Entropy along expanding foliations

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by  Jiagang Yang

ENTROPY ALONG EXPANDING FOLIATIONS
JIAGANGYANG
Abstract. The (measure-theoretical) entropy of a diffeomorphism along an
expandinginvariantfoliationistherateofcomplexitygeneratedbythediffeo-
6 morphismalongtheleaves ofthe foliation. We provethatthisnumber varies
1 uppersemi-continuouslywiththediffeomorphism(C1topology),theinvariant
0 measure(weak*topology)andthefoliationitselfinasuitablesense.
2 Thishasseveralimportantconsequences. Foronething,itimpliesthatthe
set of Gibbs u-states of C1+ partiallyhyperbolic diffeomorphism isan upper
n semi-continuousfunctionofthemapintheC1topology. Anotherconsequence
a isthatthesetsofpartiallyhyperbolicdiffeomorphismswithmostlycontracting
J ormostlyexpandingcenter areC1 open.

1
2

]
S 1. Introduction
D
Partially hyperbolic diffeomorphisms were proposed by Brin, Pesin [11] and
.
h Pugh, Shub [28] independently at the early 1970’s, as an extension of the class
t ofAnosovdiffeomorphisms[3,4]. Adiffeomorphismf ispartially hyperbolic means
a
m that there exists a decomposition TM =Es⊕Ec⊕Eu of the tangent bundle TM
into three continuous invariant sub-bundles Es and Ec and Eu such that Df |Es
[ x x x
is uniform contraction, Df |Eu is uniform expansion and Df |Ec lies in between

1 them:
v kDf(x)vsk 1 kDf(x)vck 1
≤ and ≤
4 kDf(x)vck 2 kDf(x)vuk 2
0
for any unit vectors vs ∈Es and vc ∈Ec and vu ∈Eu and any x∈M.
5
5 Partially hyperbolic diffeomorphisms form an open subset of the space of Cr
0 diffeomorphisms of M, for any r ≥ 1. The stable sub-bundle Es and the unstable
. sub-bundle Eu are uniquely integrable, that is, there are unique foliations Fs and
1
0 Fu whose leaves are smooth immersed sub-manifolds of M tangent to Es and Eu,
6 respectively, at every point.
1
: 1.1. Gibbs u-states. Following Pesin and Sinai [27] and Bonatti and Viana [10]
v
(see also [9, Chapter 11]), we call Gibbs u-state any invariant probability measure
i
X whose conditional probabilities (Rokhlin [31]) along strong unstable leaves are ab-
r solutely continuous with respect to the Lebesgue measure on the leaves. In fact,
a
assuming the derivative Df is Ho¨lder continuous, the Gibbs-u state always exists,
and the densities with respect to Lebesgue measures along unstable plaques are
continuous. Moreover, the densities vary continuously with respect to the strong
unstable leaves and the set of C1+ε diffeomorphisms. As a consequence, the space
of Gibbs u-states,denoted by Gibbu(·), is compact relative to the weak-*topology
in the probability space, and varies upper semi-continuously with respect to the
diffeomorphism in C1+ε topology ([9, Remark 11.15]). In this article, we build a
similar result in the C1 topology:
Theorem A. Gibbu(·) varies upper semi-continuously among the C1+ partially
hyperbolic diffeomorphisms in the C1 topology.
Date:January22,2016.
J.Y.ispartiallysupportedbyCNPq,FAPERJ,andPRONEX..
1
2 JIAGANGYANG
1.2. Physical measures. Let f :M →M be a diffeomorphism on some compact
Riemannian manifold M. An invariant probability µ is a physical measure for f if
the set of points z ∈M for which
n−1
1
(1) δ →µ (in the weak∗ sense)
n fi(z)
j=0
X
has positive volume. This set is denoted by B(µ) and called the basin of µ. A
program for investigating the physical measures of partially hyperbolic diffeomor-
phisms was initiated by Alves, Bonatti, Viana in [5, 10], who provedexistence and
finiteness when f is either “mostly expanding” (asymptotic forward expansion) or
“mostly contracting” (asymptotic forward contraction) along the center direction.
ThesetofGibbsu-statesplaysimportantrolesinthestudyofphysicalmeasures
of partially hyperbolic diffeomorphisms. The partially hyperbolic diffeomorphisms
with mostly contracting center are the C1+ partially hyperbolic diffeomorphisms
whose Gibbs u-states have center Lyapunov exponents all negative, which were
first studied in [10]. As a corollary of the semi continuation of the set of Gibbs
u-states in the C1+ε topology, the set of partially hyperbolic diffeomorphisms with
mostly contracting center forms a C1+ε open set (see [12, 1, 33, 16]).
The notation of partially hyperbolic diffeomorphisms with mostly expanding
center was provided by Alves, Bonatti and Viana ([5]). More recently, Andersson
and Va´squez proved in [2] that a partially hyperbolic diffeomorphism with every
Gibbs u-statehas center exponents allpositive has mostly expanding center. They
alsoproposedto the latter, somewhatstronger,propertyasthe actualdefinition of
mostly expanding center. We do that in the present paper. The reason is that we
aregoingto provethatthe setofdiffeomorphismssatisfyingthis conditionis open:
that is not true for the original definition in [5], as observed in [2, Proposition A].
As a corollary of Theorem A, we obtain the C1 openness of the partially hy-
perbolic diffeomorphisms withmostly contractingcenteror with mostlyexpanding
center.
Theorem B. The sets of partially hyperbolic diffeomorphisms with mostly con-
tracting center or mostly expanding center are C1 open, that is, every C1+ partially
hyperbolic diffeomorphism with mostly contracting (resp. expanding) center admits
a C1 open neighborhood, such that every C1+ diffeomorphism in this neighborhood
has also mostly contracting (resp. expanding) center.
The only known example of diffeomorphism with mostly expanding center (in
the stronger sense we use in this paper, as explained above) is due to Man˜e [24]
(see [5] and [2, Section 6]). As an application of Theorem B, we provide a whole
new class of example:
TheoremC. Letf beaC1+ volumepreservingpartiallyhyperbolicdiffeomorphism
with one-dimensional center. Suppose the center exponent of the volume measure
is positive and f is accessible, or the center exponent is negative, but the unstable
foliation is minimal. Then f admits a C1 open neighborhood, such that every C1+
diffeomorphism in this neighborhood has mostly expanding center or mostly con-
tracting center respectively, and it admits a unique physical measure, whose basin
has full volume.
Theorem C contains abundance of systems: By Avila [6], C∞ volume preserv-
ing diffeomorphisms are C1 dense. And by Baraviera and Bonatti [7], the volume
preserving partially hyperbolic diffeomorphisms with one-dimensional center and
non-vanishing center exponent are C1 open and dense. Moreover, the subset of
accessible systems is C1 open and Ck dense for any k ≥ 1 among all partially
ENTROPY ALONG EXPANDING FOLIATIONS 3
hyperbolic diffeomorphisms with one-dimensional center direction, by Burns, Ro-
driguez Hertz, Rodriguez Hertz, Talitskayaand Ures [17, 14]; see also Theorem1.5
inNi¸tic˘aandT¨or¨ok[25]. Thediffeomorphisms suchthatboththe strongstablefo-
liationandthe strongunstable foliationareminimal arealsoquite common,which
fill an open and dense subset of volume preserving partially hyperbolic diffeomor-
phismswithone-dimensionalcenterandsomefixedcompactcenterleaf,thisfollows
from a conservative version of the results of [8].
1.3. Partialentropyforexpandingfoliations. Letf beaC1diffeomorphism,a
foliationF isf-expanding ifitisinvariantunderf andthederivativeDf restricted
to the tangent bundle of F is uniformly expanding. The partial entropy of an
invariant probability measure along an expanding foliation is a value to measure
thecomplexityofthemeasuregeneratedonthis foliation,whichwewilldescribein
Section 2.4 (see also [35]). Let µ be any invariant measure of f, denote the partial
entropy of µ along the foliation F by h (f,F). Then TheoremA is implied by the
µ
upper semi-continuation of the partial metric entropy.
Theorem D. Let f be a sequence of C1 diffeomorphisms which converge to f in
n
the C1 topology, and µ invariant measure of f which converge to an invariant
n n
measure µ of f in the weak* topology. Suppose F is an expanding foliation of f
n n
for each n and F →F in the sense of Definition 2.2, then
n
limsuph (f ,Fu)≤h (f,Fu).
µn n n µ f
The research on the regularity of entropy has a long history, one can find more
references from [37, 26, 23, 34, 36]. Our proof of Theorem D is inspired by the
dimensiontheoryofinvariantmeasures(see [38,21,22,13]), andthe inversetothe
Pesin entropy formula ([19, 20, 21]).
Outline of the work. In section 2 we give the necessary material which will be
used throughout the text. And in Section 3 we build a sequence of measurable
partitions, which is used in Section 4 to prove Theorem D. Section 5 is devoted to
the proofs of Theorems A and B. The proof of Theorem C is divided into Sections
6 and 7.
2. Preliminary
Throughoutthis subsection, let f be a diffeomorphismof manifoldM,and µ an
invariant probability measure of f.
2.1. Volumepreservingpartiallyhyperbolicdiffeomorphism. Wesayapar-
tially hyperbolic diffeomorphism is accessible if any two points can be joined by a
piecewise smooth curve such that each leg is tangent to either Eu or Es at every
point.
Pugh,Shubconjecturedin[29]that(essential)accessibilityimpliesergodicity,for
a C2 partially hyperbolic, volume preserving diffeomorphism. In [30] they showed
that this does hold under a few additional assumptions. The following result is a
special case of a general result of Burns, Wilkinson [15]:
Proposition 2.1. Every C1+ε volume preserving, accessible partially hyperbolic
diffeomorphism with one-dimensional center is ergodic.
2.2. Continuation of foliation. In this subsection we explain the convergence
between foliations that appeared in Theorem D.
LetF be a foliationofM with dimension l,that is, every leafis a l-dimensional
smoothimmersedsubmanifold. AnF-foliation box issomeimageB ofatopological
4 JIAGANGYANG
embedding Φ : Dd−l × Dl → M such that every plaque P = Φ({x} × Dl) is
x
contained in a leaf of F, and every
Φ(x,·):Dl →M,y 7→Φ(x,y)
is a C1 embedding depending continuously on x in the C1 topology. We write
D =Φ(Dd−l×{0}), and denote this foliation box by (B,Φ,D).
Take a finite cover of M consists of F-foliation boxes {(B ,Φ ,D )}k .
i i i i=1
Definition 2.2. We say a sequence of l-dimensional foliations F converge to F if:
n
• for each n, there exists a finite cover of M by F -foliation boxes
n
{Bn,Φn,D }k ;
i i i i=1
• for each 1 ≤ i ≤ k, the topological embeddings Φn : Dd−l × Dl → M
i
converge uniformly to Φ in the C0 topology;
i
• for every x∈D (1≤i≤k), Φn(x,·):Dl →M defined by
i
y 7→Φn(x,y)
isaC1 embeddingwhichconvergestoΦ(x,.)intheC1 topologyasn→∞.
2.3. Measurable partitions and mean conditional entropy. Let B be the
Borel σ-algebra on M. In this subsection, we recall the properties of measurable
partitions, more details see [31, 32].
Definition 2.3. A partition ξ of M is called measurable if there is a sequence of
finite partitions ξn, n∈N such that:
• elements of ξ are measurable (up to µ-measure 0);
n
• ξ =∨ ξ , that is, ξ is the coarsest partition which refines ξ for each n.
n n n
For a partition ξ andx∈M, we denote by ξ(x) the element of ξ which contains
x. For any measurable partition, we may define conditional measure on almost
every element.
Proposition 2.4. Let ξ be a measurable partition. Then there is a full µ-measure
subset Γ such that for every x ∈ Γ, there is a probability measure µξ defined on
x
ξ(x) satisfying:
• Let B be thesub-σ-algebra of B which consist unions of elements of ξ, then
ξ
for any measurable set A, the function x→µξ(A) is B -measurable.
x ξ
• Moreover, we have
(2) µ(A)= µξ(A)dµ(x).
x
Z
Remark 2.5. Let π be the projection M → M/ξ, and µ be the projection of
ξ ξ
measure µ onto M/ξ by the map π . Then equation (2) can be written as:
ξ
(3) µ(A)= µξ (A)dµ (B)
B ξ
Z
where B denotes the element of ξ and µξ the conditional measure on B.
B
Let ξ be a measurable partition and C ,C ,... be the elements of ξ of positive
1 2
measure. We define the entropy of the partition by
φ(µ(C )), if µ(M \∪ C )=0
(4) H (ξ)= k k k k
µ
(P∞, if µ(M \∪kCk)>0
where φ:R+ →R is defined by φ(x)=−xlogx.
ENTROPY ALONG EXPANDING FOLIATIONS 5
If ξ and η are two measurable partitions, then for every element B of η, ξ
induces a partition ξ on B. We define the mean conditional entropy of ξ respect
B
to η, denoted by H (ξ |η), as the following:
µ
(5) Hµ(ξ |η)= Hµη(ξB)dµη(B).
B
ZM/η
Definition 2.6. For measurable partitions {ζ }∞ and ζ, we write ζ ր ζ if the
n n=1 n
following conditions are satisfied:
• ζ <ζ <...;
1 2
• ∨∞ ζ =ζ.
n=1 n
Lemma 2.7. [[32, Subsection 5.11]] Suppose {η }∞ , η and ξ are measurable
n n=1
partitions, such that η րη and H (ξ |η )<∞, then
n µ 1
H (ξ |η )ցH (ξ |η).
µ n µ
Definition 2.8. Let ξ be a measurable partition, we put
h (f,ξ)=H (ξ |fξ+),
µ µ
where ξ+ =∨∞ fnξ.
n=0
Remark 2.9. A measurable partition ξ is said to be increasing if fξ < ξ. For an
increasing partition ξ,
h (f,ξ)=H (ξ |fξ).
µ µ
2.4. Expanding foliations. Throughoutthissubsection,F denotesanexpanding
foliation of f. We are going to give the precise definition of the partial metric
entropy of µ along the expanding foliation F, which depends on a special class of
measurable partitions:
Definition 2.10. We say a measurable partition ξ of M is µ-subordinate to the
F-foliation if for µ-a.e. x, we have
(1) ξ(x)⊂F(x) and ξ(x) has uniformly small diameter inside F(x);
(2) ξ(x) contains an open neighborhood of x inside the leaf F(x);
(3) ξ is an increasing partition, meaning that fξ ≺ξ.
Ledrappier,Strelcyn[20]provedthatthePesinunstablelaminationadmitssome
µ-subordinate measurable partition, the same proof can also be applied on general
expandingfoliations. Becauseinthefollowingproof,weneedauniformconstruction
of these partitions for a sequence of diffeomorphisms and measures, we providing
the construction in Section 3.
The following result (for the subordinate partitions constructed as in Section 3)
is contained in Lemma 3.1.2 of Ledrappier, Young [21]:
Lemma 2.11. Given any expanding foliation F, we have h (f,ξ )=h (f,ξ ) for
µ 1 µ 2
any measurable partitions ξ and ξ that are µ-subordinate to F.
1 2
This allows us to give the following definition:
Definition 2.12. The partial µ-entropy h (f,F) of the expanding foliation F is
µ
defined by h (f,ξ) for any µ-subordinate partition ξ constructed as in Section 3.
µ
3. Construction of subordinate measurable partitions
Let f be a sequence of diffeomorphisms which converge to f in the C1 topol-
n 0
ogy, and F an expanding foliation of f such that F converge to F . And
n n n 0
{(Bn,Φn,D )}k and {(B ,Φ ,D )}k are the foliation boxes of F and F re-
i i i i=1 i i i i=1 n
spectively as in the Definition 2.2. For simplicity, we assume that each plaque of
every foliation box has diameter less than one.
6 JIAGANGYANG
Themainaimofthissectionistoconstructmeasurablepartitionµ -subordinate
n
toexpandingfoliationF foreachninauniformway,whichisdealtinLemma3.2.
n
The construction can be divided into two steps: The first step is to choose a finite
partition A of M such that
• every element of A is contained in some foliation chart,
• the neighborhood of ∂A has small measure for µ and for every µ where
n
n≥1 (see (9)).
Let AF (resp. AFn)) be the partition such that every element is the intersection
betweenanelementofAandalocalF (resp. F )plaqueofthe correspondingfoli-
n
ation box. Then the second step is to show that ∨∞ fi(AF) (resp. ∨∞ fi(AFn))
i=0 i=0 n
is subordinate to F (resp. F ).
n
Take r ≪ 1 a Lebesgue number of the open covering {B }k , that is, there is
0 i i=1
a function
(6) I :M →{1,...,k} such that B (x)⊂B .
r0 I(x)
When n is sufficiently large, by the definition of convergencyof foliations, we have
(7) B (x)⊂Bn .
r0 I(x)
After removing a finite sequence, we assume (7) holds for every n≥1.
We need the following proposition whose proof we postpone to Appendix A
Proposition 3.1. Let {ν }∞ be a sequence of probability measures on M. Then
n n=0
for any 0 < λ < λ′ < 1 and R > 0, there is a finite partition A of M and C > 0
n
for every n∈N, such that
• the diameter of every element of A is less than R,
• ν (B (∂A)) ≤ C (λ′)i, for every n,i ∈ N, where B (∂A) denotes the r
n λi n r
neighborhood of ∂A.
Take a>1 such that for any x∈M and n≥1,
1
(8) kDf−1| k< .
n TxFn(x) a
Applying Proposition 3.1 for
• ν =µ for any n≥0, where we write µ =µ;
n n 0
• R=r ;
0
• λ= 1 and 1 <λ′ <1,
a a
we obtain a partition A and C >0 (n∈N) such that diam(A)<r and
n 0
(9) µ (B (∂A))≤C (λ′)i, for n,i≥0.
n λi n
Recall that every element of the partition AF (resp. AFn) is the intersection
between A and a local F (resp. F ) plaque in the corresponding foliation box.
n
Lemma 3.2. ∨∞ fi(AF) is subordinate to F and ∨∞ fi(AFn) is subordinate to
i=0 i=0 n
F for every n>0.
n
Proof of Lemma 3.2: We only prove the first part of this lemma, the proof of the
second part is similar.
Because µ is f invariant, by (9), we have that
∞ ∞
µ(fj(B (∂A)))= µ(B (∂A))<∞.
(1)j (1)j
a a
j=1 j=1
X X
Hence, there is a µ full measure subset Z and a function I : Z →N, such that for
every x∈Z and any j >I(x), x∈/ fj(B (∂A)), or equivalently,
(1)j
a
(10) f−j(x)∈/ B (∂A).
(1)j
a
ENTROPY ALONG EXPANDING FOLIATIONS 7
Because µ(∂A) = µ( B (∂A)) = 0, after removing a zero measure subset,
(1)j
a
we can assume that for every x ∈ Z and any j ∈ Z, fj(x) ∈/ ∂A. This hypothesis
T
implies that for every m>0, ∨m fj(AF)(x) contains an open neighborhood of x
j=0
inside the leaf F(x).
Thenthis lemma follows fromthe claimthat ∨m fj(AF)(x)=∨I(x)fj(AF)(x)
j=0 j=0
for any m≥I(x), since this implies that ∨∞ fj(AF)(x)=∨I(x)fj(AF)(x).
j=0 j=0
To prove this claim, we only need observe that every plaque of each foliation
box (B ,Φ ,D) has diameter bounded by 1. Suppose by contradiction that there
i i
is m ≥ I(x), such that ∨m fj(AF)(x) 6= ∨m+1fj(AF)(x). This implies that
j=0 j=0
fm+1(∂A)∩∨m fj(AF)(x)6=∅, i.e.,
j=0
dF(fm+1(∂A),x)≤1,
where dF denotes the distance inside a leaf of the foliation F. Then
1
d(f−(m+1)(x),∂A)≤dF(f−(m+1)(x),∂A)≤( )m+1,
a
which contradicts with (10), i.e., f−(m+1)(x)∈/ B (∂A).
(1)m+1
WeconcludetheproofofthisLemma,andhencea,completetheconstruction. (cid:3)
From the construction, it is easy to show that:
Lemma 3.3. Hµn(AFn |fn(AFn))<∞.
Proof. Bythepreviousconstruction,thediameterofeveryelementofAisbounded
by r0, which is sufficiently small. Then every element B of the partition fn(AFn)
is contained in a plaque of some foliation box. Moreover, the partition (AFn)B of
B induced by AFn coincides to the partition of B induced by A, which is uniform
finite. Because the metric entropy of a partition is bounded by the logarithm of
the number of its components, by (5), we have that
Hµn(AFn|fn(AFn))=ZM/fn(AFn)H(µn)Bfn(AFn)((AFn)B)d(µn)fn(AFn)(B)
is bounded by the logarithm of the number of the components of A. The proof is
finished. (cid:3)
4. Approach of partial entropy
In this section we give the proof of Theorem D.
For simplicity, we denote by f =f, µ =µ, F =F, and the foliation boxes
0 0 0
{(B0,Φ0,D )}k ={(B ,Φ ,D )}k .
i i i i=1 i i i i=1
Let {(Bn,Φn,D )}k be the foliation boxes of F as in the Definition 2.2, and A
i i i i=1 n
and AFn be the partitions constructed in the previous section.
4.1. First approach: Inthe subsection,weusethe partitionAFn tocalculatethe
partial metric entropy of µ along the expanding foliation F .
n n
Proposition 4.1. For every n≥0, we have
1 1
h (f ,F )=lim H (∨m f−j(AFn)|AFn)=inf H (∨m f−j(AFn)|AFn).
µn n n m µn j=1 n m µn j=1 n
Proof. By the property of conditional entropy ([32, Subsection 5.9]),
H (∨m f−j(AFn)|AFn)=H (f−1(AFn)|AFn)+···
µn j=1 n µn n
+H (f−m(AFn)|∨m−1f−j(AFn)).
µn n j=0 n
8 JIAGANGYANG
Because µ is f invariant, it follows that
n n
H (∨m f−j(AFn)|AFn)=H (AFn|f (AFn))+...
µn j=1 n µn n
+H (AFn|∨m−1fm−i(AFn))
µn j=0 n
(11)
m
= H (AFn|∨i fj(AFn)).
µn j=1 n
j=1
X
Since ∨i fj(AFn) is an increasing sequence, by Lemmas 2.7 and 3.3, we have
j=1 n
H (AFn|∨i fj(AFn))ցH (AFn|∨∞ fj(AFn))=h (f ,F ).
µn j=1 n µn j=1 n µn n n
Then by (11):
1 1
lim H (∨m f−j(AFn)|AFn)=inf H (∨m f−j(AFn)|AFn)=h (f ,F ).
m µn j=1 n m µn j=1 n µn n n
(cid:3)
4.2. Secondapproach. Inthissubsection,weusetheconditionalentropybetween
two finite partitions to approach the partial µ -entropy of the expanding foliation
n
F . We begin by the following easy observation, where Am =∨m f−jA.
n n j=0 n
Lemma 4.2. For every m>0, 1≤i≤k and x∈B ,
i
(12) ∨m f−j(AFn)(x)=Am(x)∩AFn(x).
j=0 n n
Proof. Denote by F (x) the local plaque of foliation F (x) which contains x.
n,loc n
Suppose by contradiction that there is y ∈ Am(x) such that y ∈ F (x) but
n n,loc
∨m f−j(AFn)(y)6=∨m f−j(AFn)(x). Let 0<k ≤m be the number such that
j=0 n j=0 n
• yj = fj(y) and xj = fj(x) belong to the same elements of AFn for every
0≤j <k;
• yk and xk belong to different elements of AFn.
BecauseAhassmalldiameter,AFn(yk−1)=AFn(xk−1)alsohassmalldiameter,
whichimpliesthatfn(AFn(yk−1))iscontainedinFn,loc(xk). Thenbythedefinition
of AFn:
y ∈F (x )∩A(x )=AFn(x ),
k n,loc k k k
a contradiction to the assumption. (cid:3)
4.2.1. New partitions: LetC ≤C ≤... be a sequenceoffinite partitionsonDi
i,1 i,2
such that
(A) diam(C )→0;
i,t
(B) for any i,t≥0 and any element C of Ci,t: µn(∪x∈∂CAFn(x))=0.
Foreveryi,t≥0,the partitionC inducesapartitionC˜ onthefoliationbox
i,t n,i,t
Bn:
i
C˜n,i,t ={∪x∈CAFn(x); C is an element of Ci,t}.
Remark 4.3. Because diam(C )→0, for any x∈Bi, C˜ (x)→A (x).
i,t n n,i,t Fn
For an element P of Am, suppose that I | = i (see (6) on the definition of the
n P
function I(x)), which implies that P ⊂ Bn. Then C˜ induces on P a partition
i n,i,t
P : {P ∩C˜; C˜ is an element of C˜ }. And for every m,n,t≥0,
t n,i,t
Am ={P ; where P ∈Am}
n,t t n
is a new partition of the ambient manifold M. In the following we identify some
properties for the new partition, which are important for the further proof.
Lemma 4.4. For any n,m≥0:
ENTROPY ALONG EXPANDING FOLIATIONS 9
(i) Am ր ∨m f−iAFn;
n,t j=0 n
t→∞
(ii) Am <Am <∨m f−iAFn;
n n,t j=0 n
(iii) µ (∂Am )=0.
n n,t
Proof. From the construction of the partition Am , (i) and (ii) follow immediately.
n,t
Moreover,
∂Amn,t ⊂∂Amn ∪ki=1∪C∈Ci,t ∪x∈∂CAFn(x).
By the assumption (B) above,[µn(∪x∈∂CAFn(x)) = 0. Note also that by (9),
µ (∂Am)=0. The proof is complete.
n n
(cid:3)
The following proposition is the key for the approach:
Proposition 4.5. Hµn(Amn,t |A0n,t) ց Hµn(∨mj=0fn−jAFn |AFn).
t→∞
Proof. We first claim that for t <t ,
1 2
Am ∩A0 =Am ∩A0 .
n,t1 n,t2 n,t2 n,t2
ObservethateverycomponentB ∈Am ∩A0 is the intersectionofcomponents:
n,t1 n,t2
C ∈ Am, D ∈ C˜ , E ∈ A0 and F ∈ C˜ . Because C˜ is finer than C˜ ,
n n,i,t1 n n,i,t2 n,i,t2 n,i,t1
we have D ⊃F, which implies that
B =C∩D∩E∩F =(C∩F)∩(E∩F)
is a component of the partition Am ∩A0 , as claimed.
n,t2 n,t2
Fix t =1, the above claim implies in particular that
1
H (Am |A0 )=H (Am ∨A0 |A0 )
µn n,t2 n,t2 µn n,t2 n,t2 n,t2
(13) =H (Am ∨A0 |A0 )
µn n,1 n,t2 n,t2
=H (Am |A0 ).
µn n,1 n,t2
Applying Lemma 4.4 (i) on m = 0, A0 ր AFn. Because both partitions Am
n,t n,1
and A0 are finite, H (Am |A0 )<∞. By (13) and Lemma 2.7, we have
n,1 µn n,1 n,1
H (Am |A0 )=H (Am |A0 )
µn n,t n,t µn n,1 n,t
ցH (Am |AFn)
µn n,1
=H (Am ∨AFn |AFn).
µn n,1
Then the proof follows from the claim that Am ∨AFn =∨m f−jAFn.
n,t j=0 n
It remains to prove the claim, which is a corollary of Lemma 4.4 (ii) by taking
t=1: On one hand, Am <∨m f−iAFn, which implies that
n,1 j=0 n
(14) Am ∨AFn <∨m f−iAFn ∨AFn =∨m f−jAFn.
n,1 j=0 n j=0 n
On the other hand, Am <Am . Then
n n,1
Am∨AFn <Am ∨AFn.
n n,1
By Lemma 4.2,
(15) ∨m f−jAFn <Am ∨AFn.
j=0 n n,1
We conclude the proof of the claim by (14) and (15). (cid:3)
Corollary 4.6. H (Am |A0 )>h (f ,F ).
µn n,t n,t µn n n
Proof. This is a consequence of Propositions 4.1 and 4.5. (cid:3)
10 JIAGANGYANG
4.3. Proof of Theorem D. By Proposition 4.1, for any ε > 0, there is m suffi-
0
ciently large, such that
1 ε
(16) H (∨m0 f−jAF|AF)− ≤h (f,ε)
m µ j=0 3 µ
0
By Proposition 4.5, we may further take t >0 large, such that
0
ε
(17) H (Am0 |A0 )− ≤H (∨m0 f−jAF |AF)
µ 0,t0 0,t0 3 µ j=0
Becausef convergetof intheC1 topology,andthefoliationsF convergetoF
n n
(seeDefinition2.2),eachcomponentofP ∈Am0 isconvergedbythecorresponding
component P of the partition Am0 in0 the0,Ht0ausdorff topology. Note that µ
n n,t0 n
converge to µ in the weak* topology, and by Lemma 4.4 (iii), µ(∂P )=0, hence,
0
lim µ (P )=µ(P ).
n n 0
n→∞
Because Am0 is a finite partition, we have
0,t0
(18) lim H (Am0 |A0 )=H (Am0 |A0 ).
n→∞ µn n,t0 n,t µ 0,t0 0,t0
Then there is n large enough, such that for any n≥n ,
0 0
ε
H (Am0 |A0 )− ≤H (Am0 |A0 ).
µn n,t0 n,t0 3 µ 0,t0 0,t0
Combining (16) and (17), for any n≥n , one has
0
H (Am0 |A0 )−ε≤h (f,F).
µn n,t0 n0,t0 µ
By Corollary 4.6, for any n≥n ,
0
h (f ,F )−ε≤h (f,F).
µn n n µ
Because ε can be taken arbitrarily small, we conclude the proof of Theorem D.
5. Gibbs u-states of partially hyperbolic diffeomorphisms
Let f be a C1+ partially hyperbolic diffeomorphism with invariant splitting on
the tangentbundle: T M =Es⊕Ec⊕Eu. Denote by Fu the unstable foliationof
x x x x
f which is tangent to the unstable bundle Eu, and write Jacu(x)=detDf | .
Eu
x
5.1. Preliminaries for Gibbs u-states. Denote by Gibbu(f) the set of Gibbs
u-states of f. The proofs for the following basic properties of Gibbs u-states can
be found in Bonatti, D´ıaz and Viana [9, Subsection 11.2]:
Proposition 5.1. (1) Gibbu(f)isnon-empty,weak*compactandconvex. Er-
godic components of Gibbs u-states are Gibbs u-states.
(2) The support of every Gibbs u-state is Fu-saturated, that is, it consists of
entire strong unstable leaves.
(3) For Lebesgue almost every point x in any disk inside some strong unstable
leaf, every accumulation point of 1 n−1δ is a Gibbs u-state.
n j=0 fj(x)
(4) Everyphysicalmeasureoff isaGibbsu-stateand,conversely,everyergodic
P
u-statewhosecenterLyapunovexponentsarenegativeisaphysicalmeasure.
In the following, we show an upper bound for the partial entropy along the
expanding foliation Fu, which is similar to the Ruelle inequality.
Proposition 5.2. Let µ be an invariant probability measure of f, then
h (f,Fu)≤ logJacu(x)dµ(x).
µ
Z