Annals of Mathematics
The Metric Entropy of Diffeomorphisms: Part I: Characterization of Measures Satisfying
Pesin's Entropy Formula
Author(s): F. Ledrappier and L.-S. Young
Source: Annals of Mathematics, Second Series, Vol. 122, No. 3 (Nov., 1985), pp. 509-539
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1971328 .
Accessed: 15/09/2014 23:43
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact support@jstor.org.
.
Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of
Mathematics.
http://www.jstor.org
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
Annals
ofMathematics,
122(1985),509-539
The metricentropyof diffeomorphisms
Part I: Characterizationof measures
satisfyingPesin's entropyformula
By F. LEDRAPPIERand L.-S. YOUNG*
This is the firstarticlein a two-part
seriescontainingsomeresultsin smooth
ergodictheory.We begin by givingan overviewof these results.Let M be a
and let m
compact Riemannianmanifold,let f: M -- M be a diffeomorphism,
be an finvariantBorel probabilitymeasureon M. There are variousways of
measuringthe complexityof the dynamicalsystemgeneratedby iteratingf.
Kolmogorovand Sinai introducedthe notionof metricentropy,writtenhm(f)
This is a purelymeasure-theoretic
invariantand has been studieda good deal in
abstractergodictheory(see e.g. [Ro 2]). A more geometricway of measuring
chaos is to estimatethe exponentialrate at whichnearbyorbitsare separated.
These ratesofdivergenceare givenby thegrowthratesof Dfn (the derivativeof
f composedwithitselfn times).They are called the Lyapunovexponentsof f
and are denotedin thispaperby { Xi(x): x e M, i = 1,. . ., r(x)}. (See (1.1) for
precisedefinitions.)
The relationship
betweenentropyand exponentshas been studiedbefore.A
well-knowntheoremof Margulisand Ruelle [Ru 2] says thatentropyis always
boundedabove by the sum of positiveexponents;i.e.,
hm(f) < f
( *)
Xt(x)dimEi(x)dm(x)
where dimEi(x) is the multiplicity
of Xi(x) and a'= max(a, 0). Pesin shows
that( * ) is in factan equalityif f is C2 and m is equivalentto the Riemannian
measureon M. This is sometimesknownas Pesin's formula[P 2].
Our aim here is to further
the studyof relationsof thistype.In Part I we
identifythosemeasuresforwhichequalityis attainedin (*) by theirgeometric
properties.Part II is mainlydevotedto provinga formulathat is valid forall
* Part of
this work was done while the authorswere visitingthe MathematicalSciences
ResearchInstituteof Berkeley,California.Partiallysupportedby NSF GrantNo. MCS 8120790
and AFOSR-83-0265.
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
510
F. LEDRAPPIER AND L.-S. YOUNG
invariantmeasures.This generalizedformulacontains,in some sense,the above
mentionedresultsof Margulis,Ruelle and Pesin. It involvesthe notion of
dimensionand leads to certainvolumeestimates.These resultsare announcedin
[LY].
Fromhere on our discussionwillbe confinedto the subjectof PartI.
We attemptto givea briefhistory
leadingto thisproblem.Recallthatin the
thereis an
or of AxiomA attractors,
ergodictheoryof Anosovdiffeomorphisms
properties:
invariantmeasurethatis characterized
by each of the following
(1) Equality holds in (*). (In the literaturesuch a measureis sometimes
referredto as the equilibriumstate of a certainfunctionconnectedwith the
derivative
of f.)
are absolutelycontinuous
(2) Its conditionalmeasureson unstablemanifolds
withrespectto Lebesgue.
(3) Lebesgue a.e. point in an open set is generic with respect to this
measure.
(4) This measureis approximableby measuresthat are invariantunder
suitablestochasticperturbations.
in its own
Each one of thesepropertieshas been shownto be significant
is the factthattheyare all equivalentto one
right,but perhapsmorestriking
another.Many of theseideas are due to Sinai, Bowen and Ruelle. For further
information
and detailswe referthe readerto [A], [B], [Ki], [Ru 1], [S 1], [S 2]
and [S 3].
At about the same time that progresswas being made on uniformly
hyperbolicsystems,Oseledec [0] proved an ergodictheoremforproductsof
matricespavingthe way foranalyzingdynamicalsystemsof moregeneraltypes.
Pesin then set up the machineryfortranslating
thislineartheoryof Lyapunov
exponentsinto non-linear
resultsin neighborhoods
of typicaltrajectories[P 1].
Using these new tools he began to develop an ergodic theoryfor arbitrary
diffeomorphisms
preservinga measureequivalentto Lebesgue measure[P 2].
(The entropyformulawe alludedto earlieris amongthesefirstresults.)Part of
his theoryhas sincebeen extendedand applied to dynamicalsystemspreserving
onlya Borel measure.(See e.g. [Ka] and [Ru 3]; see also [M].)
In view of thesedevelopments,
it was naturalto ask if some of the major
resultsforuniformly
hyperbolicsystemswould remainvalidin the moregeneral
frameworkof all C2 diffeomorphisms.
In particular,it was conjecturedthat
properties(1) and (2) above were equivalent.That is, givena diffeomorphism
measurem, is it the case thatPesin's formulaf
preservinga Borel probability
holds if and only if m has absolutelycontinuousconditionalmeasures on
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
I
511
unstable manifolds?It is to this conjecturethat we shall address ourselvesin
Part I.
Resultspartiallyconfirming
thisconjecturewere obtainedearlier.That (2)
implies (1) is an extensionof Pesin's theoremand was proved in [LS]. The
reverseimplicationwas proved by the firstauthor [L] under the additional
stipulationthatthe systembe at leastnonuniformly
hyperbolic.We now remove
thisassumption,confirming
the above conjecturein fullgenerality.
To carryout thislast step,we have foundit necessaryto considerexplicitly
the roleplayedby zeroexponents.Indeed,a good portionofourproofconsistsof
an attempt to obtain some controlover these nonhyperbolicparts of the
dynamicalsystem.
This paper proceedsas follows:Definitions
and precisestatements
of results
are givenin Section1. In Section2 we discussthe estimatesand constructions
associatedwithpartialnonuniform
Two partitions
are describedin
hyperbolicity.
Section 3. They are used to estimatethe variousentropies.Section4 contains
some technicallemmas.These togetherwithall the previousconstructions
are
used in Section5 to provethe mainproposition.
The proofsof the theoremsare
thencompletedin Section6.
It is our pleasure to thankour familiesand a long list of friendsand
colleagues whose interestand supportcontributedto the existenceof this
manuscript.The firstauthorwishesalso to acknowledgethe hospitalityof the
of Maryland.
University
fortheentirepaper
Standinghypotheses
A. M is a C' compactRiemannianmanifoldwithoutboundary;
B. f is a C2 diffeomorphism
of M ontoitself;
measureon M.
C. m is an finvariantBorelprobability
1. Definitions
and statements
of results
(1.1) For x E M, let TxMdenotethe tangentspace to M at x. The point x
is said to be regular if there exist numbersX1(x) > ... > r(x)(x) and a
decompositionof the tangentspace at x into TxM = El(x) ED...* Er(x)(x)
such thatforeverytangentvectorv = 0 E Ei(x),
= Xi(x)
lim -1loglDfxnvlj
and
r(x)
lim -logIJac(Dfx)I = >.X(x)dim Ei(x).
n?0
i+=
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
512
F. LEDRAPPIER AND L.-S. YOUNG
By a theoremofOseledec [0], theset F" ofregularpointsis a set offullmeasure.
The numbersXi(x), i = 1,..., r(x), are called the Lyapunovexponentsof f at
of Xi(x). The functionsx '-* r(x), Xi(x)
x; dim Ei(x) is called the multiplicity
and dim Ei(x) are invariantalongorbits,and so are constantalmosteverywhere
if m is ergodic.
( 1.2) Define
Es(x)
=
E
xi<o
Ei(x),
Ec(x) = Eio(x) whereXio(x) = 0,
Eu(x) = @ Ei(x)
Xi>o
and
Ws(x) = {y E M: limsup-log d (f'x, f'y) < 0
where d is the Riemannianmetricon M. The set WS(x) is called the stable
manifold at x. For x E F', if dimEs(x) # 0, then Ws(x) is an immersed
submanifoldof M of class C2, tangent at x to Es(x). The collection
{ Ws(x), x E F') is sometimesreferredto as the "stable foliation"of f. The
unstable manifoldat x, denotedby Wu(x), and the "unstable foliation"are
definedanalogouslyusing f-' instead of f. (See [Ru 3] or [FHY] for more
details;see also ? 2.2 and 4.2.) If W is an immersedsubmanifold
of M, thenit
inheritsa Riemannianstructure
fromM. We denotethe corresponding
Riemannian measureon W by lLw.
(1.3) Let V be the Borel a-algebraon M completedwithrespectto m.
Then (M, A2,m) is a Lebesguespace; i.e., it is isomorphic
to [0, 1] withLebesgue
measureuniona countablenumberofatoms.A measurablepartition( of M is a
partitionof M such that,up to a set of measurezero,the quotientspace M/l is
separatedby a countablenumberof measurablesets (see [Ro 1]). The quotient
space ofa Lebesguespace withitsinherited
probability
space structure
is againa
Lebesgue space. An importantpropertyof measurablepartitionsis thatassociated witheach A,thereis a canonicalsystemofconditionalmeasures:That is, for
everyx in a set of fullm-measure,
thereis a probability
measuremt definedon
((x), the elementof ( containingx. These measuresare uniquelycharacterized
(up to sets of m-measure0) by the followingproperties:If Mt is the sub-awhose elements are unions of elements of A, and A c M is a
algebra of
measurable set, then x '-* mt(A) is 4-measurable and m(A) = fmt(A)m(dx).
.
(1.4) Let ( be a measurablepartitionof M.
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
I
513
if for
Definition1.4.1. We say that t is subordinateto the Wu-foliation
m-a.e. x, we have
1. ~(x) c WU(x) and
topologyof
2. ((x) containsa neighborhood
of x open in the submanifold
WU(x).
is not a
Note that in generalthe partitioninto distinctWu-manifolds
measurablepartitionand thatin orderforthe notionof conditionalmeasureson
unstable manifoldsto make sense it is necessaryto work with measurable
partitionssubordinateto Wu.
Definition1.4.2. We say that m has absolutelycontinuousconditional
measureson unstablemanifoldsif foreverymeasurablepartitiont subordinate
to WU, mt is absolutelycontinuouswithrespectto tLWu(x)fora.e. x.
of a compact
(1.5) THEOREM A. Let f: M <-- be a C2 diffeomorphism
measurem. Thenm has
a Borelprobability
RiemannianmanifoldM preserving
absolutelycontinuousconditionalmeasureson unstablemanifoldsif and onlyif
hm(f)
=
fAX+(x)dimEj(x)m(dx)
wherea += max(a,O).
is
implication
A in PartI. The reverse
We provethe"if" partofTheorem
essentially
due to Sinaiand is provedin preciseformin [LS]. (See [M] foran
fromPartII.)
alternate
approach.Thisresultalsofollows
formula
in TheoremA is
Remark.We showin factthatwhentheentropy
that
functions
are
strictly
positive
by
thedensities
given
satisfied,
dm$/dtiWU(x)
manifolds.
6.2.)
are C' alongunstable
(See Corollary
areunionsof
ofa whoseelements
(1.6) DefineMu tobe thesub-a-algebra
RecallthatthePinsker
a-algebra
entireWU-manifolds;
analogously.
As is defined
<-,
A
thatif
of
sets
such
of
of f: (M, AR,
is
the
a
consisting
sub-a-algebra
m)
a) = 0.
a = {A,M - A), thenhm(f,
ofa compactRiemanB. Letf: M <-- be a C2 diffeomorphism
m. Then
measure
a Borelprobability
preserving
nianmanifold
THEOREM
U
t_
S _
thePinskera-algebraof f.
meansthatforeveryA1 E
For sub-a-algebras
1-2
1, 2 C a
one has A2 E -2 suchthatm(ApAA2)= 0 andviceversa.
measures
invariant
B wasshowntobe trueforsmooth
byPesin[P].
Theorem
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
514
F. LEDRAPPIER AND L.-S. YOUNG
We have chosento proveTheoremsA and B by reducingthe problemsto
their respectiveergodiccases (see ?6). While not at all essential,this line of
especiallywhere notationis concerned.
approach simplifiesthe presentation,
Thus along withthe standinghypothesesstatedat the beginningof thispaper,
we now declarethe following
forSections2-5:
Additionalhypothesis
D. m is ergodic.
2. Lyapunovchartsand relatedconstructions
We let
X+= min{fXxi > 0},
A- = max{Xi,Xi < },)
u = dimEu,
c = dimEC,
s = dimEs
and assumethat u > 0.
(2.1) Lyapunovcharts.In thissubsectionwe summarizesome resultsfrom
fromthatin [P1]. See theappendix
slightly
differs
Pesin theory.Our formulation
of thispaper formoredetails.
As always,we let d be the Riemannianmetricon M. For
(x, y,z) E RU X Rc X RsI
we define
IyIy,
Izs}
j(x,y,z)j = max{IxIx,
where 1 I I ic and Is are the Euclidean normson R11,Rc and Rs respectively.The closed diskin Ru of radiusp centeredat 0 is denotedby Ru(p) and
R(p) = Ru(p) X Rc(p) X Rs(p).
Let 0 < F < X+/100, - A-/100 be given.We shalldefinein a nonautonoof each regularpoint.
mous way a changeof coordinatesin some neighborhood
the local chartand the estimateswill varywith
The size of the neighborhood,
x E F'. Firstthereis a measurablefunction1: F' -> [1, Io) such thatl(f-x) <
efl(x). Then there is an embedding Dx: R(l(xf') -l),M with the following
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
515
I
properties:
= x; D4FD(O)takes Ru, RC and RS to Eu(x), Ec(x) and Es(x)
i) DxFO
respectively:
ii) Let fIx= Ax ' o f o Dx be the connectingmap betweenthe chartat x
and the chart at fx, defined whereverit makes sense, and let 1X =
Then
- lxo f-1 o (DXbe definedsimilarly.
e +-EIvI< jDfj(O)vI
e-EIvI < IDL(O)vI< e~jvj
forv E R',
IDLx(O)vI< e"Ejvj
forv E Rs.
forv E RC
and
iii) If L(g) denotesthe Lipschitzconstantof the functiong, then
L(fx
L(fX
-
-
Dfx(O))
Dfx (o))
?
6,
<
and
L(Dx), L(Dfx
? 1(x).
iv) For all z, z' E1R(l(x)-<), we have
K-ld((DxzDxz?)
< jz - z'j < l(x)d(DxzDxz')
forsome universalconstantK.
It followsfromii) and iii) thatthereis a numberX > 0 dependingon E and
' ( x)
the exponentssuchthatforall x c F', IfxzI< eXizIforall z c R(e
In particular,fxR(e-X-El(x)-1) C R(l(fx)-<).
From here on, we shall referto any systemof local charts{ (Dx x GEF)
and X willbe as above.
satisfying
i)-iv) as (c, 1)-charts
(2.2) Local unstablemanifoldsand centerunstablesets. For very small
> 0, let { 4)x,x c F') be a systemof (c, l)-charts.Sometimesit is necessaryto
reduce the size of our charts.Let 0 < 6 < 1 be a reductionfactor.For x c F',
define
E
S8"(x)
that is,
=
z E R(l(x)Y):
4IDXSU( x)
IDfflxof
oOxzI < 3l(f
x) lVn
0);
consistsof those points in M whose backward orbit stays (well)
inside the domainsof the chartsat f nXforall n ? 0. It is called the center
unstableset of f at x associatedwiththecharts{(Dx} and reductionfactorS. On
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
516
F. LEDRAPPIER AND L.-S. YOUNG
S"u(x), we have
fn
-fxon
o
=
Xf
f-n+1
? . *
*ff
f-1X
We nextintroducethe local unstablemanifoldat x associatedwith { ?x}
and 3. This is definedto be the componentof WU(x) n (DXR(8l(x)-') that
containsx. The (x-l-imageof thisset in the x-chartis denotedby WXU1(x).
PROPOSITION 2.2.1. Let { X4 x E F") be a systemof (e, l)-charts.
A. If O < 3 < 1 and x IF,, then
i) Wxus(x)is thegraphof a function
gX: R'(8l(x)
1)
withgx(O)= 0 and JjDgxlj<
ii)
WXUa(X)
C
Rc+s(8l(x) 1)
3;
SgC"(x).
B. If 0 < 3 < e-X-e (where X is as in (2.1)) and x E F', then
]xWS I(x) n R(1(fx)1)
-=
WfXafr)*
are standardin unstablemanifoldtheory.We
The proofsof theseassertions
referthe readerto [FHY] fordetails.
LEMMA2.2.2. If 8 < e-X-e, thenforalmosteveryx E F',
scu(x) n (D -lWu(x) = Wxu(x).
Proof. In view of ii) in Proposition 2.2.1, it suffices to show
S C( ) < >x-1 W (x) c Wxu,(x).Let z E Sgu(x) n (Fl 1W'(x) and let du denote Riemannian distance along Wu-manifolds.Since (Dxz E WU(x),
reasons,l(f -x) i> 0 as
du(f noxz, f-nx) - 0 as n -x o. But forrecurrence
-+
E
n
oo foralmosteveryx F'. This impliesthatforalmosteveryx E F' there
is some k > 0 suchthatJx
Z E Wf8-kX6(f
kx). Let
k = k(x) be thesmallest
nonnegativeintegerfor which this happens. If k > 0, then by Proposition
whichcontradictsz E Sg"(x). So k = 0, or
2.2.1 B, tk+lz
q R(81(f-k+lx)-,)'
equivalently,z E Wu,6(x).
Considernow y E F' l (Fsgu(x) where8 < '. Let WX23(y)
4'-U- be the
contains
image of the componentof Wu(y) n 4DxR(28l(x)-1).Then IxWxWu23(y)
to as a local unstable
of y in Wu(y) and is also referred
an open neighborhood
# OxWxu(y)).A reductionfactor
manifoldat y (althoughin general0yWyu6(y)
1
in
we cannotcontrolthe unstable
is takenbecause working f ax-charts
of <
manifoldsof pointswhose backwardorbitscome too close to the boundaryof
Anothertechnicalnuisanceis that If->x f fyI 0. Aside
f-nXR(l(f -x)-).
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
I
517
fromthese,we have theanalogof(2.2.1) and (2.2.2) forWju23(y)whichwe state
ones above.
below. The proofsare almostidenticalto the corresponding
as usual.
Let { Ox} be (E, 1)-charts
x
e
andy ci"
l'
every
<
Then
for
.
<68
0
Let
A.
i) Wx238(Y) is thegraphof a function
LEMMA2.2.3.
gx
U RU(261(x)')
?
with IjDgX~yI
n Scu(x),
Rc+s(261(x) ')
->
43;
ii) Wxu28(Y) C S48u(X).
For almosteveryx E F', if y E F' n Scu(x)
B. Let 8 < min(, 4e -X-).
then
andfy E Scu(fx),
i) fxWx28(y) n R(281(fx)-1)C Wx2(h)
ii) S cu(x) n 4I- lWu(y) C WxU2>(y) C Sc8(x) n 4y-LWu(y).
We remarkthatin generalScu(x) is a rathermessyset.Amongotherthings
(see Lemma 2.2.3,
we thinkof it as containingpieces oflocal unstablemanifolds
is equal to
are
Scu(x)
zero,
of
the
exponents
A.ii)). In the case where none
WX,8(X)
(2.3) Moreestimates.We listheresomeestimatesthatwillbe used in later
Let E, 1, { (Dx x E F' } and X be as before.Whenworkingin charts,we
sections.
of the point z E R(l(x) -1). Othernotations
use z U to denotethe u-coordinate
such as zS and zCUare understoodto have obviousmeaningsas well.
2.3.1. Let 8 < e-X- and let x E F'. Then
(a) If z, z' e R(Sl1(x)-<) and Iz - z'j = I - zj 1,then
LEMMA
| fXZ -
fXZ | =
It
fXZ) U -
2 e
--2E
) U1
(!XZ
z'I-Z;
(b) If u in (a) is replaced by cu, then the conclusionholds with XA
replacedby 0;
(c) If z, z' E Scu(x), then
_- 7'z-
Itfz
?
e2 IZ
Z'I
-
Proof The proofsof (a) and (b) are directapplicationsof propertiesii) and
iii) in (2.1). We prove (c): First we claim that IZ~ -z
I = Iz - z'I. Suppose
not. Then applying (a) to ft
j1Z-
,
we have
fx-
=
(z
-
e-X--2eZ
(fx z )s
-
z'I.
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
518
F. LEDRAPPIER AND L.-S. YOUNG
Inductively,thisgives
|DnZ
n(
=Z I~
_n
Z)s -
2 e-(X-+2?)n z
Z,) s
z'I
-
forall n 2 0, whichforcesone of the points1-7nzor fxnz' to leave the chart,
z, z' E Sgu(x). Now thisargumentalso appliesto fx 1z and fixz',
contradicting
since theybelongin ScU(uf 'x). It thenfollowsfrom(b) that
Iz
-
I = Izcu
-
zul
2 e-2
lJ-jz
-
'
O
whichis the desiredconclusion.
The nextlemmainvolvessome estimatesin the chartsat x and fx where
both of thesepointsbelongin F'.
LEMMA
({O} X Rcs)
2.3.2. Assume 8 < min(4, le A E). Let y E SCU(x), z =
n W,28(y) and z' = ({O} X Rc~s) ffWx28(y) Then
Iz'j < e3eIzI.
is thegraphofa functiong x, withIIDgx
Proof.SinceWxu,2A(y)
slope of fxWXj2(Y)is < 1. This gives
Iz'I
<
,ll
<
1,
the
+
I(Jxz)csI ?i(xz)ui.
From propertiesii) and iii) of (2.1), it followsthat l(fxz)csl
< _ez,so thatjz'j < e3eIZI.
I(fxz)t1l
<
(eE + e)IzI and
(2.4) Partitionsadapted to Lyapunovcharts.In orderto make use of the
geometryof Lyapunovchartsin the calculationof entropy,it is convenientto
have partitionswhose elementslie in charts.If 37 is a partitionof M, write
+ = V00Ofn . Let -, 1,{'Ix, x e 'I and X be as before,and let 0 <6 ? 1 be
a reductionfactor.
Definition 2.4.1. A measurablepartition37 is said to be adapted to
({(x 1,6) if foralmosteveryx E F', 9+(x) C DxScu(x)
LEMMA 2.4.2. Given { Ix and 0 < 6 < 1, thereis a finiteentropy
partition 3Z such thatg? is adapted to ({ Ox 6).
of b? usingan idea of Maiies [M]. Fix
Proof We outlinethe construction
some l0> 0 and let A c F' n {l(x) < 101.Assumethat mA > 0. For x e A,
let r(x) be the smallestpositiveinteger k such that fxkE A. We define
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
4:A
R
I
519
by
ifx E A
(
Io(X)
if x
81-2e-(X+E)r(x)
A.
Then 4 is definedalmost everywhereon A and log 4 is integrablesince
fArdm= 1. Let B(x, p) = { y E M: d(x, y) < p}. By Lemma 2 in [M], thereis
a partition9 with Hm(97)< xo such that 9Y(x) C B(x, 4(x)) foralmostevery
x. We claim that this 92 is adapted to ({ IDx},6). In fact,we will show that
97+(x) C OXR(81(x)1) foralmosteveryx E UnOfnA.
Firstconsiderx E A. By choiceof 92,we have 97+(x) C 97(x) C B(x, +(x))
which is containedin eDR(6l(x)-L) because 4(x)l(x) = 610-2e-(X+?)r(x)l(x)<
positiveintegersuch
8 1( x) - 1. Suppose now thatx 0 A and n > 0 is thesmallest
that f-nx e A. Then f-ng+(X) C g+(f-nX) C B(f-nX, ,(f-x)). Now
fnB(f
nx, ,(fx n)) C IY;?xR(6l(f-nx)
c OxR( l(fnx)
C (DR(81(
X)
e-(X?+)r(fX)
ele (+)r(f
nx)eXn)
1
since n < r(f-nX). Note thatthiscomputationmakessense because forevery
c R(l(f-n+kx)-le-(X?e))
This
1 < k < n, fkn R(8l(f-nx)-le-(?+,)r(f-nx))
S
completesthe proof.
3. Construction
oftwopartitions
subordinateto the W"-foliation.Giventwo par(3.1) Increasingpartitions
titions4l and 42 of M, we say that 4l refines42 (4l > 42) if at m-a.e. x E M,
41(x) C 42(x). A partitionis said to be increasingif ( > fE.
In this subsectionwe describea familyof increasingpartitionsthat are
subordinateto the unstablefoliation.Partitionsof thistypewere used by Sinai
hyperbolicsystemsand are discussedin detailin [LS].
[S1] to studyuniformly
ties:
LEMMA3.1.1. Thereexistmeasurablepartitionswiththefollowingproper1) ( is an increasingpartitionsubordinateto Wu;
2) Voc of-nt is thepartitionintopoints;
3) the biggesta-algebracontainedin fl~ofnt is s"y.
The constructionwe sketch below not only proves Lemma 3.1.1 but
withcertainadditionalpropertiesthatwillbe usefullater.
producespartitions
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
520
F. LEDRAPPIER AND L.-S. YOUNG
Let { 1y x E F') be a systemof (E, l)-localcharts
Outlineof construction.
and let lo be a numbersuch that m(l < lo) > 0. We claim that thereis a
measurableset S withthe following
properties:
(a) mS > 0;
(b) S is the disjointunionof a continuousfamilyof embeddeddisks{ Da),
whereeach Da is an open subsetof DxWxu1(xa) forsome xa E {l ? lo};
(c) For almosteveryx E M, thereis an open neighborhoodUx of x in
Wu(x) such that foreach n ? 0, eitherf-UX n S = 0 or f-UX c Da for
some a;
is irrelevantforprovingLemma 3.1.1.) There is a
(d) (This requirement
numbery such that:
i) The du-iameterof everyDa in S is less thany and
ii) If x, y e S are suchthat y E Wu(x) and du(x, y) > y, then x and y
lie on distinctDad5isks.
The existenceofan S satisfying
(a), (b) and (c) is provedin [LS] and willnot
be repeatedhere.Property
(d) is easy to arrangeby cuttingdown the d usize of
the disksin [LS]. Let ( be the partitionof M definedby
if x e Da
if x 0S;
= ( Da
OX)
M -S
((x)A
then = ? is the partitionwe desire. It is easy to verifythat it has the
propertiesstatedin Lemma3.1.1.
0
The partitionswhose constructionwe just outlinedhave the following
There is a set S satisfying(a)-(d) such that if
alternatecharacterization:
a = VtofOt S, M - 5), thenforeveryx E M, y E ((x) ifand onlyif y E a(x)
fx e S.
and d"(f-x, ff-y) < y wheneverf
For measurablepartitions'q1 and 'q2, let Hr1(q11'q2) denote the mean
then
conditionalentropyof 'q1given'q2 Note thatif 'q is an increasingpartition,
h,,(f, q) = Hm(.tlIfl).
LEMMA 3.1.2. Let 1 and
Lemma 3.1.1. Then
2
be partitionsconstructedin the proofof
41) = hrnf 42)
hnm(f,
Proof.It sufficesto show h(f,.41 V 4)
have
h(f,
v
42)
=
h(f, 4 V fnt2)
=
H(
=
H((,IfN1
V fnf21fA1 V
V
fn+fl?
=
h(f, 1)- For every n ? 1, we
fnfl
)+
)
H(42Ifi2
V f-nf.)
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
As n
-- o,
the second term goes to 0 since
V fn+ L2)
I
521
f -'n generates. We claim that
H((jjft,). Clearly,H((jIftj V ffn+L2) < H((jIftj) for
Dn = {x: (fg1)(x) C (fn2)(x)} . Since foralmosteveryx, the
du-diameterof (1(x) is finiteand du-diam(f -nl)(x) IO as n -x o, we have
H((jIftj
all n 2 0. Let
mDn -- 1. Thus forlargeenoughn, thereis a set Dn withmeasurearbitrarily
close to 1 such thatrestricted
to Dn, f1 V fn 2 = ftj. This proves
limH( lljgj v fn+ t2) 2 H((jjft,).
(3.2) Two usefulpartitions.Let { DX,x E F') be a systemof (E, l)-charts,
let X be as beforeand let t be an increasingpartitionsubordinateto Wu
constructedas in the proofof Lemma 3.1.1, with 10, S and y havingthe same
meaning as in (3.1). Let 8 < min(4,Ae -x-y/2K) and let b? be a finite
entropypartitionadapted to ({ DX, 8). We requirethat b? refine{ S, M - S}
and anotherfiniteentropypartitionto be specifiedlater.Define
%1=
V
v '7
and
712 = g+.
These two partitions
playcentralrolesin Section5. We comparetheirproperties:
1) Both % and 2 are increasingmeasurablepartitions,
2)
and
1 > '12'
3) 2(x) C FS'u(x) and p1(x) C DxoWx(x)form-a.e.x,
4) hm~f 2) = hm(f,9) and hm(f, 1) = Hm(Ift).
Properties1) and 2) and the first
halfof3) followfromthe definitions
of rq1
and 2. The secondhalfof3) is a consequenceof Lemma 2.2.2. The first
halfof
4) is straightforward.
We provethe remainingassertion:
LEMMA
3.2.1. h(f,
f)
= Hm(Ift)-
Proof As in the argumentin Lemma3.1.2, we have
h(f,,ll)
= h(f,
=
H((Ift
V fnYA+)
V fn+L+)
+ H(+
I f-nt V fg+)
where the firsttermis < H( Ift) and the second termgoes to 0 as n -3 x.
Also,usingthe factthat H(9') < x, we have
h(f,,ql) = h(f,
V 39) ? h(f, ).
to each
(3.3) QuotientStructure.Since 1 > % we can view % restricted
as
a
of
This
has
a
72(X) (written% lI%(Wx))
subpartition 2(x).
subpartition
simple
geometricdescriptionin the x-chart:recallthatsince 2(x) C FDxSu(x) where
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
522
F. LEDRAPPIERANDL.-S.YOUNG
6 < 1, for every y E S"u(x), Wx'23(y) is the graph of a function from
Ru(281(x)-<) to RC s(281(x)-L). The restrictionof these graphs to O71x2(x)
gives a natural partitionof 1X-r1q(x). The next lemma says that this corresponds
to 'qi1I'q 2(x).
3.3.1. For almosteveryx and everyy E I' f 'q2(X),
LEMMA
xWx,?28(y)n q2(X) = q1(Y).
Proof First consider z E xWx,2J(y)n 'q2(x). We will show that z E
Since 9 refines (S, M - S} and z E 9a+(y), it sufficesto show (using the
characterization of t in (3.1)) that d u(f ny, fnZ) < -y whenever ffny E S.
This is in fact true forall n ? 0, for I(D1-yy- 1-lzI < 261(x)-1 and by Lemma
2.3.1,
f n-ly
ly - ( - |z
?<
g-noj[lz
forall n
2
O.
Togethertheseimplythat du(f-ny, f -z) < K281(x)-1 < -y.The reversecontainment followsfromLemma 2.2.3, B.ii).
o
This lemma allows us to identify the quotient space 'i2(x)/ql
with a subset
of RC`s via 1(y) '*> Wxu26(y) n {O} X RC+s. The next lemma tells us that the
map ff- 1('q2(X)): f 1('q2(x)) -> p2(X) acts like a skew product with respect to
this quotient structure.
LEMMA
3.3.2. For almosteveryx and everyy E F, n
if- (q(y))
Proof.
First
f-1(q1(Y))
f- L(q 1(y)) c q I( f 1y)because
=
%W,
ml(f1y) nfl 1('12(X)).
C p2(x)
because
p1(Y)
q I is an increasing partition. That
C fl('q2(x))
f(,l(f-ly))
nq2(X)
and
C %1(Y)
followsimmediatelyfromLemma 3.3.1 and Lemma 2.2.3, B.i).
(3.4) Transversemetrics. To use the fact that all the expansion of f occurs
along the W"-foliation, we need to show that the map induced by f on
(f- 712(X))/7ql
'q2(x)/,q1 does not expand distances. To that end we define a
metric on the quotient space q2(x)/rql for m-a.e. x. This will be referredto as a
transversemetric.
As far as we know, "canonical" systemsof transversemetrics do not exist.
First we give a point-dependentdefinition:Let x E F'. From (2.2) we know
that for every y E 'rq2(x), Wxu26(y) intersects(O) x RC's at exactly one point.
We call this point z. For y' E 2(x), let z' be the corresponding point in
-k
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
I
523
{O} X RCs. Then
',)def
d(y
d'(yy'=Iz
-
z'I.
)
Note that d'(., ) inducesa metricon q2(x)/ql, but thatin general,d'(
d',(, ) forx' E q2(x), x' # x.
To rectifythis situation,we (arbitrarily)
choose a referenceplane T and
standardizeall measurementswith respect to T. Let S be the set in the
construction
of t (see (3.1)), the partitionfromwhich ql and q2 are eventually
derived.Let E C S be a measurableset with mE > 0. Furtherassumptionson
the diameterof E will be given in (4.2). Let T be the C2 embeddingof a
diskintoM. We assumethatT, theimageof I, is transverse
(c + s)-dimensional
Da
Da in exactlyone pointif Da n E 0 0. Finally,
in
S
and
to every
intersects
we requirethatthepartition9 in thedefinition
of qI and q2 refine{ E, M-E}-.
(See (3.2).)
With this setup, we can now definea metricon q2(x)/ql for every
x E Un>0fnE. First definea functionr: Un OPE - Rc+s as follows:For
x E E n Da, let
ST(X) = T-1{
T n Da )
and in general,let
ST(X)
=
q7T(f-n(x)X)
where n(x) is the smallestnonnegativeintegersuch thatf-n(x)x E E. Then for
x E Un>0f E and y, y' E 2(x), define
dT(y, y')
= I7Y - 7TY'I
where I denotesEuclideandistancein RC's
Note that since 2 = 9+, for every n ? 0 either f-n(2(x)) C E or
C Da forsome a. This
f-n(rq2(x)) n E = 0. Also,when f-nxE E, f-n(q(x))
T
guaranteesthatdX(, ) inducesa genuinemetricon each q2(x)/,1 and thatfor
x
E q2(x),
dXj = dX1.
of our choice of T. It will become
We commenthere on the arbitrariness
thendTX
clear after(4.2) thatforgivenE, if T and T' are admissibletransversals,
is
on
the
m-a.e.
x.
it
is uniformly
Furthermore,
equivalence
equivalentto dfT' for
classes of { dT }, not the metricsthemselves,that our estimatesin Section 5
depend.
Finally, what we have done here is to representM/lq1 as a subset of
metricson q2(x)/'1 thatcorrespondto
M/'q2 X R's. and to definetransverse
Euclidean distanceon RC's. This Euclideanspace geometry
playsa rolein some
in
of our averagingarguments
as we shallsee (4.1).
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
524
F. LEDRAPPIER AND L.-S. YOUNG
4. Some technicallemmas
(4.1) A coveringlemmaand some consequences.For x E R , let B(x, r)
denote the ball of radius r centeredat x. All distancesare Euclidean in this
subsection.
BESICOVITCH COVERING LEMMA (BCL, [G]). Given a set E C Rn and an
arbitraryfunction r: E -- (0, o.) with supXeEr(x) < + ox, let J?/=
{ B(x, r(x)), x E E }. Thenthereexistsa subcover_i' C _ such thatno x in Rn
lies in morethanc(n) elementsof A1', c(n) dependingonlyon n.
measureon Rn. The nexttwo lemmasare
Now let ,i be a Borelprobability
finiteBorel measstandardwhen ,i is Lebesgue. When workingwitharbitrary
lemma
instead
of
Vitali's
lemma.(This, of
we
use
Besicovitch's
ures,
covering
course,is not new.) Let g e L1([t) and define
g6(x)
I
B(x,)
=
gd
define
For g positive,we further
g * =sup g6
and
infg6.
g=
Firstwe have the maximallemmas.
LEMMA 4.1.1.
(a) For X E R.,
p(g*>X)<
c n)
(b) Let v be definedby dv = g di. Thenfor X E R+,
v(g. < X) < c(n)X.
Proof We give a proof of (a). Part (b) is proved similarly.Let
A = {g* > X}. For each x e A, choose 6(x) such that g6(x) > X; i.e.
JB(X,6(x))g d > XiB(x, 8(x)). Letting I= {B(x, 8(x)), x E A) and choosing
A'
as in BCL we have
11(A) <
E
BE=-fWl
(B)
BE
WL( Te
g
-n
g am
erw
t
e
LEMMA4.1.2. Letg E-L'(IL). Then
g,6---g almosteverywhere.
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC
ENTROPY
I
OF DIFFEOMORPHISMS.
525
Proof This is obviousif g is continuous. UsingPart(a) of Lemma 3.1.1 we
can show that the set of functionsg forwhichthis is true is normclosed in
0
V(p)L
The next lemmais usuallystatedslightlydifferently
in the literature.For
geometricreasonswe averageover balls insteadof takingconditionalexpectationswithrespectto fixedpartitions.
4.1.3. Let (X,1i) be a Lebesgue space and let n: X -" R' be a
, togeta familyofprobability
measurablemap. Disintegrate
measures{ I. t tE Rn.
Let a be a partitionof X withH.M(a)< ox. For t E Rn and A E a, define
LEMMA
gA(t) = pt(A).
Letg6 andgA
be givenby
be functionson Rn definedas above. Letg, g6 and g: X
g(x) =
g;(x)
XA(X)gA(X)
Aea
=
XA(x)gA(7rx)
E
Aea
g*(x) =
R
and
XA(X)g*(Mx)
X
AEa
Theng , -- g almosteverywhere
on X and
-
logg*d, ?< H,(a) + logc + 1
wherec = c(n) is as in BCL.
g-
Proof Firstby Lemma4.1.2 we have gA
g a.e. on X. Note also that
-
g ,
00
-
-1
oT
a.e. on Rn and hence
logg* > s)ds
p4-
logg*dp=
-
f~~IL(A
0 Aea
n{gAor
< e-s})ds.
Now
p(A nf{gAo7T < e-s})
< ,u(A).
Also,
,I(A nf{g*Ao
< e-s})
<
{g<e-s}
< c(n)e-s
gAd(I
o er1)
by Lemma4.1.1(b).
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
526
F. LEDRAPPIER AND L.-S. YOUNG
Thus
du
f-logg*
logg,~~~~AdEi
<
fmin(c(n)e-s,,l(A))
ds
Aa
-
+ log c(n) + 1
< H,(a)
[l
by a simplecalculation.
Anotherconsequenceof BCL is the followingclassicalresult(whose proof
we omit):
LEMMA4.1.4. Let ,I be a finiteBorelmeasureon R'. Then
inf ,IB(x, E) > 0
0<E<l
E
for p-a.e. x. In particular,
limsup
?< n.
log
of local unstablemanifoldswithincenterunstable
(4.2) Lipschitzproperty
whereit is essentialto assumethat
sets. This is the onlypartofourconstruction
f is C2, or at least C1+1, as opposed to CL+a for some a > 0. We will be
in chartsand all notationsare as in Section2.
workingexclusively
Let L(RU, RC's) denote the set of linear maps fromRU to RC`s with norm
Fix x E F' and let U c R(l(x<l) be such that fxUc R(l(fx) -1). To
by
simplifynotationwe writeF = fx.For z E U, define*I: L(RU, Rc+s)
<3.
DF-graph(v) = graph(*Pv)
L(Ru, RC+s), if fig is the functionfrom
where v E L(Ru, RC+s). Given g: U
F(U) to L(Ru, RC+s) given by
Ig(Fz)
=
jg(z))
of g by (F, DF). Similarideas are
then 'Pg is called the graph transform
discussedin [HP], forinstance.
In what follows,L(.) denotesthe Lipschitzconstantof the map.
LEMMA 4.2.1. Let x and U be as above, and let g: U
->
L(Ru,Rc+s)
be
Lipschitz. Then 'Pg is Lipschitzwith
L('g)
< e-X+?+6L(g)
+ 4e2Fl(x).
to verify
thatforall y E U
Proof.By ii) and iii) of(2.1), it is straightforward
and forall u, v E L(Ru, RC+s),
| *Yu-
Iyv I < e-x
+4IU
- V.
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
527
I
Now
(*)
Ig(Fy)
-
Ig(Fy')j
?I'IY(g(y))
-
+'I'*y(g(y'))
'y(g(y'))I
side of (*) is < e-A
The firsttermof the right-hand
+4-
- *'y(g(y'))l.
g(y)-g(y')l, so that
*yWgY)) - *y(g(y,))I
IFy
-
Fy'I
Iy-W
- g(y)l
<-X++4Fg(y)
< e-X++4?
*
-
IFy
-I
Fy'I
L(g) *e2F by Lemma2.3.1(c).
A simplecalculationshowsthatthesecondtermis < 41DFY- DFYI. By iii)
of (2.1), thisis ? 41(x)Iy - y'I*Thus
IIy(g(y')) - Iy (g(y'))I
41(x)y
IFy- Fy'I
IFY - Fy'I
<
0
4e2?1(x)
is
Recall from(2.2) thatforx E F' and 8 < 4, if y E S"u(x), then
Wx2,8(y)
well defined. A well-knownfact fromunstable manifoldtheoryis that if
definedby
go: Sgu(x) L(R', RC's) is implicitly
DIx(graph go(z)) = Eu(1xz),
by (fX. Df>nn) and
Oi' O, where I, is the graphtransform
then go = limn
Ois the zero functionfromf nSgu(x) to L(Rt, RC+s).
LEMMA4.2.2. Let x E I' and 8 < 4. Thengo as definedabove is Lipschitz
with
L(go) ? Dl(x)
of x.
whereD is a constantindependent
L(Ru, RC+s) be the trivialmap and using
Proof Letting0: ynSCU(x)
that
inductively
4.2.1,
can
show
we
Lemma
n-1
L (*no)
< 4e 2e
e(- X + 6)i
E
i=O
< 41(x)e3
n-1
i=O
e
(f
-
i-x)
+
This geometricseriesconvergesas n -x oo since e < X+/100. (See (2.1).) The
O
of Ino forall n passeson to go.
uniformLipschitzproperty
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
528
F. LEDRAPPIER AND L.-S. YOUNG
We have shownthaton Sgu(x), thetangentbundleto WxU2A(Y) y E Sgu(x),
is Lipschitz with Lipschitz constant< Dl(x). It follows from this that
{ WXU,2(Y)I Y E Scu(x)} is a Lipschitzlamination,meaningthat the Poincare
map between transversals(whereverit make sense) is also Lipschitz with
to 1(x).
Lipschitzconstantproportional
We will recordthiscorollaryin a convenientform,but firstwe give further
on E and T.
specification
small) positive
Let E c S n { 1 < lo} be as in (3.4) and have (arbitrarily
let us fixsome
measure.We can takeT as before,but forthesakeofdefiniteness,
in the
T
contained
Then
is
=
x Rc+s(l(wo)-l).
wo E E and let T
Xwol(?}
of 0 in EC~s(wo). We assumethe diameterof E
expo-imageof a neighborhood
is smallenoughthatforall x E E, 1)7'w0 E R(41(x<') and )7-'T is the graph
of a functionfromRc+s(1I(x)-1) to Ru(lI(x)-l) withslope < 1/100. This is
possible to arrangesince x - Ec +s( x) is continuouson ({ <? 10 and all chart
on { I < 10)
estimatesare uniform
LEMMA 4.2.3. Let E and T be as above. Thenthereis a numberN = N(10)
such thatforall x E E,
- dx(., -) 'd
T(,
) <Ndf(*,
)
Proof:In the chartat x, we definethe Poincaremap
9: ({O} X Rc's) fl { wu(2y)
y E Sgu(x)}
O
x T
by sliding along Wxu28(y).Lemma 4.2.2 tells us that there is a number D'
independentof x suchthat
L(O),L(0-1)
< D'l(x).
of
thepointsofintersection
Thus if y, y' E 'q2(x),and z and z' are respectively
Dwith
then
1T,
and
WX'23(y)
Wxu2a&(Y')
IZ -
Z'lD-1T ?
D'dX(y y')
where I - I z-'T denotesdistancein R(I(x)-')
Thereforewe have
TWy Y'
X
d<7y,
- 17Ty-
along the submanifold y'-1T.
= 10 -10 z - (DO XZ|
7Tyf|
KDfdf(ydYf=
The otherinequalityis provedsimilarly.
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
O
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
529
I
As is evidentfromtheproof,the numberN dependsonlyon the chartsand
on 10. It is independentof i1 and %2, or the choice of E and T (providedof
is as describedbefore).
coursethateverything
5. The mainproposition
developedin Sections2, 3 and 4 we now provethat
Using the machinery
the entropyof f is equal to the entropyof f withrespectto certainpartitions
subordinateto Wu.
PROPOSITION
5.1. Supposef: M <-, is a C2 diffeomorphism
of a compact
Riemannianmanifoldand m is an ergodicBorelprobabilitymeasureon M. Let
/ > 0 be given. Thenthereis an increasingmeasurablepartition(/aof thetype
discussedin (3.1) suchthat
f3(c+ s) ? (1-)[hm(f)
-hm(f,B)-/3]
is to constructby as in (3.1) and to use it to construct 1
Proof:Our strategy
and q2 as in (3.2) with hm(f,'12) 2 hm(f) - //3. Calling the conditional
measures associated with ql and iq2 { ml} and { m2} respectively,we will show
that if BT(x, p) = (y E 2(x): d j(x, y) < p1, then
3 -liminflogmx(p)
logp
0
P--+
(1-0-?
) [ hm(f,
i2)
-
hm(fAql)
-
2/3/3]
fromthisand Lemmas
form-a.e. x. The desiredconclusionfollowsimmediately
3.1.2 and 4.1.4.
We dividethe proofinto5 parts.
on (A, ql and q2. Firstfix
the specifications
(A) We startby enumerating
e > 0. We assumethatE < /3/3,X /100 and
A /100. Let ( ?X, x E IF') be
an
as describedin (2.1). Usingthesecharts,we construct
a systemof (E, 1)-charts
increasingmeasurablepartitionbyas in theproofof Lemma3.1.1 withS. lo and
y havingthe same meaningas in thatproof.Let N = N(10) be the constantin
Lemma 4.2.3. Pick E c S nf {I < o10)accordingto (3.4) and the paragraph
beforeLemma 4.2.3. Let T be chosenlikewise.We assumethatthemeasureof E
is small enough that e - -eN4m(E) < 1. Now as in (3.2), let 8S =
min( 1, 1e- - , y/2K) and let 9 be a finite entropy partition adapted to
({ 4x }, 80). We require also that 9 refine( S, M - S } and ( E, M - E } and that
hi1(f,9) ? hm(f)- E. Finally we set ql = (A V Y' and '2 = }e+. Recall that
endowed
with q11and %2 so constructed,
'q2(x)/,qlhas a nice quotientstructure
metricd T form-a.e.x.
witha transverse
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
530
F. LEDRAPPIER
AND L.-S. YOUNG
(B) Beforeproceedingwiththe main argument,we recordsome estimates
derivedfromthe resultsof (4.1). For 8 > 0, defineg, g8, g*: M -* R by
g( y)
=
my fq2)(y),
=m BT(Y8)
JBTY
q( "
2)(Y)M2(dZ)
and
g*(y) = infg3(y).
Note thatby Lemma3.3.2 g(y) is also equal to ml,(f-,q 1)(y). We leave it to the
reader to verifythe measurability
of go. (For fixed 8, one could check for
instancethat y - JBT(Y,,)mz(f -q2)(y)m(dz)
is measurableon E.)
-*
We claim that g&
g almost everywhere on M and that
J - logg dm < ox. To see this,firstconsiderone q2-element
at a time.Fix x.
Substitute(772(x), mX2)for(X, t) in Lemma4.1.3, let '7: -12(X) RC+s be the7
in (3.4) and let a = (f' q2)jIl2(x) Then g, g8 and g * as definedabove agree
with the correspondingfunctionsin (4.1.3). We can thereforeconclude that
gg, m2-a.e.and thatf - logg*dm2 ? Hm2(f'-12) + logc + 1. Integrating
over M, thisgives f - logg*dm < Hm(f-q2j'q2) + logC + 1 < oo.
(C) The purpose of this step is to study the induced action of f on
withrespectto the metricsd' , and dX. Consider
-12(x)/4(X/71
x E M. The point x willbe subjectedto a finitenumberofa.e. assumptions.
Let
...
<
<
<
<
E
E
r1
with
0
<
r2
be the successive times t when ftx
rl.
ro
ro
Note that ro is constanton 2(x). For large n and 0 < k < n, definea(x, k) as
follows:If rj < k < rj 1, then
a(x, k)
LEMMA
= BT(fkX e-
5.2. a(x, k) n (f i12)(fkx)
C
(n- r)N2
).
f la(x, k + 1).
Proof If k # rj - 1 for any j, then we have fa(x, k) nl q2(fk~lx) =
a(x, k + 1) automatically
since d
and dF+ 1 are definedby pullingback to E.
The case when k = rj - 1 forsome j reducesto the followingconsideration: Let y E E and let r > 0 be the smallestintegersuch that fry E E. Let
to showthat
Z E (f-rq2)(Y).It suffices
d yT(fry,
frZ)
<
N2erfdT(y,
Z)
of d' see (3.4).)
First we have d'(y, z) < Ndy(y,z). (For the definition
Then fori = 1,2,..., r, Lemma 2.3.2 tellsus that dfy(fiy, flz) < e'3d,(y, z).
We pick up anotherfactorof N whenconverting
back to the d'imetricat f ry.
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
(D) We now estimate m2 BT(x,ewriteas
m
2
a(x,0)
x
=
H~
mfkxa
2 fm
(n-ro(x))) =
mX2a(xO), which we can
(x, k)
2
1)
~k=o mfk+lxa(x, k+
531
I
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
pa(x,p)
x
where p = [n(l - e)]. First note that the last term < 1. For each 0 < k < p,
m
mfkXa(x, k)
mfk+lxa(x,
fk Xf (02(
k + 1)
f
X))
+ 1))
m)fkXf-'(a(xk
by invarianceof m and uniquenessof conditionalprobabilities.
This is
m2k a(x,
mfk
k)
n a(x, k))
((f-?2)(fxk)
mfkX(f-,2)(fx)
by Lemma 5.2. If g, is definedas in (B), thenthefirstquotientin (*) is equal to
[g
where
(x, n k)( fkX)]
= e-3(n-rj(x))N2i
8(x,n,k)
and
j = #tO < i < k: fix E
When I(x)
=-log
E).
the second term in (*) is equal to e I(fkx).
m2(f -i2)(x),
Thus
p-l
logm~BT(x
e-E(nro(x)))
?
-
E
k=O
p-l
logg8(x n k)(f X)
I(fkx).
k=O
-
Multiplying by - 1/n and taking lim inf on both sides of this inequality, we
have
.3*.flogmXB(xP)
- --1
,l8 lm inf
log p
o
P --*
log m2BT(x, e1-(n-ro(x)))
=
/ *liminf
?
liminf n - oo
log -- An
n -*00lg
1 n(l-)
nk=o
1
logg8(x( n k)(fkX)
where the last limit = (1 - -)H('q2lffq2)
+
lim
n- oo
> (1 -)(hm(f)
-
?
n(l -E)]
k=0
-
I(fkx)
c). Thus Proposi-
tion 5.1 is proved if we show that
limsup n
1 [ n(l -E)]
(x]
k=O
n[ngj)E
nk)(fkX)
< (1
-
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
+
2E]
532
F. LEDRAPPIER AND L.-S. YOUNG
(E) We prove this last assertion. It follows from (B) that there
is a measurable function8(x) such that if 8 < 8(x), then - logg&(x) <
- log g(x) + E.Also,since f - logg* < + so, there is a number 81 such that if
A = {8 > 81} then JM-A- log g* ? E.
We claimthatforalmosteveryx, if n is sufficiently
large,then8(x, n, k) <
<
for
all
k
n(l
e).
First
there is N(x) such that for n ? N(x),
81
#{O < i < n: fix E E} < 2n *m(E). If n ? N(x), then
8(x, n, k) = e- (n-rj)N2j
<
Since e -
-eN4m(E)
<
e-#enN2
2nm(E)
1, 8(x, n, k) is less than 81 for n sufficientlylarge. Thus
[n(l-e)]
-
E
k=O
log g(x
n k)(fkX)
[ n(1-c)]
k=O
fkxeA
[ n(l-E)]
(-lo
gg(fkx) + e) +
E
k=O
fkx 1ZA
_
logg*(fkx)
and the lim sup we wish to estimate is bounded above by
(1 E[f
Recall
now
- logg*j
logg +e+
that g(x) = ml(f- l1)(x),
so that
f - log g = hm(f, 1). This
O
completes the proof.
COROLLARY 5.3. Withthesame hypotheses
as in Proposition
5.1, if t is any
in theproofof Lemma 3.1.1, then
partitionconstructed
hm(f, t) = hm(f).
)
Proof For any /3, hm(f,
Let
/3 -O0.
=
hm
(f, (,q) where
(A is as in Proposition 5.1.
[O
6. Proof of theorems
We fill in the gaps between
as stated
in Section
the results in Section
5 and Theorems
A and B
1.
(6.1) Proofof TheoremA: the ergodiccase. We may assumethat u > 0.
(The
reader
can verify that Theorem
A is completely
trivial if u = 0.) Let
t be
an increasing partition subordinate to Wu, constructed as in the proof of Lemma
3.1.1.
= hm(f). Let {mx} = {mx} be the condiBy Corollary 5.3, Hm(Ift)
tional probabilities associated with t and let ux be the Riemannian measure on
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
533
I
W"(x). It remainsto showthat
H((Ift)
=E
dimE1 =i mx<< ILxform-a.e.x.
At
This proofis givenin Frenchin [L]. We recallthe ideas involvedforthe sake of
completeness.
Let JU = IJac(DfIEu(x))I. By Oseledec'sTheorem,f log Ju= i X+dimE .
Suppose we knowthat mx<< lx foralmosteveryx. Then dmx= pdtx almost
everywhereforsomefunctionp. This functionmustsatisfyh(X)P(y)djux(y)= 1,
and p(y)JU(f-ly)/p(f-ly) mustbe constanton ((x) by the change of variables formula.(See [LS], Proposition4.2.) From thiswe can guess thatforall
y E
(x),
00
(f ix)
defpy) _HJU
p(x)
-nJu(f-iy)
i=1
A candidateforp thenis p(y) = A(x, y)/L(x), whereL(x) = (x)A(x y) dIlx.
boundedon ((x).
Of courseall thismakessenseonlyif A(x, y) is uniformly
LEMMA
6.1.1. For almosteveryx, y - > log zA(x,y) is a Lipschitzfunction
on 4xwuW(x).It followsfromthis thatfor each x, y - z>A(x,y) is uniformly
boundedaway from0 and + xo on ((x)-
ofthe
Proof.This is a standardcalculationrelyingon theLipschitzproperty
functionsz '-4 Dfz and z -k Eu(z) (Lemma 4.2.2) and the factthatforany two
y'l (Lemma 2.3.1(a))
2Ey
points y, y' e WXui(x),I7'ny ] nytl ? -
-
So we definep as above and definea measurev on M such thatif { vx} are
the c-conditional
measuresof v,thendvx= p dpx and v coincideswith[ on ,
the biggesta-algebracontainingsetsthatare unionsof elementsof t.
LEMMA
6.1.2.
f-
log vx(f ')(x)
dm(x)
=
flog Judm.
Proof:Defineq(x) = vx(f'-')(x). Then
q(x)
A(x, y) dtx(y)
=
L(x)
L(fx)
_______
L(x)
I
Ju(x)
Since L is a positivefinite-valued
measurablefunctionwith
flog+(L(fx)/L(x))
< flog+Ju< x,
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
534
F. LEDRAPPIER AND L.-S. YOUNG
flogq
it follows that logq is integrableand
Proposition2.2.)
=-f
logJU. (See e.g. [LS],
0
of v, it is clear that v = m when restrictedto the
From the definition
a-algebra H*I The nextlemma and an inductionshow that theyare equal on
4f-. forall n ? 0 and hence theyare equal on A.
LEMMA6.1.3. f logJudm= Hm(.(ff) impliesv = m on Of 1q.
is a countablepartition.For y E ((x), define
For m-a.e. x, (f1l)Il(x)
dP I
dmvf
(f- l()(y
vy
m(f-')(y)
By the convexityof
Note that(dv/dm)lf-1Xis well definedalmosteverywhere.
log we have
,
(dv
log dm
with
f
log
dv
dm
=
)
(dv
dm<
zlog
dm
dm = 0 ifand onlyif
)
|dm=O
dv
dm rlt
--1 m-a.e.
But we knowthat f log(dv/dm)If-1-dm= 0, forLemma 6.2.2 saysthat
- flogvX(f-1t)(x)dm = flog Judm
HM(f-'tI()
-|
flog mx(f')(x)
dm.
Thus v = m on f- . This completes the proof of the ergodic case of
[
TheoremA.
Pesin's formula,
6.1.4. Let m be an ergodicmeasuresatisfying
let t be as above, and let p be the densityof mt withrespectto [ix. Thenat
positivefunctionon ((x) satisfying
m-a.e. x, p is a strictly
COROLLARY
p(y)
-
ftJU(f-ix)
In particular,logp is Lipschitzalong Wu-leaves.
Remark. It can in factbe shownthatwhen f is C2, each WU(x) is a C2
(see e.g. [PS]) and that p is C' along Wu(x).
immersedsubmanifold
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
I
535
(6.2) Proofof TheoremA. We reduce the generaltheoremto its ergodic
case. Let g be the sub-c-algebra
of 4 consisting
ofall invariantsubsetsand let D
be a measurablepartitionsuch that ,- g. (We know that such a partition
exists.)Choose a familyof conditionalprobabilitymeasuresassociatedwith D.
Call it { m, }. Then thereis an invariantset N1 C F' withmN1= 1 suchthatfor
everyx E N1, mxis invariantand ergodic.
Suppose that hm(f) = fYEi+(x)dim Ei(x)m(dx). Since
hm(f)=
fhm(f)m(dx)
and
hm1(f)< fXX+(y)dimEi(y)mx(dy)
foreveryx E N1[Ru2],thereis an invariantset N2 c N1 withmN2= 1 suchthat
foreveryx E N2,
hm(f)
=
XI+(y)dim Ei(y)mx(dy).
Let t be a measurablepartitionsubordinateto the Wu-foliation
and let
{ m) } be a familyofconditionalprobility
measuresassociatedwith(. We verifya
couple of technicalpointsbeforeapplying(6.1) to f: (M, mX) .
Firstthereis an invariantset N3 c N2 with mN3= 1 such thatforevery
x E N3, ((y) c Wu(y) and containsa neighborhood
of y in Wu(y) formi-a.e.
y, i.e. ( is indeed a partitionsubordinateto Wu withrespectto mx forevery
x E N3. More crucialis the factthat t refinesD (see e.g. [LS] Proposition2.6).
This implies that thereis a set N4 c N3 with mN4= 1 such that for every
measuresassociatedwith( in
x e N4, { in } is a familyofconditionalprobability
the space (N4nf(x), m)
Thus forf: (M, mx) -,
x e N4, we can appeal to the proofof the ergodic
and
case
concludethat myis absolutelycontinuouswithrespectto MWU(y)for
min-a.e.
y. Moreover,dmI/dIWU(Y) is as in Corollary6.1.4.
Let A = {x: m << ALWU(X)).It is straightforward
to verifythat A is a
measurableset. We have just proved that mXA= 1 for m-a.e. x. Therefore
m(A) = 1 and theproofofTheoremA is complete.
O
COROLLARY
6.2. Corollary6.1.4 is truein thenonergodic
case.
the Pinsker
u_
s
c-algebra.The otherequalityinvolving is obtainedby substituting
f by f1.
Again we firstprovethe theoremassumingthatm is ergodic.
The case u = 0 is trivial,for then I u
the Pinsker c-algebra. So
in (3.1). Then { 4f-n } n > 0 is
suppose u > 0, and let t be a partition
constructed
(6.3) Proof of Theorem B. We shall prove that
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
536
F. LEDRAPPIER AND L.-S. YOUNG
a generatingfamilyand hm(f) = Hm(Ift). A theoremin [Ro2]tellsus thatfor(
with these properties,
whichin
the Pinskera-algebracoincideswithfnl?0oar
thiscase is 'U.
When m is not ergodic,let q, D and { m.,} be as in (6.2). There is a
measurablepartitionu suchthatIAu" fu(m), and we let { mu} be a familyof
conditionalprobability
measuresassociatedwith(u*
Since g C MU, thereis an invariantset N1 with mNN= 1 such that for
everyx E N1 we have
i) mxis invariantand ergodic,
ii) (u is a measurablepartitionof D(x) and
AULo
qu(mX)
and
iii) y-4 mu is a familyof conditionalmeasuresassociatedwith (u in the
space (D(x), mj).
Let A be a set in the Pinsker u-algebra of if: (M,m)
. Since
=
=
C
{
M
1 such
N1
A,
there
is
an
invariant
0,
set
with
N2
mN2
A))
h7i(f,
that hm(ff{A,M-A))=0
for xeN2. For such x, our argumentin the
ergodiccase showsthatA n D(x) is in Mu and by ii) and iii) above, miu(A) = 0
A is in Ad".
or 1 at mi-a.e. y. Thus min(A)= 0 or 1 at m-a.e. x and therefore
This provesthatthe Pinskera-algebrais containedin 4u. The extensionof the
is easy.
othercontainment
Appendix:Lyapunovcharts
We includeherean outlineoftheconstruction
ofLyapunovchartspartlyfor
the convenienceofthereaderand partlybecause we need a littlemorethanwhat
is usuallydone. (See forinstance[P1].) All notationis as in Section2. In addition,
we let
, .) be the usual innerproductin Euclideanspace, (( , ) )x be the
inner product on TxM given by the Riemannianstructureand 11 lixbe its
norm.
corresponding
Let E > 0 be given as in Section 2. We use as our startingpoint the
followingfact,namelythatthereis a measurablefunctionC: IF' - [1, I) such
that:
1. For everyx e F' and n ? 0,
<< C(x)ei
IlDf
-nVI
fS all vE
()-A-E//2)nil/72|IvIIx
l Ex for
IIDfx7vvIgn
I< ? C(x)e(// )nIlvIIx
IIDfxIg
forall vE
Eu(x),
Es(x),
forall v e Ec(x);
? C(x)e(E/2)nhiViix
IlDfx~nvIlf?-x
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
537
I
2. Isin(Eu(x), Ec(x))I, Isin(Ec(x),ES(x))I, Isin(ES(x),Eu(x))II C(x)';
3. Qf +x ) -< e(1/3)-c(xX).
A standardtechniqueforobtaininggood estimateson Df afterone iteration
is to introducea new innerproduct K )x on TXMforeveryx E F'. Firstwe
let
KKDf xu,DffnV))f-nX
((U
v>,
KK((Df
u, Dfi v))r
E
fl=
x0
foru, v E Eu(x)
e-2n(X+ -)
n=O
foru, v E Ec(x)
-00o
E
n=O
KKDf uDfv )
foru, v E Es(x).
2n(A- + E)
e~nX~
Then we extend K , ))'x to all of TxM by demandingthat the subspaces
withrespectto KK
Et'(x), Ec(x) and Es(x) be mutuallyorthogonal
Recall that our objective here is to define a measurable function
1: F'
-*
[1, oc) and a familyof maps { 4x: R(l(xf-)
-*
M, xe
'P) so that these
coordinatechartsand theirconnectingmaps have propertiesi)-iv) as statedin
(2.1). Let Lx: TxM -* Ru+c+s be a linearmap takingEu(x), Ec(x) and Es(x)
onto Ru x {0} x {0}, {0} x Rc X {0} and {O} x {0} x Rs respectivelyand
satisfying
(Lxu, Lxv) = (1u, v))X
foreveryu, v E TxM.Setting
(X=
expx oL
one immediately
verifies
i) and ii) in (2.1).
Nextwe wantsomeboundson Iv
IK
xforv 0 0 E TxM,whereII IL is
I/ IIvII
First we considerv E E't(x) or Ec(x) or
the norm derived fromKK ,))'.
Es(x). Obviously llvii'> llvii.A directcalculationusing the propertiesof the
functionC( x) showsthat
-< C0C(X)11V11x
1v11X,
where C0 = (2EZ oe%-En)1/2. For arbitraryv E TxM, write v = vu + vc + Vs
respectingthe decompositionEU(x) e EC(x) e ES(x). It is easy to check that
llvllx< 3iIvil'. Observethat llvii
? iivull IsinSl - sin 21where 01 is the angle
between vu + vc and vs and
02
is the angle between vu and vC. Since IsinOl,
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
538
F. LEDRAPPIER AND L.-S. YOUNG
sin021 C(X)
we have
IIVIL' C0C(x)[hIvulix+ IIvCII + llvSII]
< 3CoC(x)3llvllx.
Finallywe are in a positionto showthatproperties
iii) and iv) are satisfied
if
we let l(x) = C. C(X)3 where C is a constantthe magnitudeof whichwill be
obviousfromthe nextdiscussion:
Thereis number8 > 0 suchthatforall x E IF%expxrestricted
to { lIvII< 8}
1
< 2. If C is largeenough,we are
is a diffeomorphism
withIIDexpxIH,
IIDexp
assuredthatL - 'R(l(xf1) C {jv II < 8) and thatpropertyiv)holdswithK = 6.
Since
A=
Lfxoexpj-' o foexpX o L-1,
and the second derivatives
of exp,exp- 1 and f are uniformly
boundedin x, the
Lipschitzsize of Dix is essentiallydeterminedby the normof LfX.Thus for C
sufficiently
largewe have
L(Dfx)
< 1(x)E.
Also,
IDLx(z)- DLx()I < 1(x)eIzI < E
forz E R(l(x) -). Similarconsiderations
forf-x1 guaranteethe properties
listed
in iii).
Remarks.1) The above construction
requiresonlythatf be C' +a forsome
a > 0. Needless to say, with this hypothesispropertyiii) in (2.1) has to be
modifiedaccordingly.
2) The reader can verifyeasily that if TxM= El(x) E ... EEr(x)(x) is the
to exponentsX1(x),5... ,(x)(x)X
decompositioninto subspaces corresponding
then the same trickused for Ec(x) above can be performedon each Ei(x)
separatelyto obtaina normII - II", withthe propertythatforeach v E i(X),
e(Xi-,-)jjvjj"
< JjDfxvjlfx < e(Ai+ E)lv
are extremely
Chartswiththeseproperties
usefulin PartII.
3) If the measure m is not ergodic,the constructiondescribedin this
appendixcan be carriedout on invariantsetsof the form
F(E, XA+X-)
where XA,
=
{x E F/: min Xi(x) ? XA, max Xi(x)
-2
? OOc and mF(,, XA,X) > 0.
UNIVERSITEDE PARIs VI, PARIS,FRANCE
MICHIGAN STATE UNIVERSITY, EAST LANSING, MICHIGAN
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions
)
THE METRIC ENTROPY OF DIFFEOMORPHISMS.
I
539
REFERENCES
D. V. ANOsov,Geodesic flowon closed Riemannianmanifoldswithnegativecurvature,
Proc. SteklovInst.of Maths.90 (1967).
R. BOWEN, EquilibriumStates and the Ergodic Theoryof Anosov Diffeormorphisms,
[B]
SpringerLectureNotesin Math.470 (1975).
and J-C.Yoccoz, A proofof Pesin's stable manifoldtheorem,in
[FHY] A. FATHI, M. HERMAN
GeometricDynamics,SpringerLectureNotesin Math. 1007 (1983), 117-215.
of Integralsin R , SpringerLectureNotes in Math. 481
M. DE GUzMAN, Differentiation
[G]
(1975), 2.
[HP] M. W. HIRSCH and C. PUGH,Stable manifoldsand hyperbolicsets. Proc. Sym. in Pure
Math. 14, A.M.S., Providence,RI (1970).
Publ.
[KA] A. KATOK, Lyapunovexponents,entropyand periodicorbitsfordiffeomorphisms,
Math. I.H.E.S. 51 (1980), 137-174.
of some smoothdynamicalsystems,Math.
[Ki] Yu. KIFER,On small randomperturbations
USSR-Izv 8 (1974) 1083-1107.
F. LEDRAPPIER, Proprieites
ergodiquesdes mesuresde Sinai,Publ. Math.I.H.E.S. 59 (1984),
[L]
163-188.
[LS] F. LEDRAPPIER and J.-M.SIMELCYN, A proofof the estimationfrombelow in Pesin's
entropyformula,Ergod.Th. & Dynam.Sys. 2 (1982), 203-219.
Bull. A.M.S. 11
[LY] F. LEDRAPPIER and L.-S. YOUNGThe metricentropyof diffeomorphisms,
(1984), 343-346.
Ergod.Th. and Dynam. Sys. 1 (1981), 95-102.
R. MANSE,A proofof Pesin'sformula,
[M]
numbersfor
V. I. OSELEDEC, Multiplicativeergodictheorem.Lyapunov Characteristics
[0]
Trans.Mosc. Math. Soc. 19 (1968), 197-221.
dynamicalsystems,
to non-zerocharacteristics
[P1] YA. B. PESIN, Families of invariantmanifoldscorresponding
exponents,Math.of the USSR, Izvestija10 (1978), 1261-1305.
, Characteristic
Lyapunov exponentsand smoothergodic theory,Russ. Math.
[P2]
Surveys32:4 (1977), 55-114.
in preparation.
[PS] C. PUGHand M. SHUB, ErgodicAttractors,
ideas of measuretheory,A.M.S. Transl.(1) 10 (1962),
[Rol] V. A. ROHLIN, On the fundamental
1-52.
with invariant
[Ro2] V. A. ROHLIN, Lectures on the theoryof entropyof transformations
measures,Russ. Math. Surveys22:5 (1967), 1-54.
Amer.J. Math. 98 (1976),
[RUl] D. RUELLE, A measureassociatedwith AxiomA attractors,
619-654.
, An inequalityforthe entropyof differentiable
maps, Bol. Soc. Bras. Math. 9
[RU2]
(1978), 83-87.
, Ergodic theoryof differentiable
dynamicalsystems,Publ. Math. I.H.E.S. 50
[RU3]
27-58.
(1979),
YA. G. SINAI, Classical systemswithcountableLebesgue spectrumII, A.M.S. Transl.68
[S1]
(1968), 34-88.
Func. Anal. and its Appl. 2 (1968),
and C-diffeomorphisms,
, Markovpartitions
[S2]
64-89.
, Gibbs measuresin ergodictheory,
Russ. Math. Surveys27:4 (1972), 21-69.
[S3]
[A]
This content downloaded from 128.122.114.35 on Mon, 15 Sep 2014 23:43:39 PM
All use subject to JSTOR Terms and Conditions