- ne -n8'mgnax
(ii) mq ((x) ?
Proof of Lemma 12.2.1. Let (v', v', v') denote the coordinates in TxM
respecting the splittingEu(x) e EC(x) e Es(x). Via the expc-map,this definesa
coordinate systemin a neighborhoodof x. Our strategyis to divide the manifold
into small sets, select a representativefromeach set and work with the coordinate systemsassociated with these points.
By Lusin's theorem,there exists a compact set A3with mA3 > 1 - (E/3)
such that on A3, the four functions Eu(x), EC(x), Es(x) and 1(x) are continuous. Let L = maxfl(x), x E1A3). Choose
8 > 0 small enough that if
d(x, x')
S,
<
then
(i)
lviji < 38, and
is defined whenever
expX2'expj(v)
(ii) expX,lexpx{ va = constant) has slope < (1/4KL) relativeto Ea(x') fora = u.
c and s. Let 20 be a partition of A3 into sets of diameter smaller than 8 and
For2
x A3, write x =U
+ xC+ xS
choose a point z(q) E q for every q eQ0.
where xia = (exp-'(x))x)' q(x)) a = u, c, s. We may assume that 'd(x, x') <
< 2d(x, x') for x' E q(x).
x- -x'I
SUBLEMMA 12.2.2. Let 8 be sufficiently small. For x E A3, define
V(xn)
=
{y
e q(x):
8
<
d(fixfiy)
If y1, Y2 e V(x . n) satisfy I 1-Y2C
I
nb <
-
forallj,
l(fiX)2
j <
< (e - n/12KL ), then 1j1 - Y21 <
Proof of Sublemma
12.2.2. First, if max(Jjj
- ',
the conclusion is clearly true.
P1, I15-s
If not, then 1yc - `1 < (1/4KL)max(ljlu
(exp['lyi),
i = 1,2, we have jvc - vc < (1/KL)max(lvu
IY1 -
'I,
4 KL I
na}.
e-
n.
Ysl)
s1). Writing v'
-
<
=
vul, Ivs - vj) by
choice
=
,
our
of &. Set wi
where 4IXis the (E, 1)-local chart at x. We have:
'J'Yi
w2c < max(jwu
wc-
and therefore either w1
-
w21
-
w2u, jw
wul, or
WI-
wl-),
w21
=
M
-
w21.
wul. This property is preserved by fx which
expands in the u direction by at least eX -2e (see Lemma 2.3.1a). Since Y1, Y2 lie
in V( x, n), we can apply this expansion [na] times and thus
Suppose
Iw1 -
w21
=
jwu
=wu
-
-
-
jwu - wul ? Sex en.
In the other case, Iw1
-
w21 =
Iws - wsl and the same argument applied to
fx- gives
WI-
w21
< Se-Xen.
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570
F. LEDRAPPIER AND L.-S. YOUNG
Changing coordinates again we obtain in both cases yl - Y21 <
small 8 gives Sublemma
2LS max(eX, e-A )e-n. The choice of a sufficiently
12.2.2.
El
We now choosethe partition9 in Lemma 12.2.1. Let 9, be the partition
at the end of (12.1). Let 92 be the partitionintoM\ A3 and 20 on A3. Using
2.4, we can finda finiteentropypartition!3 and a functionno such thatif
n > no(x), then(g3)I-b(X)C
V(x, n). We take9 = Y'1 v A2 V g3 and choose
>
No such that m { no < No} 1 - Et/3.
We now definethe partitions2n n > No on A3 (l {no < No). Since 9
refines.20 the entireset 9(x), and hence n'b(x), lies in q(x). We shall
subdividethe atoms of gpna that meet A3 n {no < No) one at a time. Fix
x E A3nA no < NO). Let Qnlgnb(X) be a partitionsuch that if y and Y2
belong in the same elementof .n then I1' - 'I < e-n/12KL. By Sublemma
12.2.2, these elementsof 2n have diameterless than 2e-. Clearly,we can
arrange to have the cardinalityof 2 I,pAn(X) to be less than Clenac where
C1 = (24KL)3c.
Let
(X: m(q(X)
nGina
(X))
r/
e
< e
- e
)m9
b(x
Then, mAn < (e'/3)e-n(l - e-e). Setting A2 = (A3 n {no < NO})\UnAn,
we have mA2 ? 1 - c' and Lemma12.2.1 is provedwithC = (3C1/e'(1 -e)).
12.1. Set a = (1/XA- 2e), b = (1/- X - 2e).
(12.3) ProofofProposition
Considera set A 2, mA2 > 1- e' an integerNo and a finiteentropypartition9
with the propertyin the statementof Lemma 12.2.1. By (4), (7) and (8) there
exista set A4 cA 1 withmA4 > 1 - 2e' and an integerN1 suchthatif x E A4
and n > N1, then
(a)
mx(Y'lb(x))
(b)
Mj(gmon"(x))
(c)
Set A5=
(d)
(e)
nb(x))
2
?
e
<
e-n
>
en(a~b)(h?2e)
h2e
N1).If z EA
n A4 N2= max(No,
diamqn(z) < 2e-n
5
and n 2 N2,then
and
mqn(z) ? C-le-nee-
e-n(a+b)(h+2e)
By (6) and (4.1.2), we can choose A6 C A5 withmA6 > 1 - 4E', and N3 ? N2
such thatif y E A6 and n 2 N3,then:
(f)
mY(A5 n Bu(y, e-n)) ?
e -n(8+e)
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571
II
METRIC ENTROPY OF DIFFEOMORPHISMS,
A C A6 withmA > 1 - 5e' and
By (3) and again (4.1.2), we can choosefinally
2
N4,
n
then:
and
N4 2N3 such thatif x EAA
(g)~
~~~M
m(AV n B s(x,e- n)) 2 le -n(8-+e
The set of pointsforwhichthe inequalityin Proposition12.1 holdswill be
thisfinalset A. Fix x in A and set = limSUpn - 1/n logmB(x,4e-n). There
manyn suchthat
existinfinitely
.
mB(x,4&-) ? e(h)
Fix such an n, assumingthat n ? N4 and 4C < en8.Considerthe number
A5 n B(x, 2e -n)}.
N = #{atomsofQn intersecting
The minimummeasureoftheseatomsimposesan immediateupperboundon N.
More precisely,from(d), (e) and (h), we get
N<
(N)
CenEena en(a+b)(h+2e)e-n(8-e).
moreinvolved.Basically,
The estimatesleadingto a lowerboundare slightly
measuresof gpona-and
conditional
we use our upper estimateof the appropriate
nb-atoms. To begin, since x E A, we have (g). Then for every y in
{S(x) n A6 n B(x, e-n), we have by (a):
< e
msx(Yg2nb(y)) = ms(go'2lnb(y))
Thus:
A6 n B(x,e-)}
#{atomsof 90)-?b intersecting
>
len(e
e)enb(h28).
and choose y e p. n A6 n B(x, e-n). We
Let us fixone of theseatomspm
have (f), and forany z in 71(y)n A5 n B(y, e-n), by (b):
<
mY(glona(z)) = mz(glona(z)) e-na(h-2?)
Thus if n( X) denotes the number of atoms of sona intersectingthe set
X n A5 n B(y,e-n), then:
n(- (y))
2
'e -n(l+
+ e)e na(h -2e)
since 7q(y)C 9?2O(y) C Pu,we have
Furthermore,
n(p.)
2
n(q7(y)).
Let p5 be one of these atoms, pu n ps is now an atom of g-nab intersecting
A5 n B(y, e-n) forsome y withd(y, x) < e-n. Therefore
(**)
N
E
2
{pu: pu A6.nB(xe-)*
?
4e
0}
n(pu)
n(8 +8 +2e)en(a+b)(h-28)
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572
F. LEDRAPPIER AND L.-S. YOUNG
Comparing(*) with(**), we completethe proofof Proposition12.1 and hence
thatof TheoremF, providedwe justify(5) and (6) in Section12.1.
(12.4) Two lemmasfrommeasuretheory.In this section,we prove two
generallemmason conditionalmeasures,leadingto (5) and (6) in Section12.1.
in a Lebesguespace (X, i), 9 is a finite
Suppose ( is a measurablepartition
entropypartitionand px is a systemof conditionalmeasuresassociatedwith (.
We denote
= -
1(9/0
logyx9(x).
The followinglemmatellsus that(1) implies(5):
12.4.1. Let f: (X, 1u) <- be an ergodicautomorphism
of the Lebesguespace (X, p), withfiniteentropyh,(f ). Let ( be an increasingmeasurable partitionsuch that H(l/f 'l)= h,(f) and let 9 be a finiteentropy
partition.Then, at p-a.e.x:
LEMMA
1
lim- I( 90/( V9?0))
=
ht(
I
f)
Proof.(Comparewith9.3.1). First:
1
_ (=
n1n~
n((Vg?D?))(X)=
and the secondtermconvergesm-a.e.and in L' towardH(9?/Ai(9? V fwhichis clearly < htL(S.f). The limit,beingconstant,is also the limitof
)
nH( 4n1V
1H(gon/(S.V900J<D)
gon( (V gZ
))-
1 H( n( (~
o)
We have
lim -H(nVg/(
( V.
O)
=
lim -H(t
V o/n) > H(l/f-
)
and
1
lim H(tln/((Vgn J) = H(I(/(f - 1lV 9+ 00))
V Then:
Let 9 be a finiteentropypartitionsuchthat = f1- V2.
V9+ 00)) < H(9/92 V 9+k)
H(((f
= h,(9VP, f) - h,(9, f)
< h,(f) -
h,,(9,
f).
Thereforethe limitin questionhas to be largerthan H(I/f- ') - h,(f) +
h,(g, f), and thiscompletesthe proof.
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METRIC ENTROPY OF DIFFEOMORPHISMS,
II
573
The followinglemmatellsus that(2) implies(6):
12.1.2. Let ,u be a probabilitymeasureon 9qq(1), ( a measurable
partitionand ,usa systemof conditionalmeasuresassociatedwiththepartition
(. Suppose A is such that
LEMMA
l
limjsup
log(s ')
form-a.e.s.
?L
Ten
log usBq(s, E)
2imsup loge
Proof Fix a
>
<?
m-a.e.
0 and call
[tBq(t,e -n) ? e-n(A+a)}.
Our hypothesisis liminfnyBn(a) = 1 forall a. Call
Bn(
An(a)
)
tt:
=
ts: juBq(s,
=
2e-n)
e-n(A+2a)?
<
We have:
pL(An(a) n Bq(t, e-n))
If tLs(An(a) n Bq(t, e-n))
and thus:
>
fus(An(a) n Bq(t, e-n))ju(ds).
=
0, thenthereexistss' E s(s) n An(a) n Bq(t, e-n)
n Bq(t , e-n))
JUS(An(G)
< 1Us,(B q(s, 2e-n))
< e- n( + 2a).
Therefore,if t E Bn(a), then
j(An(a) n Bq(t, e -n)) < e-n'juBq(t, e- n).
Using the Besicovitchcoveringlemma (see 4.1) we concludethat JU(An(a)n
Bn(a)) < e -nc(q). Therefore,u-a.e.point s belongsto a finitenumbersof sets
O
An and thiscompletestheproof.
UNIVERSITE DE PARIS VI, PARIS, FRANCE
MICHIGAN STATE UNIVERSITY, EAST LANSING, MICHIGAN
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