Geometric Expansion, Lyapunov Exponents and Foliations

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Geometric expansion, Lyapunov exponents and foliations

Radu Saghin

, Zhihong Xia

aCentre de Recerca Matematica, Apartat 50, Bellaterra, 08193, Spain bDepartment of Mathematics, Northwestern University, Evanston, IL 60208, USA Received 28 May 2007; received in revised form 2 June 2008; accepted 22 July 2008

Available online 26 July 2008

Abstract

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an application, we show examples of systems with persistent non-absolute continuous center and weak unstable foliations. This generalizes the remarkable results of Shub and Wilkinson to cases where the center manifolds are not compact.

©2008 Elsevier Masson SAS. All rights reserved.

Keywords:Partially hyperbolic; Fubini’s nightmare; Homology of foliations

1. Introduction

LetM be an n-dimensional compact Riemannian manifold. Let f ∈Diff^r(M)be aC^r diffeomorphism on M, r1. We will also consider volume-preserving diffeomorphisms in our examples. LetWbe ak-dimensional foliation ofMwithC¹leaves. We say that the foliation is invariant underf, iff maps leaves ofW to leaves. We first define volume growth off on leaves. We will also assume that the leaves are orientable. For anyx∈M, letW (x)be the leaf throughx and letWr(x)be thek-dimensional disk onW (x)centered atx, with radiusr.

For most of the paper we will assume that the leaves ofW have uniform exponential growth under the iterates off, i.e., there are constantsλ >1 andC >0 such that

df_xⁿvCλⁿv

for allx∈M, allv∈T_xW (x)and alln∈N, whereW (x)is the leaf ofW through the pointx.

Examples of these expanding foliations can be found in hyperbolic and partially hyperbolic diffeomorphisms.

A mapf ∈Diff^r(M)is said to bepartially hyperbolicif there is an invariant splitting of the tangent bundle ofM, T M=E^s⊕E^c⊕E^u, with at least two of them non-trivial, and there existα > α>1,β > β>1 andC >0,D >0, C>0,D>0 such that

✩ Research supported in part by National Science Foundation.

* Corresponding author.

E-mail addresses:[email protected] (R. Saghin), [email protected] (Z. Xia).

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.07.001

(1) E^uis uniformly expanding:

Df^k(v_u)Cα^kv_u, ∀v_u∈E^u, k∈N, (2) E^sis uniformly contracting:

Df^k(v_s)Dβ^−kv_s, ∀v_s∈E^s, k∈N, (3) E^udominatesE^c, andE^cdominatesE^s:

D(β)^−kv_cDf^k(v_c)C(α)^kv_c, ∀v_c∈E^c, k∈N.

We remark that in some papers it is allowed that the bounds for the expansion rates depend on the points.

The unstable distributionE^uis integrable and it integrates to the unstable foliation. The unstable foliation of a hy- perbolic or partially hyperbolic diffeomorphism is certainly uniformly expanding. Likewise, the stable distributionE^s of a hyperbolic or partially hyperbolic diffeomorphism can be integrated to obtain the stable foliation which is uni- formly expanding forf⁻¹.

The growth rate of an uniformly expanding foliation can be measured in several different ways. We first define the geometric growth rate. This is related to the volume growth used by Yomdin and Newhouse for the study of entropy of diffeomorphisms (see [12,4]). The difference is that we consider onlyk-dimensional disks on the leaves of the foliation. Let

χ (x, r)=lim sup

n→∞

1

nlog Vol fⁿ

Wr(x) .

χ (x, r)measures the volume growth of the disc of radiusrinW (x)centered atx. Let χ=χ (r)=sup

x∈M

χ (x, r).

Then,χis the maximum volume growth rate of the foliationWunderf.

Lemma 1.1.The volume growthχof the foliationW is independent ofrand of the Riemannian metric onM.

Proof. Letr, r>0. For anyx∈M, becauseWr(x)is compact, there existy1, y2, . . . , yl∈Wr(x)such that Wr(x)⊂

l

i=1

W_r(yi).

Then Vol

fⁿ

Wr(x)

l

i=1

Volfⁿ

W_r(yi) , so

χ (x, r) max

1ilχ (yi, r)χ (r).

This holds for everyx∈M, soχ (r)χ (r). In the same way one can prove thatχ (r)χ (r)so χ (r)=χ (r), ∀r, r>0,

soχis independent ofr.

Now suppose that we have another Riemannian metric on M, the volume of a diskD with respect to this new metric is denoted Vol(D), and the volume growth ofW_r(x)with respect to this new metric is denotedχ(x, r). There exist a constantC >0 such that

C⁻¹Vol(D)Vol(D)CVol(D), ∀D⊂M.

Then

log Vol(fⁿ(W_r(x)))

n −logC

n log Vol(fⁿ(W_r(x)))

n log Vol(fⁿ(W_r(x)))

n +logC

n

so

χ (x, r)=χ(x, r),

which means that the volume growth is independent of the Riemannian metric. 2

The geometric growth is hard to compute and its dependence on the points and on the map itself is not very clear. We will define a topological growth rate for the foliation. This will depend on the homology that the invariant foliation carries and the action induced byf on the homology. Typically this topological growth will be much easier to compute and it is a local constant for maps in Diff^r(M). It turns out, as we will show, the geometric growthχ (x, r)and topological growth are the same for foliations carrying certain homological information. As a consequence,χ (x, r)is independent ofx andr and remains the same under small perturbations. We will define this homological invariant using De Rahm currents.

The third type of growth rate for an invariant foliation is measured by the Lyapunov exponents in the tangent spaces of the leaves of the foliations. The Lyapunov exponents are positive in the leaves of an expanding foliation.

We can integrate, over the manifold, the sum of all Lyapunov exponents in the leaves and we call this integral the Lyapunov growth. We will show that, if the foliation is absolutely continuous, the Lyapunov growth is smaller than the geometric growth. As a consequence, if the Lyapunov growth is larger than the geometric growth, then the foliation must be singular.

Shub and Wilkinson showed some remarkable examples where the center foliations, whose leaves are circles, per- sistently fail to be absolutely continuous in some partially hyperbolic volume preserving diffeomorphisms (see [10]).

Moreover, every center leaf intersects a full measure set in a set of measure zero. They call these types of foliations pathological. Using our results, we will give examples of persistent pathological foliations with non-compact center leaves.

In Section 2 we define the topological growth of a foliation, we discuss about some properties and we show how to relate it to the volume growth in some situations. In Section 3 we talk about the Lyapunov growth and relate it to the volume growth in the case of absolutely continuous foliations. Section 4 contains the examples of non-absolutely continuous foliations.

As another application of the homological invariants we defined here, Hua, Saghin and Xia proved certain conti- nuity properties of topological entropy for partially hyperbolic diffeomorphisms, see [3].

2. Topological growth of a foliation

In this section, we will define the topological growth of a foliation. For this we will associate some homologies to the foliation and then we will analyze the action off_∗ on these homologies. The natural objects used to define homologies of foliations are the closed currents supported on the foliation (see [5,11]). This approach was used for example in the study of entropy of axiom A diffeomorphisms (see [9,7]). Here we will restrict our attention to some specific currents supported on the foliation, which are related to the dynamics off.

LetW be ak-dimensional foliation ofM, invariant underf. For any positive integern, anyx∈Mandr >0, we define thedynamical currentsCn(x, r):

Cn(x, r)(ω)= 1 Vol(fⁿ(W_r(x)))

fⁿ(W_r(x))

ω,

for anyk-formωonM. These currents depend onxandr. Also they depend on the Riemannian metric onM. The currentsC_n(x, r)are uniformly bounded so there must be subsequences with weak limits. LetCbe such a limit, i.e., we have a sequencen_i→ ∞such that for anyk-formωwe have limi→∞C_ni(x, r)(ω)=C(ω).

A currentCis said to beclosedif for any exactk-formω=dα, we haveC(ω)=C(dα)=0. IfCis closed, it has a homology class[C] =hC∈Hk(M,R). This homology class is non-trivial if there exist a closedk-formωsuch that C(ω)=0.

We would like to investigate the conditions under which the sub-sequential limits of the currents C_n(x, r)are closed. In general,C_n(x, r)itself is not closed. From Stokes’ theorem, we have:

C_n(x, r)(ω)= 1 Vol(fⁿ(Wr(x)))

fⁿ(Wr(x))

dα

= 1

Vol(fⁿ(W_r(x)))

∂fⁿ(Wr(x))

α

= 1

Vol(fⁿ(W_r(x)))

∂W_r(x)

(f^∗)ⁿα.

If the above sequence approaches zero as n→ ∞, then every sub-sequential limit of the currents Cn(x, r) is closed. In many situations, the volume growth offⁿ(Wr(x))is larger than the lower dimensional volume growth of its boundary. One of them is when the foliationW is one-dimensional:

Proposition 2.1.Suppose the one-dimensional foliationW is invariant underf and uniformly expanded by it. Then every limit of the sequenceC_n(x, r)is closed.

Proof. Let ω=dα be an exact 1-form on M. Then α is a 0-form, or a real valued function on M and hence

∂Wr(x)(f^∗)ⁿα is the difference of that function evaluated at the two end points of fⁿ(W_r(x)) and therefore it is uniformly bounded. On the other hand,W is uniformly expanded byf, so

nlim→∞Vol fⁿ

Wr(x)

= ∞

thusCn(x, r)(ω)→0 asn→ ∞and consequently all the limit currents are closed. 2

If the foliationW is one-dimensional, there is also an intuitive way to see the limit currents ofCn(x, r), related to the Schwartzmann asymptotic cycles (see [8]). In this casefⁿ(W_r(x))is just aC¹curve on the manifold, oriented with the same orientation of W. One can join the endpoints with anotherC¹ curve of length smaller or equal to the diameter ofM, denotedl_n, to get a piecewiseC¹closed curve, denoted byd_n. We will use the notation|γ|for the length of a piecewise C¹ curveγ. Now leth_n= ^[_|^d_dⁿ_n^]_| ∈H₁(M,R)be the homology class of d_n divided by the length ofd_n. Hereh_nwill depend of course of the choice of the curvel_nand the Riemannian metric onM. However asymptoticallyCn(x, r)(ω)is like(hn,[ω])if the foliationWis uniformly expanding:

h_n,[ω]

= 1

|d_n|

d_n

ω

= 1

Vol(fⁿ(Wr(x)))+ |ln|

fⁿ(Wr(x))

ω+

ln

ω

= Vol(fⁿ(W_r(x)))

Vol(fⁿ(W_r(x)))+ |l_n|C_n(x, r)(ω)+ 1

Vol(fⁿ(W_r(x)))+ |l_n|

ln

ω.

But becauseW is expanded byf we get that limn→∞Vol(fⁿ(Wr(x)))= ∞, and|ln|is uniformly bounded, so

nlim→∞

Vol(fⁿ(Wr(x))) Vol(fⁿ(W_r(x)))+ |l_n|=1,

nlim→∞

1

Vol(fⁿ(W_r(x)))+ |l_n|

ln

ω=0, and from this we get:

nlim→∞

h_n,[ω]

= lim

n→∞C_n(x, r)(ω).

This means that the homologies of the (closed) limit currents ofC_n(x, r)are exactly the limits of the homologiesh_n. Another situation when the limit currents are closed is when we have some convenient uniform bounds for the expansion rates onW:

Proposition 2.2.Suppose thek-dimensional foliationW is invariant underf and the following condition holds:

sup

v^k−1∈Λ^k−1T W

f_∗^Nv^k−1

v^k−1 < inf

v^k∈Λ^kT W

f_∗^Nv^k v^k

for some natural numberN >0. Then every limit current ofC_n(x, r)is closed.

Proof. Forn=aN+b,n, a, b,∈N, we have the following inequalities:

Vol fⁿ

W_r(x)

=

Wr(x)

Jacfⁿ|Wr(x)=

Wr(x)

Jacf^aN+b|Wr(x)

C₁Vol W_r(x)

· inf

v^k∈Λ^kT W

f_∗^Nv^k v^k

a

:=C₁Vol W_r(x)

β₁^a; Vol

∂fⁿ Wr(x)

=

∂Wr(x)

Jacfⁿ|∂W_r(x)=

∂Wr(x)

Jacf^aN+b|∂W_r(x)

C₂Vol

∂W_r(x)

· sup

v^k−1∈Λ^k−1T W

f_∗^Nv^k−1 v^k−1

a

:=C₂Vol

∂W_r(x) β₂^a,

for someC₁, C₂>0, and then for the exact formω=dαwe get:

C_n(x, r)(ω)=

1 Vol(fⁿ(Wr(x)))

∂fⁿ(Wr(x))

α CVol(∂fⁿ(W_r(x)))

Vol(fⁿ(W_r(x))) C β₂ β₁

a

for someC, C>0. But from the hypothesis we haveβ1> β2, so

nlim→∞Cn(x, r)(ω)=0,

and all the limit currents are closed. 2

We remark that the condition in Proposition 2.2 is an open condition, it is verified also for small perturbations off andW. This is for example the case whenf is close to a linear map on the torusTⁿandW is any of the expanding foliations close to the linear one. We will consider this case in more details later.

Definition 2.3.We say that ak-dimensional invariant foliationW carries a non-trivial homologyhC∈Hk(M,R)if for somex∈M,r >0 the currentsCn(x, r)defined above have a closed sub-sequential limitCandhC= [C] =0.

We say that ak-dimensional invariant foliationW carries a unique non-trivial homology (up to rescale) if for all x∈M,r >0, all sub-sequential limits of the currentsC_n(x, r)are closed and their homologies are unique up to scalar multiplication and are uniformly bounded away from zero.

We remark that there is no natural size for the currents supported on a foliation or for the homologies of the closed ones. So the homology can be unique only up to rescale, because its size depends on the Riemannian metric.

A closed current is non-trivial if there is a closedk-formωsuch thatC(ω)=0. The homology class of a non-trivial closed current is non-trivial. One way to show that the closed currentCis non-trivial is to show that there is a closed k-formωsuch thatωis non-degenerate onW, i.e.ωis non-degenerate onT_xW (x)for anyx∈M.

Proposition 2.4.SupposeC is a limit current ofC_n(x, r)which is also closed, and there exist a closed k-formω which is non-degenerate on thek-dimensional foliationW. ThenC is non-trivial, i.e.[C] =0.

Proof. When we have a non-degeneratek-form on the leaves ofW, by compactness of the manifold, there exists a constantc >0 such that

D

ω

cVol(D)

for any diskDon the leaves ofW, and therefore C_n(x, r)(ω)= 1

Vol(fⁿ(W_r(x)))

fⁿ(W_r(x))

ω c.

This implies thatC(ω)c >0, orCis non-trivial. 2

We remark that actually one can see from the proof that if there is a closed form non-degenerate on the foliation, then the set of homologies carried by the foliation must be bounded away from zero. The next proposition discusses the homologies carried by an invariant expanding foliation.

Proposition 2.5. LetW be an expanding invariant foliation, letH_x,r⊂H_k(M,R)be the set of homologies of the closed limit currents ofC_n(x, r), and let

H=

x∈M, r>0

H_x,r⊂H_k(M,R)

be the set of homologies carried byW. Then

(1) H_x,ris closed and bounded andRH_x,ris invariant under f_∗:H_k(M,R)→H_k(M,R);

ifH_x,ris bounded away from zero, thenRH_x,ris closed invariant.

(2) Hspans a linear space, invariant underf_∗:H_k(M,R)→H_k(M,R).

Proof. First we observe that the mapf naturally induces an action on the currents, defined by:

f_∗C(ω)=C(f^∗ω)

for anykcurrentC andkformω. Obviously, ifCis closed, thenf_∗C is closed too and [f_∗C] =f_∗[C] ∈H_k(M,R).

Obviously the set of limit currents ofCn(x, r)is closed, so the subset of closed limit currents is also closed, and thenH_x,rmust be closed. IfH_x,ris bounded away from zero, obviouslyRH_x,ris also closed.H_x,ris bounded because the sequenceC_n(x, r)is bounded.

Now we want to prove the invariance. If H_x,r = ∅then there is nothing to prove. Let a current C be a closed sub-sequential limit ofC_n(x, r), so[C] ∈H_x,r. Then

C(ω)= lim

i→∞

1 Vol(fⁿⁱ(W_r(x)))

fⁿⁱ(Wr(x))

ω,

for anyk-form onM. Therefore (f_∗C)(ω)= lim

i→∞

1 Vol(fⁿⁱ(W_r(x)))

fni(W_r(x))

f^∗ω

= lim

i→∞

1 Vol(fⁿⁱ(W_r(x)))

f⁽ⁿⁱ⁺¹⁾(Wr(x))

ω

= lim

i→∞

Vol(f⁽ⁿⁱ⁺¹⁾(Wr(x)))

Vol(fⁿⁱ(Wr(x))) · 1

Vol(f⁽ⁿⁱ⁺¹⁾(W_r(x)))

f⁽ⁿⁱ⁺¹⁾(W_r(x))

ω.

Since the ratio Vol(f⁽ⁿⁱ⁺¹⁾(W_r(x)))/Vol(fⁿⁱ(W_r(x)))is uniformly bounded, both from above and away from zero, there is a convergent subsequence. Without loss of generality, we may assume that the sequence actually converges and there is a constantλ >0 such that

ilim→∞

Vol(f⁽ⁿⁱ⁺¹⁾(Wr(x))) Vol(fⁿⁱ(Wr(x))) =λ.

This implies that

ilim→∞Cn_i+1(x, r)(ω)=1

λ(f_∗C)(ω),

for anyk-formω, sof_∗C/λis also a sub-sequential limit of the currentC_n(x, r), then λ⁻¹f_∗C

=λ⁻¹f_∗[C] ∈H_x,r

is also a homology of a closed limit current ofCn(x, r), soRHx,ris invariant.

The second statement follows immediately from the first one. This proves the proposition. 2

Now suppose that the foliationW carries a unique non-trivial homology. Leth_C= [C] ∈H_k(M,R), whereCis a limit current as defined above. The next proposition shows thath_Cis actually an eigenvector of the induced linear map byf on the homology ofM.

Proposition 2.6.LetW be ak-dimensional expanding invariant foliation that carries a unique non-trivial homol- ogyh_C. Thenh_Cis an eigenvector of the induced linear map:

f_∗:H_k(M,R)→H_k(M,R).

Proof. From Proposition 2.5 we know that H spans a subspace of H_k(M,R)invariant underf_∗. BecauseW has unique non-trivial homology, this invariant subspace must beRh_C. Then obviouslyh_Cmust be an eigenvector of the induced linear map:

f_∗:Hk(M,R)→Hk(M,R). 2

Keeping the assumption thatW has unique non-trivial homology, letλ_W be the eigenvalue off_∗corresponding to an eigenvectorh_Cassociated toW, as in Proposition 2.6. We callλ_W thetopological growthof the foliationW. We will see below that the topological growth and the volume growth are the same for a foliation that carries a unique non-trivial homology, except that the volume growth we defined here is an exponent, while the topological growth is a multiplier.

Theorem 2.7.LetW be an expanding invariant foliation that carries a unique non-trivial homologyhW. LetλW be the topological growth of the foliation. Then the volume growth defined before,

χ (x, r)=lim sup

i→∞

1 ilog

Vol fⁱ

Wr(x)

= lim

i→∞

1

ilog Vol fⁱ

Wr(x)

=lnλW, for anyx∈Mand anyr >0.

Proof. The volume of a piece of leaf in a foliation depends on the Riemannian metric defined onM. So in general, the volume does not grow uniformly with each iteration. We will rescale the volume at each step so that there will be uniform growth. Leth_W∈H_k(M,R)be a homology carried byW. Letω_W be a closedk-form such that the pairing betweenh_W and[ω_W]is nonzero. For anyx∈Mandr >0, we choose a sequence of numbersd_i,i∈Nsuch that

ilim→∞diCi(ωW)=

hW,[ωW] .

Moreover, there are numbers 0< c₁c₂ such that d_i can be chosen with c₂⁻¹d_i c₁⁻¹. Then, because of the uniqueness of homologies carried by the foliation, every limit current of{d_iC_i}i∈Nmust have the homologyh_W. This implies that the relation

ilim→∞d_iC_i(ω)=

h_W,[ω]

holds for every closed formω, so also forf^∗ω. Therefore, h_W,[f^∗ω]

= lim

i→∞

di

Vol(fⁱ(Wr(x)))

fⁱ(W_r(x))

f^∗ω

= lim

i→∞

d_i Vol(fⁱ(W_r(x)))

fⁱ⁺¹(Wr(x))

ω

= lim

i→∞

d_i⁻₊¹₁Vol(fⁱ⁺¹(W_r(x)))

d_i⁻¹Vol(fⁱ(Wr(x))) · d_i+1 Vol(fⁱ⁺¹(Wr(x)))

fⁱ⁺¹(W_r(x))

ω

= lim

i→∞

d_i⁻₊¹₁Vol(fⁱ⁺¹(W_r(x))) d_i⁻¹Vol(fⁱ(W_r(x))) ·

h_W,[ω] . Therefore

ilim→∞

d_i⁻₊¹₁Vol(fⁱ⁺¹(Wr(x))) d_i⁻¹Vol(fⁱ(W_r(x))) =

hW,[f^∗ω] /

hW,[ω]

=

f_∗h_W,[ω] /

h_W,[ω]

=λ_W. This implies that

χ (x, r)=lim sup

i→∞

1 i log

Vol fⁱ

W_r(x)

= lim

i→∞

1

i log Vol fⁱ

W_r(x)

= lim

i→∞

1 i log

d_i⁻¹Vol fⁱ

W_r(x)

= lim

i→∞

1 i log

d₀⁻¹Vol W_r(x)

· _i

j=1

d_i⁻¹Vol(fⁱ(W_r(x))) d_i⁻₋¹₁Vol(fⁱ⁻¹(W_r(x)))

= lim

i→∞

1 i

i

j=1

log d_i⁻¹Vol(fⁱ(W_r(x))) d_i⁻₋¹₁Vol(fⁱ⁻¹(W_r(x)))

=logλ_W.

Here we used the elementary fact that if limi→∞a_i=a, then

ilim→∞

1 i

i

j=1

a_i=a.

This proves the theorem. 2

Now we will discuss some situations where the above results apply and remind some related results.

The closed currents supported on a foliation (Ruelle–Sullivan currents) appear in [11,7]. They are equivalent to transverse measures to the foliation. They may not exist always, for example in the case of the center-unstable foliation of an Anosov flow. One of the most famous existence results is due to Plante, who proves that there are closed currents

(or transversal measures) if the foliation is oriented and there are leaves with sub-exponential growth (see [5]). An uniformly expanding foliation has all the leaves with sub-exponential growth, so it has transversal measures. The topological growth was studied for the case of uniformly hyperbolic transitive diffeomorphisms, where the logarithm of it must be equal to the entropy (see [7,9]). In this case the unstable and stable foliations have unique non-trivial homology. It is still an open question weather there are non-transitive uniformly hyperbolic diffeomorphisms, so actually all these results apply for all the known Anosov diffeomorphisms.

For the stable and unstable foliations of partially hyperbolic diffeomorphisms, the closed currents exist, but they are not always non-trivial. For example, for the time one map of an Anosov flow, every closed current supported on the stable or the unstable foliation has zero homology. This is because the map is homotopic to the identity, so the action on the homology is the identity, and this makes impossible to have non-trivial closed currents on the unstable foliation, because they would have to be expanded. However, in other known situations, the stable and unstable foliations of partially hyperbolic diffeomorphisms have unique non-trivial homology. This is the case for skew products over Anosov systems, some derived from Anosov maps (skew products over), linear partially hyperbolic maps of the torus and small perturbations.

We will present in more detail the case of maps on then-torusTⁿclose to a linear map. Consider ann×nmatrixA with determinant one and with integer entries. The matrixAinduces a toral automorphism:T_A:Tⁿ=Rⁿ/Zⁿ→Tⁿ defined byT_Ax=AxmodZⁿ. We will consider the standard Riemannian metric on the torusTⁿ. If all eigenvalues are away from the unit circle, thenTA is a hyperbolic toral automorphism. If the eigenvalues ofAare mixed, with some on the unit circle and some away from unit circle, thenTA is partially hyperbolic. Letλi, 1in, be the eigenvalues ofAcounted with their multiplicity, ordered such that

|λ1||λ2|· · ·|λn−k|1<|λn−k+1|· · ·|λn|.

In both hyperbolic or partially hyperbolic cases, letE^u be thek-dimensional unstable distribution ofTA onTⁿ. At each pointx∈Tⁿ,E^u(x)⊂TxTⁿ is the unstable subspace fordTA:TxTⁿ→TxTⁿ, meaning that it is the space spanned by the (generalized) eigenspaces corresponding to the eigenvalues greater than one in absolute value—λn, λ_n−1,λ_n−k+1. LetW^ube the unstable foliation generated byE^u. Then it is easy to see that the condition in Proposi- tion 2.2 is satisfied forN=1, because thek-dimensional expansion inW^u, or the unstable JacobianJ_u, is constant

T_A∗v^k v^k =

n

i=n−k+1

λ_i

:=J_u, ∀v^k∈Λ^kT W^u,

and the maximalk−1-dimensional expansion is at most the product of the greatest (in absolute value)k−1 eigen- values, which is strictly smaller thanJ_u:

T_A∗v^k−1 v^k−1

n

i=n−k+2

λ_i

= J_u

|λ_n−k+1|< J_u, ∀v^k−1∈Λ^k−1T W^u.

This means that for anyx∈Mandr >0, all the limit currents ofCn(x, r)are closed. Furthermore, there exist a closed differential form

ω=dxi₁∧dxi₂∧ · · · ∧dxi_k

which is non-trivial onW^u, and by Proposition 2.4 this means that all the limit currents ofC_n(x, r)are also non-trivial.

Moreover, for anyi1, i2, . . . , ik∈ {1,2, . . . , n}, we remark that C_n(x, r)(dx_i1∧dx_i2∧ · · · ∧dx_ik)=C_i1,i2,...,ik

is independent of n, x, r (this is because T_A is linear, and W^u(x)is just a plane with the same direction for ev- eryx), so all the limit currents will have the same homology, denoted byh_u. This of course will be an eigenvector of TA∗:Hk(Tⁿ,R)→Hk(Tⁿ,R), corresponding to the unstable subspaceE^uforA, and the eigenvalue will be exactlyJu

and it will be a simple eigenvalue. So the topological growth ofW^uisJuand the volume growth is logJu.

Now suppose thatf is a map which is C¹ close to T_A. Then f is also partially hyperbolic, and the unstable distributionE˜^uwill beC⁰close toE^u(this is because a dominated splitting depends continuously on the point and the map, there is a simple proof using invariant cone fields) and can be integrated to obtain a new unstable foliationW˜^u. We claim that the unstable foliation off has also unique non-trivial homology, the same as the one ofT_A, and the same topological and volume growth.

Proposition 2.8. If f is sufficiently C¹ close to T_A, then the unstable foliation W˜^u of f has unique non-trivial homology, the topological growth is equal toJ_u, and consequently the volume growth of every unstable disk islogJ_u. Proof. We remind that the condition on the rates of growth alongW^uneeded for Proposition 2.2 is an open condition, if it holds forT_AandW^uit also holds forf C¹close toT_AandW˜^usuch thatTW˜^u= ˜E^uisC⁰close toT W^u=E^u, so all the sub-sequential limits of the currentsC_n(x, r)are closed for anyx∈M,r >0. Also, the existence of a non- degenerate closed form on a foliation is an open condition, ifωis non-degenerate onW^u andW˜^u is such thatTW˜^u is C⁰ close toT W^u, then ω is also non-degenerate onW˜^u. By Proposition 2.4 we get that all the limit currents ofC_n(x, r)are non-trivial. This implies thatH_x,ris bounded away from zero for anyx∈M,r >0.

Furthermore we observe that the map f is homotopic to the linear mapT_A and hence the induced map on the homology is exactly the same as that of the linear map. Denote byJ˜_u(x)the new unstable Jacobian forf which now will depend of course on the pointx∈M. For aC¹small perturbationJ˜uis close toJu. By Proposition 2.5, for any x∈Mandr >0, the setRHx,ris a non-trivial closed invariant subset ofHk(Tⁿ,R). SupposeRHx,r=Rhu. Because RHx,r is non-trivial closed invariant, RHx,r must contain some homologyh˜ which is not in Rhu. BecauseJu is a simple eigenvalue,h˜cannot be a (generalized) eigenvector forJ_u, so if we writeh˜in a basis formed of generalized eigenvectors we will find at least one non-trivial component in a direction of an eigenvector corresponding to some eigenvalueλ˜=J_u. Then

|˜λ||λ_nλ_n−1· · ·λ_n−k+2λ_n−k|< J_u, and

jlim→∞f_∗^−jh˜^1j 1

˜ λ.

We can assume without loss of generality thath˜∈H_x,r, so there is a limit current C= lim

i→∞C_ni(x, r)

such that[C] = ˜h. Following the proof of Proposition 2.5, withf_∗C replaced byf_∗^−jC, we get that eventually for a subsequence we have

λ(j )f_∗^−jC= lim

i→∞C_ni−j(x, r), where

λ(j )= lim

i→∞

Vol(fⁿⁱ(W_r(x))) Vol(f^ni−j(W_r(x)))

= 1

Vol(f^ni−j(Wr(x)))

fni−j(Wr(x))

J˜_u(x)J˜_u f (x)

· · · ˜J_u

f^j−1(x) .

By choosingf sufficiently close toT_A, we may assume thatJ˜_u(x)is close enough toJ_uso for someα >1 we have J˜u(x) > α|˜λ|for allx∈M. From this we get:

λ(j ) >

α|˜λ|j

.

Thenλ(j )f_∗^−jCis also a limit ofC_n(x, r)and λ(j )f_∗^−jC

=λ(j )f_∗^−jh˜∈H_x,r.

But, taking the limit asj tends to infinity we get:

jlim→∞λ(j )f_∗^−jh˜ lim

j→∞

λ(j )

|˜λ|^j lim

j→∞α^j= ∞, but this is a contradiction becauseH_x,ris bounded.

This proves thatW˜^uhas unique non-trivial homology,hu, so the topological growth isJuand the volume growth of every diskW˜r(x)must be logJu. 2

3. Lyapunov exponents

The expansion of an invariant foliationW can also be described by the Lyapunov exponents. In this section, we will consider this analytical description and show its relations with the geometric expansion we described in the first section.

Let f be a diffeomorphism of M with an invariant probability measure μ. then for μ-a.e.x ∈M, there exist real numbersλ₁(x) >· · ·> λ_l(x)(ln); positive integersn₁(x), . . . , n_l(x)such thatn₁(x)+ · · · +n_l(x)=n; and a measurable invariant splittingT_xM=E¹_x⊕ · · · ⊕E^l_x, with dimension dim(E_xⁱ)=n_i(x)such that

jlim→∞

1

j logD_xf^j(v_i)=λ_i(x), wheneverv_i∈E_xⁱ,v_i=0.

These numbersλ₁(x), . . . , λ_l(x)are called the Lyapunov exponents ofx ∈M. If the probability measure μis ergodic, then these exponents are constant for a.e. (μ)x∈M. The existence of these Lyapunov exponents is the result of Oseledec’s Multiplicative Ergodic Theorem.

LetEbe an invariant sub-bundle ofT M. For example it can beT W, the tangent spaces of leaves are preserved under the map. i.e., for anyx∈M,Dxf (TxW (x))=Tf (x)W (f (x)). For any invariant probability measureμand for a.e. (μ)x∈M, a subset of the Lyapunov splittingEⁱ_x,i=1, . . . , l, spansE_x. LetΛ_E(x)be the sum (counting mul- tiplicityn_i(x)) of the Lyapunov exponentsλ_i(x)corresponding to the sub-bundlesEⁱ_xwhich are insideE_x.Λ_E(x)is defined a.e. (μ) and it is also given by the formula:

ΛE(x)= lim

j→∞

1

j logΛ^kDxf^j|Λ^kEx.

We also define the integrated Lyapunov exponent ofEto be ΛE=

M

ΛE(x) dμ.

Because of the Birkhoff Ergodic Theorem applied to the additive real-valued cocyclex→logΛ^kD_xf^j|Λ^kE_x, the integrated Lyapunov exponent ofEis also equal to the integral over the manifoldMof the logarithm of the Jacobian off restricted to the sub-bundleE:

ΛE=

M

logΛ^kDxf|Λ^kExdμ.

Whenμis ergodic,Λ_E(x)=Λ_E a.e. (μ). IfE=T W we will denoteΛ_W(x)=Λ_{T W}(x)andΛ_W=Λ_{T W}.

For the following result, we need to define the concept of absolute continuity. For simplicity, we use a stronger version of absolute continuity. For anyx∈M, letD₁andD₂be sufficiently small(n−k)-dimensional smooth disks transverse toWr(x)for somer >0. One can locally define a map, called holonomy map for the foliation, fromD1

toD2,y1→y2withy1∈D1andy2=D2∩Wr(y1). The holonomy map is said to be absolutely continuous if it maps sets of measure zero inD1to sets of measure zero inD2. The foliation is said to be absolutely continuous if the holon- omy maps are absolutely continuous. If a foliation is absolutely continuous, a full measure set for a smooth measure intersects almost all leaves in full measure. Here the measure on the leaves is the Riemannian volume restricted toW, and almost all leaves is with respect to Riemannian volume on transversals.

The next result is a standard way to prove non-absolute continuity of foliations:

Lemma 3.1.Letf ∈Diff_μ^r(M)be a diffeomorphism onM, preserving a smooth volumeμ. LetWbe ak-dimensional foliation ofM, invariant underf and

χ (x, r)=lim sup

i→∞

1 iln Vol

fⁱ W_r(x)

, and let

χ=χ (r)=sup

x∈M

χ (x, r).

Finally, letΛ_Wbe the integrated Lyapunov exponent of the foliationWfor the invariant measureμ. If the foliationW is absolutely continuous, then

ΛWχ .

Proof. LetA⊂Mbe the set of Lyapunov generic points. i.e., for anyx∈A, there exist the sum of the Lyapunov exponents forxonT_xW and is equal toΛ_W(x). This is a full measure set with respect toμ. We have that

ΛW=

M

ΛW(x) dμ,

so there exists a positive measure set B⊂A⊂M such that for anyx∈B we haveΛ_W(x)Λ_W. The absolute continuity ofW implies that there exists at least a leafW (x)for somex∈Msuch thatW (x)intersectsB in a set of positive measure (actually there is a positive set of such leaves). Denote bym_W the Riemannian volume onW (x)and fix a diskWr(x)such thatmW(Wr(x)∩B) >0.

Let Jacy(fⁱ)= Λ^kDyfⁱ|Λ^kTyWbe the Jacobian aty∈Mof the functionf restricted toW (y). Ifyis a Lyapunov regular point (y∈A), we have

Λ_W(y)= lim

i→∞

1 ilog

Jacy

fⁱ .

Then for any small >0, for anyy∈Wr(x)∩B⊂Athere existNy∈Nsuch that for alliNywe have 1

ilog Jacy

fⁱ

ΛW(y)−, or

Jacy

fⁱ

e^i(ΛW(y)−).

LetBN ⊂Wr(x)∩B be the set of pointsy such thatNyN. ThenBN is an increasing sequence of sets and the union isWr(x)∩B which has positive measure, so there is anN∈Nsuch thatmW(BN) >0. It follows that for any y∈B_Nand anyi > Nwe have

Jacy

fⁱ

e^i(ΛW(y)−)e^i(ΛW−).

We will use this to estimate the volume offⁱ(W_r(x))fori > N:

Vol fⁱ

Wr(x)

=

fⁱ(Wr(x))

dmW

=

W_r(x)

Jacy

fⁱ dmW

B_N

Jacy

fⁱ dmW

m_W(B_N)e^i(ΛW−). Therefore,

χχ (x, r)=lim sup

i→∞

1

i log Vol fⁱ

W_r(x)

Λ_W−.

Since >0 is arbitrary, we haveχΛ_W. 2

A simple corollary of the above lemma and its proof is the following: