michael shub...abstract. in this paper we eliminate some of the most nettle-some hypotheses of our...

$: Michael Shub...Abstract. In this paper we eliminate some of the most nettle-some hypotheses of our previous article, \Stably ergodic dynamical systems and partial hyperbolicity" [22],$
Abstract. In this paper we eliminate some of the most nettle-some hypotheses of our previous article, “Stably ergodic dynamicalsystems and partial hyperbolicity” [22], and expand the domain ofstably ergodic, partially hyperbolic dynamical systems to includeall partially hyperbolic affine diffeomorphisms of compact homo-geneous spaces which have the accessibility property. Our maintool is a new concept – julienne quasi-conformality of the stableand unstable holonomy maps. An important feature of this is thatthe holonomy maps preserve all julienne density points.

STABLE ERGODICITY AND JULIENNEQUASI-CONFORMALITY

CHARLES PUGH AND MICHAEL SHUB

July 2, 1998

Date. May 22, 1997.Charles Pugh thanks IBM, Rockefeller University, and the Instituto de

Matematica Pura e Aplicada for hospitality and support during part of the timethis paper was written.

Michael Shub was supported in part by an NSF grant.1

2 C.C. PUGH AND M. SHUB

1. Introduction

Let M be a compact differentiable manifold without boundary andlet m be a smooth probability measure on M . We consider the spaceDiff2

m(M) of C2 diffeomorphisms of M which are measure preserving(we also say volume preserving) with respect to the measure m. If thereis a neighborhood U of f in Diff2

m(M) consisting entirely of m-ergodicdiffeomorphisms, then f is stably ergodic .

The persistence of positive measure sets of invariant tori due to Kol-mogorov, Arnold, Moser, Herman and others (see Herman [14] andYoccoz [28]) shows that stable ergodicity cannot in general be an openand dense property in Diff2

m(M).These invariant tori have no hyperbolic behavior at all. In contrast,

a uniformly hyperbolic diffeomorphism f ∈ Diff2m(M) is stably ergodic.

See Anosov’s thesis [2]. In a series of papers, [13, 21, 22, 23, 24], we havebeen investigating the mixed situation which is partially hyperbolic.

Recall that f : M →M is partially hyperbolic if it is non-triviallynormally hyperbolic in the following sense. The tangent bundle of Mis a Tf -invariant direct sum

TM = Eu ⊕ Ec ⊕ Es,

the stable and unstable bundles Es and Eu are non-zero, and Tf con-tracts and expands them more sharply than it does the center bundleEc. That is, there is a Riemann structure on TM such that T uf ex-pands Eu, T sf contracts Es, and

supp‖T sp f‖ < inf

pm(T cpf) and sup

p‖T cpf‖ < inf

pm(T up f),

where T uf, T cf, T sf are the restrictions of Tf to Eu, Ec, Es, and m(T )refers to the conorm of T , inf{|Tv| : |v| = 1}. The center bundle ispermitted to be zero, in which case f is an Anosov diffeomorphism.

The center bolicity of f is the ratio

b =‖T cf‖m(T cf)

.

We say that f is center bunched if b is close to 1. In a typical case,b < 1.09 suffices. See Section 4 for details. The examples to which ourstable ergodicity theory applies most readily are perturbations of caseswhere b = 1, so we find center bunching a reasonable hypothesis.

Our main themes have been(1) A little hyperbolicity goes a long way in guaranteeing stably er-

godic behavior.(2) Some hyperbolicity may be necessary for stable ergodicity.

STABLE ERGODICITY AND JULIENNE QUASI-CONFORMALITY 3

With respect to the second see the papers of Brezin and Shub [4]and Diaz, Pujals, and Ures [10]. With respect to the first, in [22] wehave statedConjecture 1. Stably ergodic diffeomorphisms are open and denseamong the partially hyperbolic C2 volume preserving diffeomorphismsof M .

An approach to this conjecture breaks it down into two parts, usingthe notion of accessibility.

Given x, y ∈M and given the splitting TM = Eu⊕Ec⊕Es for f , wesay that y is us-accessible from x if there is a piecewise differentiablepath joining x to y always tangent either to Eu or Es. Clearly, acces-sibility is an equivalence relation. If there is only one equivalence class(every y is accessible from every x) we say that the splitting Eu ⊕ Es

(or the diffeomorphism f) has the accessibility property. If every gin some neighborhood of f has the accessibility property, f is stablyaccessible. If the only measurable sets which are saturated by theaccessibility equivalence relation have measure zero or one, we say thatEu ⊕Es, or f , has the essential accessibility property. Obviously,stable accessibility implies accessibility implies essential accessibility.Conjecture 2. Stable accessibility is an open and dense propertyamong C2 partially hyperbolic diffeomorphisms, volume preserving ornot. Openness is obvious.Conjecture 3. Partially hyperbolic C2 volume preserving diffeomor-phisms with the essential accessibility property are ergodic.

In [22] we proved Conjecture 3 with the addition of quite a fewtechnical hypotheses. It is the goal of this paper to weaken the mostnettlesome of these technical hypotheses concerning the bunching ofthe hyperbolic part of the spectrum, and thus to conclude the stableergodicity of a much wider collection of diffeomorphisms: according toTheorem C, below, all affine diffeomorphisms of compact homogeneousspaces with the accessibility property are stably ergodic. In essence,bunching of the unstable, center, and stable spectra is reduced to centerbunching.

Recall that Eu and Es are tangent to unique foliations Wu and Ws

which have C1 leaves, but that since the bundles Eu and Es are onlyHolder continuous, the foliations may fail to be smooth. If Eu⊕Ec, Ec,and Ec ⊕ Es are also tangent to continuous foliations with C1 leavesWcu, Wc, and Wcs and if Wc and Wu subfoliate Wcu, while Wc and Ws

subfoliate Wcs, then we say that f is dynamically coherent.

Theorem A. If f ∈ Diff2m(M) is a center bunched, partially hyper-

bolic, dynamically coherent diffeomorphism with the essential accessi-


bility property then f is ergodic.

As a consequence we get

Corollary a. Center bunched partially hyperbolic diffeomorphisms whichare stably dynamically coherent and stably accessible are stably ergodic.

Proof. According to [22], small perturbations preserve the propertiesof partial hyperbolicity and center bunching. Therefore, every pertur-bation of f in Diff2

m(M) satisfies the hypotheses of Theorem A and isergodic.

We make two remarks. The first concerns the hypothesis in Corol-lary a that the diffeomorphism be stably dynamically coherent. Per-haps dynamical coherence is itself stable under perturbations of thediffeomorphism; we do not know. A general, stable condition thatimplies stability of dynamical coherence is plaque expansivity of thecenter foliation. For, as we showed in [15], a plaque expansive, nor-mally hyperbolic foliation is structurally stable as a foliation, and thisimplies that dynamical coherence is unaffected by perturbations of thediffeomorphism. Also, if the center foliation happens to be C1, or ifthe diffeomorphism acts isometrically on the center leaves, then plaqueexpansivity is automatic.

Second, the hypothesis of partial hyperbolicity in Theorem A canbe weakened somewhat. It is only necessary to assume that one ofthe two bundles Eu, Es is non-zero. This happens if the spectrum ofTf splits into just two parts, say an unstable part that represents thestrong expanding behavior of f , and a remaining, indecomposable partthat represents the weak expanding, neutral, and contracting behaviorof f . Thus, Tf leaves invariant a splitting TM = Eu ⊕ Ec. Since Mis compact, there must be some true contraction to balance the expan-sion in Eu. Some vectors in Ec get contracted by Tf , but there neednot exist in this pseudo-center subbundle Ec a continuous, invariant,purely contracting subbundle. The proof of Theorem A in this specialcase, however, is quite simple. Since Es = 0, a us-path lies wholly inan unstable leaf. That is, accessibility equivalence classes are singleunstable leaves. Now suppose that f fails to be ergodic. Then thereis an invariant set A of intermediate measure, 0 < m(A) < 1, and bythe usual use of the Birkhoff Ergodic Theorem, it consists, except for azero set, of almost whole unstable leaves. According to [20], the unsta-ble foliation is absolutely continuous, even when Es = 0. Thus, whenwe form the unstable saturate of A, Satu(A), by taking the union ofall unstable leaves that meet A in sets of positive leaf measure, we see


that A differs from Satu(A) by a zero set. By construction Satu(A) issaturated by the accessibility equivalence relation, and this contradictsthe hypothesis in Theorem A of essential accessibility.

The proof of Theorem A, proceeds by a careful examination of themeasure theoretic regularity properties of the unstable holonomy maphu, which is defined by sliding along the leaves of the unstable foliation.See Section 4. Also, of course, we have corresponding properties of thestable holonomy map hs. Two facts are already known about hu.

(a) hu is a bi-Holder homeomorphism, but is not in general Lipschitz.(b) hu has a continuous, positive Radon Nikodym derivative with

respect to Lebesgue measure.These facts were originally proved by Anosov in [2], (b) being his maintechnical tool to establish ergodicity. It implies that hu is absolutelycontinuous. See Also [20]. To give it a name we say that a homeomor-phism satisfying (b) is RN-regular. Automatically, the inverse of anRN -regular homeomorphism is RN -regular.

At issue is how the holonomy map hu affects density points. (Recallthat p is a Lebesgue density point of a set S if the measure theoreticconcentration of S in the ball of radius r at p converges to 1 as r → 0.See also Section 2.) In low dimensions or when the hyperbolic spectrumis tightly bunched, we showed in [13, 22] that hu is density pointpreserving: hu carries every Lebesgue density point of a measurableset S to a Lebesgue density point of h(S). It is elementary that anRN -regular homeomorphism h preserves almost all density points. Inour proof of stable ergodicity, however, “almost all” is not enough. Weneed “all”, and as we showed by example in [13], “all” is not impliedby RN -regularity.

It is natural to ask whether hu satisfies some standard, additionalproperty that implies it is density point preserving. For example,one easily checks that a bi-Lipschitz homeomorphism preserves densitypoints. However, according to (a), above, holonomy maps are not ingeneral bi-Lipschitz. A different criterion is quasi-conformality. In [12]Gehring and Kelley prove that a bi-quasi-conformal homeomorphismis density point preserving. It turns out, though, that

bi-RN + bi-quasi-conformal ⇒ bi-Lipschitz,

so we can not hope that the Gehring-Kelley Theorem applies to holo-nomy maps in general. (This implication is a nice exercise in pictorialmeasure theory.)

At present, we are unable to say whether holonomy preserves Lebesguedensity points in general. Rather, using shapes called juliennes in place


of round balls (see Section 4), we re-define the density point conceptand prove three things.

(a) Almost every Lebesgue density point is a julienne density point,and vice versa.

(b) The holonomy maps are quasi-conformal with respect to juli-ennes.

(c) The holonomy maps preserve (all) julienne density points.In fact, (b) implies (c), which is a generalization of the Gehring-KelleyTheorem. See Section 8. From all this, we draw the conclusion that

Juliennes are the natural shapes to usewhen analyzing holonomy maps.

We began studying juliennes in [13] and continued to investigatetheir properties in [22]. The difference in our current approach is thatwe control the shape of the juliennes dynamically rather than onlythrough Holder estimates on the Eu and Es bundles. This is carriedout in Sections 4 and 6.

In some ways it seems a shame to give up the infinitesimal infor-mation embodied in Eu and Es and to rely on more ad hoc geometricconstructions. Anosov (pages 127-129 and 167 of [2]) voices a similarsentiment regarding the existence and uniqueness of stable and unsta-ble manifolds: instead of ad hoc dynamical arguments, he imaginesthere should be a construction that uses non-differentiable Frobeniusintegrability conditions, and avoids iteration.

It can be shown that the holonomy maps have a kind of flag differ-entiability. Perhaps julienne quasi-conformality can be derived directlyfrom this infinitesimal property.

Once we have proven the density point preservation properties forstable and unstable holonomy maps, the proof of Theorem A proceedsas in [13, 22] along classical lines dating from Hopf [16], Anosov [2], andothers, using us-accessibility and the essential Wu and Ws saturationof invariant sets. There are, however, some subtleties to this proof, oneof which we discuss now.

The center foliation, when it exists, integrates the center plane fieldEc. It is even less regular than the stable and unstable foliations, whichalways exist. As has been shown by A. Katok in an example written upby J. Milnor [17], the center foliation for a smooth system can have thefollowing measure theoretically singular property: for some measurableset S of full measure, each center leaf meets S in at most one point.This phenomenon has been referred to as “Fubini’s nightmare”, or, byMilnor, as “Fubini Foiled”. Even though the center leaves are smooth,and the center plane field is Holder continuous, the center foliation is


unavoidably pathological.

Theorem A allows us to deduce C2 stable ergodicity for many affinediffeomorphisms of compact homogeneous spaces, as we now explain.See also Section 10. We make the following

Standing Lie Group Hypotheses.(a) G is a connected Lie group and B ⊂ G is a proper, closed sub-

group.(b) G/B is compact.(c) Haar measure projects to a finite measure m on G/B, invariant

under left translation, xB 7→ gxB.(d) f : G/B → G/B is an affine diffeomorphism. That is, it is

part of a commutative diagram

Gf = Lg ◦ A−−−−−−−−−−−−−−−→ G

πy yπ

G/Bf−−−−−−−−−−−−−−−→ G/B

where A is an automorphism of G, Lg is left multiplication bysome fixed element g ∈ G, and, to make the projection welldefined, A(B) = B.

A simple example is G = Rm and B = Zm. Then G/B is the torusTm. The map Lg is an affine translation of Rm to itself, A is an m×minteger matrix with determinant ±1, and f sends the coset [x] to thecoset [g + Ax].

The hypotheses of Theorem A for a general affine diffeomorphismf are easily checked at the Lie algebra level. The Lie algebra of G isg = TeG, where e is the identity element of G. The adjoint action of gon g is given by the derivative at the identity of g-conjugation,

Ad(g) : g→ g

X 7→ TgRg−1 ◦ TeLg(X),

where Rg−1 is right multiplication by g−1, Rg−1 : h 7→ hg−1. Inducedby f is a Lie algebra automorphism

a(f) = Ad(g) ◦ TeA.It determines when we can apply Theorem A to f as follows. Split


g = TeG according to the generalized eigenspaces of a(f),

g = gu ⊕ gc ⊕ gs,

where the eigenvalues of a(f) on gu, gc, gs have modulus > 1, = 1, and< 1 respectively. As we show in Section 10, these eigenspaces are Liesubalgebras. See also [22]. The hyperbolic subalgebra of f is thesmallest Lie subalgebra h such that gu ∪ gs ⊂ h ⊂ g. It is an idealin g. See [22]. Let b be the Lie algebra of B, b ⊂ g.

Theorem B. Suppose that f is an affine diffeomorphism of the com-pact homogeneous space G/B. Then

(a) f is partially hyperbolic if and only if h /⊂ b.(b) If f is partially hyperbolic then it is center bunched and dynam-

ically coherent.(c) f has the accessibility property if and only if g = b + h.

We deduce

Theorem C. The affine diffeomorphism f : G/B → G/B is stablyergodic among C2 m-preserving diffeomorphisms of G/B if (merely)the hyperbolic Lie subalgebra h is large enough that g = b + h.

Here are some special cases of Theorem C. For their proofs and thatof Theorem C, see Section 10.

Corollary b. Suppose that G is a simple Lie group (its Lie algebra hasno non-trivial ideals), B is a uniform discrete subgroup of G, the ho-mogeneous space G/B is compact, and g ∈ G is given. Left translationby g on G/B is stably ergodic among C2 volume preserving diffeomor-phisms if and only if Ad(g) has at least one eigenvalue of modulusdifferent from 1.

Corollary c. Let B be a uniform discrete subgroup of SL(n,R) andlet M ∈ SL(n,R) be given. Left multiplication by M is stably ergodicamong C2 volume preserving diffeomorphisms of SL(n,R)/B if andonly if M has at least one eigenvalue of modulus different than 1.

These corollaries provide examples of what we mean by a little hyper-bolicity going a long way toward guaranteeing stable ergodicity. Thehyperbolic part of M can be low dimensional, the center part highdimensional, and nevertheless we get stable ergodicity.

See [22] for more equivalent conditions to stable ergodicity for SL(n,R)and semi-simple Lie groups.

The first case of Corollary c not covered by [22] is given by SL(3,R)and a matrix M with two eigenvalues of modulus different than one and


the third equal to one. In this case the hyperbolic bunching conditionsof [22] are violated.

To our knowledge ergodicity in the context of partial hyperbolicityand accessibility was first considered by Brin and Pesin [6]. Brin in[5] goes further in this direction. It seems quite likely that the openand dense set of skew products he considers in [5] which are stablyergodic among skew products are also stably ergodic in Diff2

m, but wehave not verified it. The preprint of Burns and Wilkinson [8] is relevanthere. Further work on skew products is contained in papers by Adler,Kitchens, and Shub [1], Field and Parry [11], and Parry and Pollicott[19]. For stable ergodicity of time one maps of geodesic flows see thethe papers of Grayson Pugh and Shub [13], Wilkinson [27], and Burns,Pugh, and Wilkinson [7].

We thank Dennis Sullivan for several helpful conversations aboutdensity points. It was in his seminar at CUNY that we first be-came accustomed to such concepts as bounded distortion and quasi-conformality in relation to density points.

2. Density points

The technical basis for our main theorems is differentiation of inte-grals. See M. de Guzman’s book, Differentiation of integrals in Rn,[9] and Chapter 1 of E. Stein’s book, Harmonic Analysis [26]. In thissection we review some of the ideas involved.

Let X be a locally compact metric space and let m be a regular, non-atomic, locally finite, Borel measure on it. The measure of a measurableset S ⊂ X is the infimum of the measures of open sets that containS. Points have zero measure. Compact sets have finite measure. Forexample, m could be a smooth measure on a manifold. A family V =⋃Vp, indexed by p ∈ X, is a Vitali basis if each Vp is non-empty,the sets V ∈ Vp are measurable with non-zero measure (hence non-zero diameter), each contains p, and there exist in Vp sets of arbitrarilysmall diameter. The simplest good example is the family B of all n-dimensional balls, of all radii, centered at all points of Rn.

The concentration, or conditional measure, of a measurable set Ain a measurable set V of non-zero measure, is the ratio

[A : V ] =m(A ∩ V )

m(V ).

Fix a Vitali basis V. A point p ∈ X is a V-density point of A if

lim[A : V ] = 1,


the limit being taken as V ∈ Vp shrinks to p. The basic fact aboutdensity points is theLebesgue-Vitali Theorem. With respect to the Vitali basis B ofn-balls, almost every point p of a measurable set A ⊂ Rn is a densitypoint of A.

In this paper we need to generalize the Lebesgue-Vitali theorem toVitali bases more dynamically natural than B. We must work not withballs, but with juliennes, small sets that are long and thin. They looklike slivered vegetables. At smaller and smaller scales, our juliennesbecome less round, thinner, and more elongated. See Section 4 in thispaper. One of our main results is that despite the juliennes’ strangeshape, almost every point of a Lebesgue measurable set is a juliennedensity point. See Theorem 8.4.

The Vitali basis V differentiates the integral of an integrablefunction φ : X → R if, for almost all points p ∈ X,

φ(p) = lim1

m(V )

∫Vφ(x) dx.

The limit is taken as the set V ∈ Vp shrinks to p; i.e., diamV → 0. If Vdifferentiates the integral of every φ in some set Φ of locally integrablefunctions, we say that V is a Φ-differentiation basis.

The Lebesgue-Vitali Theorem can be restated as the fact that B isa χ-differentiation basis, where χ is the set of characteristic functionsof measurable subsets of Rn. For

[A : V ] =1

m(V )

∫VχA(x) dx.

In fact more is true. As Lebesgue showed, B is an L1loc-differentiation

basis.If V is a χ-differentiation basis, it is called a density basis. A result

of Busemann and Feller shows that being a density basis is equivalentto being an L∞loc-differentiation basis. See [9], page 72.

The geometric characterization of density bases in Rn is a centralpoint in much of analysis (see [26]), and has a long history (see [9]).

When balls are replaced by cubes (whose edges need not be parallelto the coordinate axes), the same result holds: The cubic Vitali basisis a density basis. But even when cubes are replaced by n-dimensionalrectangles the situation becomes subtle and is not completely under-stood. There is a trade-off between the eccentricity of the rectanglesand how much their edge directions vary.

First consider the case in which the edge directions do not vary at all.The Vitali basis Ra of n-dimensional rectangles whose edges are axis


parallel is a density basis, but it is not an L1loc-differentiation basis.

See [9], pages 74 and 96. To refine Ra so that it becomes an L1loc-

differentiation basis we impose a restriction on the eccentricity of therectangles. A sufficient condition is that for some constant K > 0 andall rectangles R of diameter ≤ 1,

eccentricityR ≤ (diameterR)−K .

For this type of condition implies the “doubling property” of Stein [26],page 8, or the “volumetric engulfing property” discussed in Section 3below.

It is easy to see that the particular choice of coordinate axes is ir-relevant. If Ra′ is the Vitali basis of rectangles whose edges are alwaysparallel to some fixed, finite set of directions in Rn then Ra′ is a den-sity basis, and if the eccentricities of its rectangles are bounded bya fixed, negative power of their diameter then Ra′ becomes an L1

loc-differentiation basis.

On the other hand, when the edge directions of rectangles in a Vi-tali basis are drawn from an infinite set, the situation is subtler. Asis shown by Nikodym’s “paradoxical set” [18], the Vitali basis R ofall n-dimensional rectangles is not a density basis. Nor does R be-come a density basis if one imposes a continuity restriction on the edgedirections. See Chapter 5 of de Guzman’s book [9].

The Vitali bases discussed so far are linear. Rectangles are linearimages of the cube. As we stated before, however, the Vitali basesthat arise naturally in non-linear dynamics consist of juliennes – non-linear rectangloids with varying edge directions. Despite the obstaclesdescribed above, juliennes turn out to be satisfactory Vitali bases due,it seems, to three factors.

(a) The elongation axes of a julienne are Holder controlled, not merelycontinuously controlled.

(b) The eccentricity of a julienne and its diameter are both exponen-tial functions of a common number n, the number of dynamicaliterations.

(c) The non-linearity, nesting, and shape-scaling properties of thejulienne are governed by a fixed, smooth dynamical system.

One might hope to get an abstract geometric characterization ofdensity bases, say for a basis V of rectangles in R2, in these sorts ofterms. If the eccentricity of rectangles in V is no greater than a fixednegative power −K of the diameter, if the edge directions of rectanglesin Vx are a θ-Holder function of x, and if there is some favorable relationbetween K and θ (say K(1− θ) < 1), does it follow that V is a density


basis?

We conclude this section by discussing a transformation property ofdensity points.

Let (X,m) and (X ′,m′) be measure spaces of the type we have beenconsidering (locally compact, etc.) and let V,V′ be Vitali bases onthem. Let h : X → X ′ be a homeomorphism. If h bijects V to V′, andhas a positive, continuous Radon Nikodym derivative it is easy to seethat V-density points biject to V′-density points. We need a slightlybetter version of this result. We say that the homeomorphism h is K-quasi-conformal with respect to V,V′ if for all small V ∈ Vp, thereare small V ′, V ′′ ∈ V′h(p) such that

V ′ ⊂ h(V ) ⊂ V ′′ andm′(V ′′)

m′(V ′)≤ K.

Ordinary quasi-conformality in Rn is the same as quasi-conformalitywith respect to the Vitali basis of round balls and Lebesgue measure.

Proposition 2.1. Assume that h and h−1 are quasi-conformal withrespect to the Vitali bases V,V′. If h has a continuous, positive RadonNikodym derivative then it bijects V-density points to V′-density points.

Proof. Suppose that p ∈ X is not a density point of A ⊂ X. Then,for some ε > 0 and some V ∈ Vp with arbitrarily small diameter,[A : V ] ≤ 1 − ε. By continuity of the Radon Nikodym derivative andsmallness of V , [h(A) : h(V )] ≤ 1− ε/2. Since h(V ) occupies at least aportion 1/K of V ′′, the concentration of h(A) in V ′′ is bounded awayfrom 1. That is, h(p) is not a density point of h(A).

We have shown that h carries non-density points to non-densitypoints. The same is true for h−1. Hence h and h−1 also biject den-sity points.

3. Volumetric Engulfing

In this section we prove a version of the Lebesgue-Vitali density pointtheorem. We will apply the result to juliennes in Section 8.

As in the previous section, let X be a locally compact, metric mea-sure space, and let m be a regular, non-atomic, locally finite, Borelmeasure on it. Let V be a Vitali basis on X. We also assume that V isfiltered by a rank function

rank : V→ N,compatible with diameter in the sense that rank(V ) → ∞ if and onlyif diam(V ) → 0. That is, given ε > 0, there is an integer N such that

V Væ Væ

Væ

Væ

^ V


if rank(V ) ≥ N then diam(V ) < ε, and conversely, given N ∈ N thereis an ε > 0 such that if diam(V ) < ε then rank(V ) ≥ N .

Definition. V is volumetrically engulfing if for some constant L,and for each V ∈ V there is a measurable set V containing V such that

(a) m(V ) ≤ Lm(V ).(b) If V, V ′ ∈ V have non-empty intersection and rank(V ) ≤ rank(V ′)

then V ′ ⊂ V .(c) If m(V ) → 0 then rank(V ) → ∞. That is, given N ∈ N there

exists δ > 0 such that m(V ) < δ implies rank(V ) ≥ N .

See Figure 1.V is closed if every V ∈ Vx ∈ V is a closed subset of Rn.

Figure 1. The set V engulfs all V ′ ∈ V that meet Vbut are not much larger than V .

Theorem 3.1. If V is a volumetrically engulfing closed Vitali basis ofX and S is a measurable subset of X then almost every point of S isa density point of S with respect to V. In other words, V is a densitybasis.

Proof. For each β < 1 consider the set

B = Bβ = {x ∈ S : lim inf [S : V ] < β}.The liminf is taken as V ∈ Vp shrinks to p. We claim that B hasmeasure zero. From this the theorem follows, for we can take a sequenceβn = 1 − 1/n that tends to 1. Almost every point of S belongs toS ′ = S \ ⋃Bβn , and every x ∈ S ′ is a density point of S.

Let ε > 0 be given. It is enough to show that B has measure < ε.Cover B by an open set U such that

m(U) < m(B) +ε (1− β)

β.


Define a Vitali covering of B by

W = {V ∈ V : V ⊂ U, diam (V ) ≤ 1, and [S : V ] < β}.

Since B ⊂ S,

[B : V ] ≤ [S : V ].

Apply the standard method of proof of the Lebesgue-Vitali Theo-rem. First choose a V1 ∈ W with minimum rank. All V ∈ W haverank(V ) ≥ rank(V1). Then choose a V2 ∈ W with minimum rankamong those members of W that do not meet V1. Continue induc-tively. This produces a sequence of disjoint sets Vn contained in U .Their total measure is no more than the measure of U . In each Vn, theconcentration of B is < β. There may be some points of B which failto be covered by the sets Vn. Call this not-covered set

B′ = B \⋃Vn.

We claim that B′ has outer measure zero. (We are still in the process ofshowing that B has measure < ε.) Since the series

∑m(Vn) converges,

its tail,∑n>N m(Vn), tends to zero as N → ∞. To prove that B′ has

outer measure zero, then, it is enough to show that for each N ∈ N, B′

is contained in⋃n>N Vn. For since V is volumetrically engulfing, the

series∑n>N m(Vn) is no greater than L

∑n>N m(Vn), which tends to 0

as N →∞.

Since∑m(Vn) converges, its terms tend to zero. According to (c) in

the definition of volumetric engulfing, rank(Vn)→∞.

Fix an N and consider a point x ∈ B′. Since x /∈ ⋃Nn=1 Vn, since

the sets Vn are closed, and since the sets in Vx have arbitrarily smalldiameter, there is a set V ∈ W which contains x and is disjoint fromV1, . . . , VN . The set V was available for choice, but was never chosen.It has finite rank. Therefore the reason it was never chosen is that forsome first Vi, i > N , Vi ∩ V 6= ∅. Since we always choose a set withminimum rank, and since V was available to be chosen, we see thatrank(Vi) ≤ rank(V ). Because V is volumetrically engulfing, the set Viengulfs V . That is,

B′ ⊂⋃n>N

Vn,

and it follows that B′ has outer measure zero.

Except for the zero set B′, B is covered by the disjoint sets Vn, and


in each its concentration is at most β. Thus

m(B) =∑

m(Vn ∩B) ≤ β∑

m(Vn)

≤ β m(U) < β(m(B) +

ε (1− β)

β

)= β m(B) + ε (1− β).

Therefore, (1−β)m(B) < ε (1−β) and m(B) < ε. Since ε is arbitrary,it follows that B has measure zero, and as observed at the outset, thisimplies that almost every point of S is a density point with respect toV.

Corollary 3.2. If V is a volumetrically engulfing closed Vitali basisof X and S ⊂ X is measurable, then for almost every point x ∈ X, theconcentration of S in V converges to the characteristic function of Sat x as V ∈ Vx shrinks to x,

lim [S : V ] =

1 if x ∈ S0 if x /∈ S.

Proof. Theorem 3.1 applies equally to S and to its complement Sc =X\S. Thus, for almost every point x ∈ X,

if x ∈ S then [S : V ]→ 1 as V ∈ Vx shrinks to x.

if x ∈ Sc then [Sc : V ]→ 1 as V ∈ Vx shrinks to x.

Measurability implies that [S : V ] + [Sc : V ] = 1, and the corollaryfollows.

It is easy to phrase the volumetrically engulfing concept in terms ofdiameter, not rank. Namely, for some L,K,

(a) m(V ) ≤ Lm(V ).(b′) If V, V ′ ∈ V have non-empty intersection and diam(V ′) ≤ K diam(V )

then V ′ ⊂ V .(c′) If m(V )→ 0 then diam(V )→ 0.Thus, our volumetric engulfing condition is the same as assumption

(i,ii)* of [26], page 8. This makes Theorem 3.1 a consequence of thecorollary of [26], page 13. The proof given there involves maximalfunctions.

4. Juliennes

Juliennes are foliation product sets constructed as follows. Let Wu,Wcu,Wc,Wcs,Ws

be dynamically coherent foliations. At p ∈M we will choose

X ⊂ W uloc(p) D ⊂ W c

loc(p) Y ⊂ W sloc(p)

C

z

Wa(c)

Wb(cæ)

c

cæ

Waíb(C)


with carefully estimated small sizes. Then the center unstable juli-enne, the center stable julienne, and the solid julienne are

J cu = W c∩u(X ∪D)

J cs = W c∩s(D ∪ Y )

J solid = W u∩s(J cu ∪ J cs),

where the notation W a∩b(C) refers to the set of all intersection pointsof local a-manifolds and local b-manifolds that pass through C. Thatis,

W a∩b(C) = {z : for some c, c′ ∈ C, z ∈ W aloc(c) ∩W b

loc(c′)}.

Equivalently,

W a∩b(C) = W aloc(C) ∩W b

loc(C).

See Figures 2 and 3. We refer to W a∩b(C) as the (local) a∩b-saturateof C.

Figure 2. W a∩b(C) fills out C by the local foliations Wa,Wb.

We call J ∩ W aloc(z) the a-fiber of the julienne through the point

z ∈ J . In particular, if z ∈ D, the unstable fiber of J cu through z isdenoted as X(z), while the stable fiber of J cs is Y (z). If z = p thenX(z) = X is the main unstable fiber of J cu while Y (z) = Y is themain stable fiber of J cs. We call the center fiber D the center coreof the julienne.

Proposition 4.1. Juliennes are natural in three ways. They are lo-cally maximal, they iterate naturally under f , and they behave correctlyunder holonomy.

D

Y

X

Jcu

Jcs


Figure 3. The solid julienne is the u ∩ s-saturate of J cu ∪ J cs.

Proof. Juliennes are of the form

J = W a∩b(C)

for ab = cu and C = X ∪D, ab = cs and C = D ∪ Y , or ab = us andC = J cu ∪ J cs. By local maximality we mean that

J = W a∩b(J).(1)

Repeating the operation C 7→ W a∩b(C) does not enlarge a julienne. Tocheck local maximality we write

W a∩b(J) = W aloc(J) ∩W b

loc(J)

= W aloc(W

aloc(C) ∩W b

loc(C)) ∩ W bloc(W

aloc(C) ∩W b

loc(C))

⊂ W aloc(W

aloc(C)) ∩ W b

loc(Wbloc(C)) = J.

The reverse inclusion, J ⊂ W a∩b(J), is valid for any set J , whichcompletes the proof of (1).

By naturality under f iteration we mean that for J = W a∩b(C), wehave

f(J) = W a∩b(f(C)),(2)

a fact that is clear from invariance of the a- and b-foliations.Naturality under holonomy applies primarily to the center unstable

and center stable juliennes. We assert that these juliennes are productsin the following sense.

(a) The center holonomy gives a homeomorphism Y (z)→ Y (z′) be-tween any two stable fibers of J cs. It also gives a homeomorphismX(z)→ X(z′) between any two unstable fibers of J cu.


(b) The unstable holonomy gives a homeomorphism between any twocenter fibers of J cu, while the stable holonomy gives a homeomor-phism between any two center fibers of J cs.

These holonomy facts are a direct consequence of our dynamical co-herence assumption: the unstable and center foliations sub-foliate thecenter unstable foliation, while the center and stable foliations sub-foliate the center stable foliation.

Our choice X,D, Y involves several estimates on Holder exponents,contraction rates, etc. We start by defining

µ = min{m(Tf),m(Tf−1)}(3)

ν = max{‖T sf‖, ‖T uf−1‖}(4)

γ = min{m(T cf),m(T cf−1)}.(5)

The assumption of partial hyperbolicity implies

0 < µ ≤ ν < γ ≤ 1.

The behavior of f in the stable direction is a contraction whose strengthis between µ and ν, the behavior of f in the unstable direction is anexpansion whose strength is between 1/ν and 1/µ, and the behavior off in the center direction is an isometry up to a factor between γ and1/γ.

The Fiber Contraction Theorem of [15], pages 30 and 81, and [25],page 45, converts this expansion/contraction information into regular-ity statements. For example, to estimate the Holder exponent of Ecu

we proceed as follows. Let L be the vector bundle over M whose fiberat p ∈M is the space of linear maps S : Ecu

p → Esp,

Lp = L(Ecup , E

sp).

The graph transform sends S ∈ Lp to (Tf)#S ∈ Lf(p) where

graph((Tf)#S) = (Tf)(graph(S)).

This gives a bundle map

L(Tf)#−−−−−−−−−−−→ L

πy yπM

f−−−−−−−−→ M

in which the fiber is contracted more sharply than νγ−1, while the basecontraction is at worst µ. (We use the operator norm on fibers andthe Riemann metric on the base.) The zero section of L is its unique


invariant section, and it is θ-Holder if the fiber contraction θ-dominatesthe base contraction in the sense that

νγ−1 µ−θ < 1.(6)

(In the actual proof of this, to avoid circular reasoning we smoothly ap-

proximate Ecu, Es by bundles Ecu, Es, we replace L by L = L(Ecu, Es),

and we restrict the graph transform to the unit disc bundle of L.)Since our expansion/contraction hypotheses are symmetric, the same

estimate is valid for the Holder exponent of Ecs, and hence for thecenter bundle Ec = Ecu ∩ Ecs. In fact the estimate holds also for theunstable and stable bundles, and, manipulating (6), we see that all fivebundles Eu, . . . , Es are θ-Holder if

θ <log(νγ−1)

log µ.(7)

From now on, θ will denote the Holder exponent of Ec, 0 < θ ≤ 1.It is no smaller than the θ estimated in (7), but by good fortune (forinstance, if f is affine) it might be larger. In terms of this θ we willassume the following center bunching condition

ν < γ2+2/θ.(8)

Thus, γ is so near 1 that even a potentially high power of it re-mains greater than ν. Center bunching means that hyperbolic behaviorstrongly dominates center behavior.

Perturbations of the affine diffeomorphisms discussed in Section 1satisfy the center bunching hypothesis. See Section 10.

It may be of interest to express the center bunching condition in away that depends only on µ, ν.

Proposition. If µ = νr, r ≥ 1, then sufficient for the center bunchingcondition (8) is that γ is so close to 1 that

log γ

log ν<

3 + 2r −√

(3 + 2r)2 − 8

4.

The proof is left to the reader. Interpreting this inequality numer-ically shows that our current center bunching condition is modest incomparison to the bunching conditions in [22]. For example, when thehyperbolic part of the spectrum is quarter pinched

µ = ν2 =1

4

0 µ ∆ © 1

ß 1+∫

©_ )

) † ©ß ß

∆/å

åß/©

=

0 ø 1

∫


then a satisfactory γ is

γ ≥ .91.

In particular the dimensions of the stable and unstable bundles nolonger are part of the estimates.

Next, we sharpen (8). Choose α < γ and 0 < β < θ so that

ν < α2+2/β.(9)

Then set

σ =ν

α.(10)

Therefore ν < γσ and we can choose τ between them,

ν < τ < γσ.(11)

From (9) and (11) we deduce(σ

γ

)1+β

< τ <σ

γ.(12)

For (σ

γ

)1+β

= ννβ

(αγ)1+β< ν

νβ

α2+2β< ν < τ < γσ <

σ

γ.

See Figure 4 which summarizes the relations among the constants andexponents.

Figure 4. Constants denoted with a bar are imposed bythe diffeomorphism; those denoted with a dot are chosenafterward.


Now we fix a point p ∈M and specify sequences

Xn = f−n(W u(fn(p), τn)) ⊂ W uloc(p)

Dn = W c(p, σn) ⊂ W cloc(p)

Yn = fn(W s(f−n(p), τn)) ⊂ W sloc(p).

According to the general julienne definition we get sequences of juli-ennes at p

J cun = W c∩u(Xn ∪Dn)

J csn = W c∩s(Dn ∪ Yn)

J solidn = W u∩s(J cun ∪ J csn ),

We refer to this n as the rank of the julienne, and to 1/n as its size.As n→∞ the juliennes shrink down to p.

The collections of these juliennes are denoted

Jcsp = {J csn : n ∈ N} Jcup = {J cun : n ∈ N} Jp = {Jn : n ∈ N}.Letting p range over M gives a solid julienne Vitali basis

J =⋃p∈M

Jp.

Similarly if q ranges over W cs(p) or W cu(p) we get center stable andcenter unstable julienne Vitali bases,

Jcs =⋃

q∈W cs(p)

Jcsq Jcu =⋃

q∈W cu(p)

Jcuq

for the leaves W cs(p),W cu(p).

Theorem 4.2. J is volumetrically engulfing with respect to Riemannmeasure on M ; Jcs and Jcu are volumetrically engulfing with respect toRiemann measure on W cs(p) and W cu(p).

Corollary 4.3. The julienne Vitali bases are density bases: almostevery point of a measurable set is a julienne density point.

Proof. This a direct application of Theorem 3.1, which states that avolumetrically engulfing Vitali basis is a density basis.

Theorem 4.4. The unstable and stable holonomy maps are juliennequasi-conformal: the unstable holonomy is quasi-conformal with re-spect to the Vitali basis Jcs, while the stable holonomy is quasi-conformalwith respect to Jcu.

The proofs of Theorems 4.2 and 4.4 appear in Sections 6, 7, and 8.

B

r

r


5. Julienne shape

In this section we study the shape of center stable juliennes. Every-thing we say has its counterpart for center unstable juliennes. A centerstable julienne is a foliation product


where Dn is small round disc in W c(p) and Yn is a small set in W s(p).It is unclear, a priori, what a julienne looks like, beyond the fact thatit is small. For although f−n(Yn) = W s(f−n(p), τn) is round, there isno reason to expect that its fn image Yn is round or quasi-round. Thesame applies to the pre-julienne f−n(J csn ). It is the foliation productof a small set f−n(Dn) ⊂ W c(f−n(p)) and a round disc f−n(Yn) =W s(f−n(p), τn). The julienne and pre-julienne each have one roundfactor and one non-round factor.

A set that can be expressed as

T =⋃x∈B

W s(x, r)

for some B ⊂ W cloc(p) and some r > 0 is a center stable tube

T = T cs(B, r)

with base B and stable radius r. The stable fibers of a tube are bydefinition round. The shape of its base can be arbitrary. See Figures 5,6.

Figure 5. A center stable tube with one dimensionalstable fibers and two dimensional base.

r


Figure 6. A center stable tube with two dimensionalstable fibers and one dimensional base.

Proposition 5.1. The pre-julienne f−n(J csn ) is tube-like. It consistsof the center core f−n(Dn), through which pass s-fibers. The s-fibersare approximately round discs of radius τn. More precisely, given ε > 0,if n is large, then the pre-julienne is bracketted between tubes with basef−n(Dn) and stable radii (1± ε)τn.

The proof of this proposition involves the center bunching and Holderassumptions used to choose the julienne shape. The next lemma gen-eralizes easily to any foliation whose tangent bundle is θ-Holder con-tinuous, where 0 < β < θ ≤ 1.

Lemma 5.2. The center holonomy h : W s(p) → W s(q), h(p) = q, isapproximately isometric at small β-Holder scale in the followingsense. Given ε > 0 there is a δ > 0 such that if

r1+β ≤ ρ ≤ r < δ(13)

and if q ∈ W c(p, r), then h carries a round disc D of radius ρ in W s(p)to a nearly round disc h(D) of radius ρ in W s(q). More precisely, ifD = W s(p, ρ) then h(D) is bracketted between round discs

W s(q, (1− ε)ρ) ⊂ h(D) ⊂ W s(q, (1 + ε)ρ).

Proof. The point to note here is that ρ can be quite a bit smallerthan r, but not arbitrarily smaller; it can never be less than r2. Discswhose radii are too small may be grossly distorted in shape by thecenter holonomy map h. See Figure 7, which indicates that the shapeand size of h(D) is under good control at one scale, but still may bepathological at a smaller scale. We recapitulate the proofs of Lemmas

Ws(q, (1+‰)®) Ws(q, (1-‰)®)

h(D)

Ec(r)p

Esp

Wc(p, r)


Figure 7. At small β-Holder scale h(D) is approxi-mately round.

1 and 2 in Section 4 of [22]. In an exponential chart at p, the centerleaf through a point z near p is the graph of a function

φz : Ecp(r)→ Es

p.

See Figure 8. The function φz(x) is differentiable with respect to x. The

Figure 8. Locally, in exponential charts, leaves are graphs.

norm of its derivative ∂φz/∂x is the slope of the leaf in the exponentialchart. By assumption ∂φz/∂x is θ-Holder continuous, and at the point(z, x) = (p, 0), ∂φz/∂x = 0. Thus

‖∂φz∂x‖ ≤ Krθ,

where (z, x) varies in the r-neighborhood of (p, 0). Geometrically, thismeans that the center leaves are quite flat, quite horizontal.

Ws(q, (1+‰/2)®)

p

D

Ws(p) Ws(q)


Consider the horizontal holonomy H. It is translation parallel to theplane Ec

p. Continuity of Es implies that H has the following property.Given ε > 0 there is a δ > 0 such that if ρ ≤ r < δ then

H(D) ⊂ W s(q, (1 + ε/2)ρ).

See Figure 9. No underbound for ρ is needed here.

Figure 9. At small scale, the horizontal holonomy isapproximately isometric.

Over a distance r, the center leaf diverges from the horizontal planeparallel to Ec

p by at most

Krθr = Kr1+θ,

a quantity that is negligible in comparison to ρ. For when r is small,(9) and (13) imply that

r1+θ ¿ r1+β ≤ ρ.

Hence,

h(D) ⊂ W s(q, (1 + ε)ρ).

Applying the same reasoning to h−1, we see that

h−1(W s(q, (1− ε)ρ)) ⊂ D,

and it follows that h(D) is bracketted between the round discsW s(q, (1± ε)ρ) as claimed.


Proof of Proposition 5.1. The main stable fiber of the pre-julienne

f−n(J csn ) = W c∩s(f−n(Dn) ∪W s(f−n(p), τn))

is the round disc W s(f−n(p), τn), while the center core is f−n(Dn) =f−n(W c(p, σn)). Since ‖T cf‖ ≤ γ−1,

f−n(W c(p, σn)) ⊂ W c(f−n(p), σn/γn).

We will apply Lemma 5.2, with r = σn/γn and ρ = τn. We need tocheck (13), namely, (

σn

γn

)1+β

< τn <σn

γn,

and by (12), this is true. Also, when n is large, r = σn/γn is small.Lemma 5.2 then completes the proof.

6. Juliennes under unit holonomy

In this section we study the unstable holonomy h : W cs(p)→ W cs(q),q = h(p), and how it affects center stable juliennes. Everything we sayhas its counterpart for center unstable juliennes. We assume through-out that q lies in the unit unstable manifold of p,

q ∈ W u(p, 1).

Theorem 6.1. There is an integer ` such that for all large n,

J csn+`(q) ⊂ h(J csn (p)) ⊂ J csn−`(q).(14)

In words, the h image of a rank n julienne at p is bracketted betweenrank n± ` juliennes at q.

Proof. Referring to J csn+`(p) as the small julienne at p and to J csn (q)as the large julienne at q, we claim that the holonomy map h carriesthe small julienne into the large one,

h(J csn+`(p)) ⊂ J csn (q).(15)

The subtlety of the analysis is present exactly here. The shapes of thejuliennes may be awful, but in terms of holonomy they nest. Using h−1

and juggling indices, we will see that (15) implies (14).The unstable holonomy map h sends center manifolds to center man-

ifolds, and the restriction of h to each individual center manifold is ofclass C1 or better. See Theorem B of [24]. (On the full center stablemanifold h is of course much less regular than Lipschitz.) By compact-ness, the supremum, L ≥ 1, of the Lipschitz constants of all the local


unstable holonomy maps h : W c(p) → W c(q) is finite. Choose ` suchthat

σ` ≤ 1

6L.(16)

The center core disc Dn+`(p) has radius σn+`, and thus h sends it to asubset of W c(q) having radius Lσn+`, which by the choice of ` in (16)is less than σn/2. Thus

h(Dn+`(p)) ⊂ 12Dn(q),(17)

where 12Dn(q) is the half-sized sub-disc of Dn(q). The inclusion (17) is

a step toward (15). We want to show that the holonomy image of onejulienne is contained in another julienne, and (17) shows that at leastthis is true of their center cores.

Since the unstable holonomy is defined by the invariant foliation Wu,it commutes with f . Specifically, if we write

hn : W cs(f−n(p))→ W cs(f−n(q))

then

h = h0 = fn ◦ hn ◦ f−n.Thus, (15) is equivalent to

hn ◦ f−n(J csn+`(p)) ⊂ f−n(J csn (q)).(18)

Think of the holonomy map as taking an object located on one cen-ter stable manifold, and “pushing it across” to another center stablemanifold. Then (18) has two advantages over (15).

(a) The distance between the domain and target for hn (the “pushacross distance”) is much less than that for h.

(b) The domain and target of hn in (18) are more round in shape thanthose that confront h in (15). For pre-juliennes are less elongatedthan juliennes.

To prove (18) it suffices to find a tubes Tn, T′n such that

f−n(J csn+`(p)) ⊂ Tn

T ′n ⊂ f−n(J csn (q))

hn(Tn) ⊂ T ′n

(19)

Set

Bn = f−n(12Dn(q)) B′n = f−n(Dn(q)).

By Proposition 5.1 the pre-julienne f−n(J csn (q)) is tube like. It containsa tube T ′n with base B′n and stable radius 2τn/3. The distance from

(©ß)n1-2

ßn/2

h(Dn+Ô(p))

ßn q

Dn(q)

Dn(q)1-2

Bnæ

Bn

ÅBnæ

Bn

f-n(q)


12Dn(q) to the boundary of Dn(q) is σn/2. Under f−n this distance can

decrease by at worst the factor γn. Thus,

x ∈ Bn ⇒ d(x, ∂B′n) ≥ (γσ)n

2.(20)

See Figure 10. We do not assert that Bn lies in the middle of B′n, merely

Figure 10. The center cores of the julienne and pre-julienne at q and at f−n(q).

that its distance to ∂B′n decays no faster than a slow exponential. Theshape of the pair Bn ⊂ B′n can in principle be quite messy.

By Proposition 5.1 the pre-julienne f−(n+`)(J csn+`(p)) is also tube-like.It is contained in a tube Tn+`,

f−(n+`)(J csn+`(p)) ⊂ Tn+` =⋃

x∈f−(n+`)(Dn+`(p))

W s(x, 2τn+`).


Since ‖T sf‖ ≤ ν < σ, we see from (16) that

f−n(J csn+`(p)) ⊂ f `(Tn+`) ⊂⋃

x∈f−n(Dn+`(p))

W s(x, τn/3) = Tn,

a tube in W cs(f−n(p)) with base f−n(Dn+`(p)) and stable radius τn/3.hn pushes Tn across from W cs(f−n(p)) into W cs(f−n(q)). The dis-

tance between these two center stable manifolds is at most νn sincetheir original distance apart is at most 1, and ‖T uf−1‖ ≤ ν. It is un-likely that hn sends an s-fiber of Tn to an s-fiber of W cs(f−n(q)). ForEu⊕Es is probably not Frobenius integrable. Still, we can easily esti-mate how large hn(W s(x, τn/3)) is. Directly, or as a trivial alterationof the proof of Lemma 5.2,

hn(W s(x, τn/3)) ⊂ W cs(hn(x), 2νn + τn/3).(21)

(11) implies that for large n,

2νn + τn/3 < τn/2¿ (γσ)n.

By (20), hn(Tn) lies far enough away from ∂B′n that it is contained inthe local s-saturate of B′n,

hn(Tn) ⊂ W sloc(B

′n).

See Figure 11. Likewise

hn(Tn) ⊂ W cs(Bn, τn/2) ⊂ T ′n,

which implies (19), hence (18), and hence (15).It remains to deduce (14) from (15). Since the latter is valid for all

large n, we can replace n with n− `. This gives half of (14),

h(J csn (p)) ⊂ J csn−`(q).

Interchanging p and q, and applying (15) to h−1 instead of to h, we get

h−1(J csn+`(q)) ⊂ J csn (p).(22)

Applying h to both sides of (22), we get the other half of (14).

7. Juliennes engulf

In this section we verify the geometric part of the volumetric en-gulfing property. In the next section we work out the measure ratioestimate.

Theorem 7.1. Center stable juliennes have the following engulfingproperty. There is an ` ∈ N such that if two center stable juliennesJ csn+`, J

csn+`′ meet then

J csn+`′ ⊂ J csn .

Tnæ

f-n(Jcs(q))

Wcs(Bn,†n/2)

Bnhn(Tn)


Figure 11. Despite its ragged ends, hn(Tn) is containedin T ′n.

Proof. Recall that n is the rank of the julienne J csn . The theoremasserts that a rank n julienne will engulf a rank n + ` julienne, if thelatter meets the rank n+ ` sub-julienne of the former. Write


J csn+` = W c∩s(Dn+` ∪ Yn+`)

J csn+`′ = W c∩s(D′n+` ∪ Y ′n+`).

We refer to these juliennes as the big julienne, the small julienne, andthe neighboring julienne, respectively. According to Proposition 5.1,the stable fibers of a julienne of rank n are images by fn of approxi-mately round discs of radius τn. Since ‖T sf‖ ≤ ν, these fn images arecontained in stable discs of radius νnτn < νnσn, a number that is muchsmaller than the radius σn of the core disc Dn. Thus, the diameter ofa center stable julienne is approximately the diameter of its core disc,and the diameter of the neighboring julienne that we hope to engulf is

Dn

Jcsn

Jcsn+Ô

Jcsn+Ôæ

Yn


approximately σn+`. We choose ` so that

σ` ≤ 1

3.

Then the the two intersecting juliennes of rank n+` have diameter lessthan 1/3 of the diameter of the big julienne, and one of them, J csn+`,includes the point p. It follows that the neighboring julienne, J csn+`

′, isso close to the middle of the center core disc Dn, of the big juliennethat it projects into Dn under the stable holonomy map. That is

J csn+`′ ⊂ W s

loc(Jcsn ).

See Figure 12. All the s-fibers of the neighboring julienne lie on local

Figure 12. The neighboring julienne is contained in thelocal stable saturate of Dn.

stable manifolds passing through the core disc Dn of the big julienne.If we can show that all the c-fibers of the neighboring julienne lie onlocal center manifolds passing through the main stable fiber Yn of thebig julienne, then by the naturality of the julienne definition, Proposi-tion 4.1, the big julienne engulfs the neighboring julienne.

To understand the center fibers, we apply f−n to the picture. Ac-cording to Proposition 5.1 the big julienne becomes tube-like. We referto the f−n images of the other two juliennes as the small pre-julienneand the neighboring pre-julienne. Their s-fibers, being the f `-imagesof approximately round discs of radius τn+`, are contained in rounds-discs of radius ν`τn+`. Moreover, the center core of the small pre-julienne, f−n(Dn+`), is contained in the center core f−n(Dn) of the


tube-like pre-julienne, and its stable fibers are contained in discs, cen-tered at points of f−n(Dn), that have radius < τn/3. One of theses-fibers of the small pre-julienne meets an s-fiber of the neighboringpre-julienne. The latter s-fiber lies in a disc of radius < τn/3. Twointersecting discs of these sizes and positions are contained in the s-discof radius τn. See Figure 13.

Figure 13. If a small, centered sub-disc of a big discmeets a second small disc, the big disc engulfs it.

Hence, at least one, special s-fiber of the neighboring pre-julienneis contained in an s-fiber of the big, tube-like pre-julienne. By theconstruction of juliennes as foliation products, all points on an s-fiberof a julienne lie on local center manifolds that pass through the mainstable fiber of the julienne. See Proposition 4.1. Therefore, all pointsof the special s-fiber of the neighboring pre-julienne lie both on localcenter manifolds that pass through the main stable fiber of the bigpre-julienne, and on local center manifolds that pass through the mainstable fiber of the neighboring pre-julienne. Local center manifolds areunique. Therefore, all the c-fibers of the neighboring pre-julienne lie onlocal center manifolds passing through the main stable fiber f−n(Yn)of the big pre-julienne. Re-applying fn, we conclude that the sameis true of the juliennes themselves: all the c-fibers of the neighboringjulienne lie on local center manifolds passing through the main stablefiber Yn of the big julienne. This implies that the big julienne engulfsthe neighboring julienne.

Theorem 7.2. Solid juliennes have the same engulfing property: thereexists an integer L such that a rank n julienne will engulf a rank n+L


julienne, if the latter meets the rank n+ L sub-julienne of the former.

Lemma 7.3. If J, J ′ are solid juliennes and the local unstable andlocal stable saturates of J contain the main center stable and centerunstable slices of J ′,

W uloc(J) ⊃ J cs′ and W s

loc(J) ⊃ J cu′,

then J ⊃ J ′.

Proof. The proof is like that of Proposition 4.1. Take a point z′ ∈J ′. It is the intersection point for a pair of local unstable and stablemanifolds,

z′ = W sloc(x

′) ∩W uloc(y

′)

where x′ ∈ J cu′, y′ ∈ J cs′. But by assumption, x′ lies on a local stablemanifold through the main center unstable slice J cu of J , and y′ lieson a local unstable manifold through the main center stable slice J cs

of J , say

x′ ∈ W sloc(x) y′ ∈ W u

loc(y)

for some x ∈ J cu, y ∈ J cs. Since local stable and unstable manifoldsare unique, we see that z′ ∈ J .

Proof of Theorem 7.2. We use Theorems 6.1 and 7.1 repeatedly.The L = 3` we produce seems far from optimal.

Suppose that Jn(p), Jn(p′) are two rank n solid juliennes that meetat a point z. By the julienne construction, z lies on a local unstablemanifold through the main center stable slice J csn of Jn, and also on alocal unstable manifold through the main center stable slice J csn

′ of J ′n.Local unstable manifolds are unique, so the local unstable manifoldthrough z, passes through both J csn and J csn

′. Dually, the local stablemanifold through z passes through the main center unstable slices ofboth Jn and J ′n, say

x = W sloc(z) ∩ J cun x′ = W s

loc(z) ∩ J cun ′

y = W uloc(z) ∩ J csn y′ = W u

loc(z) ∩ J csn ′

See Figure 14.We apply Theorem 7.1 to the rank n center stable julienne J csn (y′)

at y′. Decreasing its rank by ` causes it to engulf J csn (p′),

J csn−`(y′) ⊃ J csn (p′).(23)

Then we apply Theorem 6.1 to the center stable julienne J csn−`(y) at y.Decreasing its rank by ` causes its unstable holonomy image to engulf

yæ yz

p

pæ

x

xæ

Jcs(pæ)n

Jcs(yæ)n


Figure 14. Locating the points p, p′, etc. Read verti-cals as center stable and horizontals as unstable.

J csn−`(y′),

hu(J csn−2`(y)) ⊃ J csn−`(y′).(24)

Finally, decreasing the rank of the center stable julienne J csn−2`(p) by afurther ` causes it to engulf J csn−2`(y),

J csn−3`(p) ⊃ J csn−2`(y).(25)

Combining (23), (24), and (25), we see that the unstable holonomyimage of J csn−3`(p) engulfs J csn (p′). In terms of saturates this means

W uloc(J

csn−3`(p)) ⊃ J csn (p′).

Dually we see that

W sloc(J

cun−3`(p)) ⊃ J cun (p′).

Then, according to Lemma 7.3, J ⊃ J ′.

We conclude this section with an extension of Proposition 5.1. A setthat can be expressed as

T =⋃x∈B

W s(x, r)

for some B ⊂ W culoc(p) and some r > 0 is a solid tube

T = T solid(B, r)

with base B and stable radius r.


Theorem 7.4. A solid pre-julienne f−n(Jn(p)) can be bracketted be-tween solid tubes

Tn ⊂ f−n(Jn(p)) ⊂ T ′n

such that Tn has base f−n(J cun+`) and stable radius rn = (µτ)`τn, whileT ′n has base f−n(J cun−`) and stable radius r′n = (µτ)−`τn.

Proof. Theorem 7.2 implies that Jn(p) engulfs juliennes Jn+`(x) whichtouch it; for example this true when x ∈ J cun+`. The main stable fiber of

f−(n+`)(J csn+`(x)) is the round, stable disc W s(f−(n+`)(x), τn+`). The f `

image of this disc contains a round, stable sub-disc of radius rn. Lettingx range over J cun+`(p), we see that f−n(Jn(p)) contains the tube Tn. Theproof that f−n(Jn(p)) is contained in the tube T ′n is similar.

8. Julienne measure

Fix a Riemann structure on TM that is adapted to the partiallyhyperbolic diffeomorphism f . It induces Riemann structures tangentto the leaves of the foliations Wu, Wcu, Wc, Wcs, Ws. These inducedRiemann structures give rise to Riemann measures m, mu, . . . ,ms onM and on the leaves of the foliations Wu, . . . ,Ws.

We need to estimate the volume of discs and tubes with respectto these measures. To do so, we first modify the exponential chartsexpp : TpM(ρ) → M as follows. (Everything will be done locally –i.e., in a modest sized neighborhood of the zero section, say of radiusρ.) The exponential pre-image of the local center unstable manifoldW cu

loc(p) is a graph,

exp−1p (W cu

loc(p)) = graph(φcup ),

where φcup : Ecup (ρ) → Es

p(ρ) is a function of at least class C1+ Holder.Write x ∈ Ecu

p (ρ), y ∈ Esp(ρ), and define

Φcup (x, y) = (x, y + φcup (x)).

Φcup is a diffeomorphism of TpM(ρ) to another neighborhood of the

origin of TpM . At the origin of Ecup , φcup and its derivative Dφcup vanish.

The derivative is taken with respect to x ∈ Ecup . Thus, at the origin of

TpM , DΦcup = I, the identity map.

The union of all the maps Φcup as p varies in M , is a homeomorphism

Φcu from TM(ρ) to another neighborhood of the zero section. It and itsfiber derivative DΦcu

p are Holder continuous on TM . The compositionexp ◦Φcu is a partially adapted exponential map. It sends each Ecu

p (ρ)to W cu

loc(p).


Repeating this type of modification for the other invariant manifoldsleads to a homeomorphism H from TM(ρ) to another neighborhood ofthe zero section covering the identity map on M such that the restric-tion of H to TpM(ρ) is a C1+ Holder diffeomorphism which carries thediscs Eu

p (ρ), Ecup (ρ), Ec

p(ρ), Ecsp (ρ), Es

p(ρ) to the exponential pre-imagesof the corresponding local invariant manifolds. The composition

e = exp ◦His the adapted exponential map. Its restriction to TpM(ρ) is aC1+ Holder local diffeomorphism ep from TpM(ρ) to M sending the discsEup (ρ), . . . , Es

p(ρ) to the corresponding invariant manifolds. Like theunadapted exponential, the tangent to ep at the origin is the identity,

T0(ep) = I.(26)

Using e, we lift all objects in M to TpM , writing a bar over the liftedobject. For example, S ⊂M lifts to S = e−1

p (S) ⊂ TpM . Likewise, themeasure m on M lifts to a measure m = e∗p(m) in TpM , the m-measure

of S being m(S). The measure m is not translation invariant withrespect to the linear structure of TpM .

The adapted Riemann structure on TM gives each tangent spacean inner product, and the inner product induces a second, natural,translation invariant measure, the Riemann measure mp on TpM . Dueto (26), one can view mp as the linear part of m at the origin of TMp.

The other measures mu, . . . ,ms lift to measures mu, . . . , ms on thesubspaces Eu

p , . . . , Esp. On these linear subspaces, and on all affine

subspaces of TpM parallel to them, we also have the Riemann measuresmup , . . . ,m

sp induced by the inner product. Again, one can view the

latter measures as the linear parts of the former measures.

Lemma 8.1. The volume of a small tube is approximately the volumeof its base times the volume of its fiber.

m(T )

mcu(B) ms(W s(p, r))≈ 1.

More precisely, for any ε0 > 0 there is a δ > 0 such that if T is a solidtube with base B ⊂ W cu(p, δ) and stable radius r < δ then∣∣∣∣∣ m(T )

mcu(B) ms(W s(p, r))− 1

∣∣∣∣∣ < ε0.(27)

Corresponding assertions are true for center unstable tubes and centerstable tubes.


Proof. According to (26), when T is small,

mp(T )

m(T )≈ 1

mcup (B)

mcu(B)≈ 1

msp(W

s(p, r))

ms(W s(p, r))≈ 1.(28)

It is therefore fair in the proof of (27) to replace the tube, its base, itsfiber, and the various measures by corresponding objects in TpM .

Since ep is adapted to the invariant foliations of f , the base B of Tlies in Ecu

p . Due to (26), given ε > 0, it actually lies in Ecup ((1 + ε)δ)

when T is small.The main stable fiber of T is e−1

p (W s(p, r)). Since ep is adapted tothe invariant foliations of f , it lies in Es

p. By (26), when T is small itis bracketted between round stable discs

Esp((1− ε/2)r) ⊂ e−1

p (W s(p, r)) ⊂ Esp((1 + ε/2)r).

Let πsp : TpM → Esp be the projection with kernel Ecu

p . Since, when

T is small, the stable fibers of T uniformly C1 approximate the mainstable fiber, their πsp-projections into Es

p are bracketted between discswith slightly relaxed radii, say Y± = Es

p((1± ε)r). In particular,

Y− ⊂ W s(p, r) ⊂ Y+.

It also follows that T is bracketted between T±,

T− ⊂ T ⊂ T+,

where

T± = {z ∈ W s(x, 2r) : x ∈ B and πsp(z) ∈ Y±}.See Figure 15. The set T± is sliced perfectly by the affine subspacesEcup × y, y ∈ Y±. In fact the intersections of T± with these affine

subspaces are the holonomy images of the base, hy(B). The stablefoliation is absolutely continuous, and so is its lift to TpM . The RadonNikodym derivative of its transverse holonomy, RN(hy), is continuous.When y = 0, hy is the identity and the Radon Nikodym derivative is1. By continuity, when y is small, RN(hy) ≈ 1. Thus, by Fubini’sTheorem

mp(T±)

mcup (B) ms

p(Y±)=

1

msp(Y±)

∫y∈Y±

mcup (hy(B))

mcup (B)

dmsp ≈ 1,

(29)

if T is small. Since Y± are round discs,

msp(Y+)

msp(Y−)

=(

1 + ε

1− ε

)s≈ 1.(30)

T+

T-

T_


Figure 15. Why Fubini’s Theorem applies to T± butnot directly to T .

Since T± bracket T and Y± bracket W s(p, r),

mp(T−)

mcup (B) ms

p(Y+)≤ mp(T )

mcup (B) ms

p(Ws(p, r))

≤ mp(T+)

mcup (B) ms

p(Y−).

Hence, (28), (29), (30) imply (27).

Next we give an estimate to control volume of a small set along anorbit of our C2 diffeomorphism f : M → M . We assume that E is aθ-Holder continuous, integrable, Tf -invariant k-dimensional subbundleof TM , θ > 0. We denote by U a small k-dimensional disc tangent toE with p ∈ U , and by mE the Riemann measure on such a disc. Therestriction of Tf to E is TEf . We have in mind the example thatU is a small center disc, a small center stable disc, or a small solidneighborhood. Then E = Ec, Ecs, TM respectively.

Theorem 8.2. The restriction of a high iterate of f to a small enoughdisc tangent to E is approximately measure theoretically con-formal in the following sense. Fix a constant κ < 1. If the disc U isso small that for i = 0, . . . , n,

diam f iU ≤ κn,


then

mE(f iU)

|detTEp fi| mE(U)

≈ 1.(31)

Proof. mE(f iU) is the integral of the absolute determinant of TEf i

over U . Thus,

mE(f iU)

|detTEp fi| mE(U)

=1

mE(U)

∫x∈U

i−1∏j=0

detTEfjxf

detTEfjpfdmE,

which, since E is θ-Holder and f is C2, is the average value on U of aproduct

∏(1 + (εj(x))θ) where

|εj(x)| ≤ C diam f iU ≤ Cκn.

Because nκnθ → 0 as n→∞, the product approximates 1.

We have chosen the juliennes so that they remain exponentially smallunder n iterates of f . According to Theorem 8.2, then, the effect off i, |i| ≤ n, on the fibers of the juliennes is approximately measuretheoretically conformal. Under f i the fiber measures get multipliedapproximately by the appropriate determinant. See (31).

Theorem 8.3. Fix an integer ` ≥ 1. Center stable juliennes at pwhose rank differs by at most ` have uniformly comparable measure.More precisely, there are constants, 0 < c ≤ C < 1 such that if n islarge then the center stable juliennes at p, J csn and J csn+` satisfy,

c ≤ mcs(Jcsn+`)

mcs(J csn )≤ C.

Proof. We first assume ` = 1. Since the center cores Dn, Dn+1 ofJ csn , J

csn+1 are round discs of known radii σn, σn+1, there are constants

0 < c1 ≤ C1 < 1 such that

c1 ≤mc(Dn+1)

mc(Dn)≤ C1.

Let Bn = f−n(Dn) and Bn+1 = f−n(Dn+1). Clearly, Bn+1 ⊂ Bn andthey are exponentially small. According to Theorem 8.2, since the ratioof the center stable measures of Dn, Dn+1 is uniformly controlled, so isthe ratio between the center stable measures of their iterates Bn, Bn+1.In fact, nearly the same control constants can be used.

Now consider the stable fibers Yn(x), Yn+1(x) of the pre-juliennef−n(J csn ), f−n(J csn+1) at x ∈ Bn. By Proposition 5.1, the former is ap-proximately equal to a round disc,

Yn(x) ≈ W s(x, τn).


The latter is bracketted between round discs,

W s(x, µτn+1) ⊂ Yn+1(x) ⊂ W s(x, ντn+1).

Thus, for some constants 0 < c2 ≤ C2 < 1, we have

c2 ≤ms(Yn+1(x))

ms(Yn(x))≤ C2.

Applying Lemma 8.1 at the point f−n(p) with B = Bn, gives a com-parison

c3 ≤mcs(f−n(J csn+1))

mcs(f−n(J csn ))≤ C3.

Iterating by fn and applying Theorem 8.2 again, we see that the centerstable juliennes J csn , J

csn+1 enjoy the same comparability of measure,

c4 ≤mcs(J csn+1)

mcs(J csn )≤ C4,

which completes the proof when ` = 1. For general ` the choicesc = c`4, C = C`

4 give

c ≤ mcs(J csn+`)

mcs(J csn )≤ C.

Theorem 8.4. The Vitali bases J, Jcu, and Jcs are volumetrically en-gulfing, and hence are density bases.

Proof. The assertion for J means that if S ⊂ M is measurable withrespect to the Riemann measure m on M then almost every point ofS is a density point with respect to J. In Theorem 7.2 we showed thata solid julienne Jn is engulfed by J ′n−` if Jn meets J ′n. In Theorem 8.3we showed that there is a uniform bound on the volume ratio betweenengulfer and engufee. Thus, Theorem 3.1 applies and J is a densitybasis.

The assertion for Jcu means that if S ⊂ W cu(p) is measurable withrespect to Riemann leaf measure mcu then almost every point of S isa density point with respect to Jcu. The proof is similar.

Theorem 8.5. The stable and unstable holonomy maps are juliennequasi-conformal, and hence any unstable holonomy map preserves allJcs density points, while a stable holonomy map preserves all Jcu densitypoints.


Proof. According to Theorems 6.1 and 8.2, if q ∈ W u(p, 1) then theunstable holonomy map h : W cs

loc(p)→ W csloc(q) satisfies

J csn+`(q) ⊂ h(J csn (p)) ⊂ J csn−`(q),

and the measure ratio between J csn+`(q) and J csn−`(q) is bounded awayfrom zero. That is, h is julienne quasi-conformal. According to Propo-sition 2.1, h preserves all Jcs density points. The dual statements holdfor a stable holonomy map.

Next, we relate density points of a set to density points of the stableprojection of that set onto a local center unstable manifold.

Theorem 8.6. Assume that A is measurable and s-saturated. Thepoint p is a density point of A with respect to the solid julienne basisJ if and only if p is a density point of A ∩W cu(p) with respect to thecenter unstable julienne density basis Jcu.

Proof. Suppose that p is a density point of A with respect to J. Then

[A : Jn(p)]→ 1

as n → ∞. According to Theorem 8.2, f−n(A) is also highly concen-trated in the pre-julienne f−n(Jn(p)),

[f−n(A) : f−n(Jn(p))]→ 1.

By Theorem 7.4 there is a tube Tn ⊂ f−n(Jn(p)) whose volume ratioin the pre-julienne is bounded away from zero. Hence

[f−n(A) : Tn]→ 1.

The base of the tube is f−n(J cun+`) and its stable radius is rn. SinceA is s-saturated, so is f−n(A). Thus f−n(A) consists of whole stablemanifolds, and its intersection with the tube Tn consists of whole stablediscs of equal radius rn. We apply Lemma 8.1 to two tubes: the tubeTn and the tube Tn ∩ f−n(A). This gives

m(Tn ∩ f−n(A))

mcu(f−n(J cun+`) ∩ f−n(A)) ms(W s(f−n(p), rn))≈ 1

while the volume of Tn itself satisfies

m(Tn)

mcu(f−n(J cun+`)) ms(W s(f−n(p), rn))

≈ 1.

Hence, the concentration of f−n(A) in the base f−n(J cun+`) must tendto 1. Applying fn to this configuration and using Theorem 8.2 again,we see that the concentration of A in the center unstable julienne J cun+`

also tends to 1. Since ` is fixed, this shows that p is a density point ofA ∩W cu(p) with respect to Jcu.


The converse is proved similarly, using the large tube T ′n from The-orem 7.4.

We combine Theorems 8.5 and 8.6 as follows.

Theorem 8.7. If A ⊂ M is measurable and essentially u-saturatedthen its set of J density points consists of whole unstable manifolds.

Proof. A set is essentially u-saturated if it differs from a u-saturatedset by a zero set. Alteration by a zero set has no effect on the set of(solid) density points, so it is fair to assume that A is u-saturated. Letp be a J density point and let q be a point in W u(p). We claim that qis a J density point of A.

Assume at first that q lies in the local unstable manifold of p. ByTheorem 8.6, p is a density point of A∩W cs(p) with respect to Jcs. ByTheorem 8.5 and the fact that A is u-saturated, q is a Jcs density pointof A∩W cs(q). By Theorem 8.6, q is a J density point of A. Thus, theset P of J density points consists of whole local unstable manifolds.Because (global) unstable manifolds are connected and consist locallyof local unstable manifolds, the set P of density points actually consistsof whole unstable manifolds.

9. Proof of Theorem A

The proof of Theorem A follows the same pattern as in [2, 13, 22]. Wesuppose that f is not ergodic and, using the Birkhoff Ergodic Theorem,we find a set A0 such that A0 is f -invariant, has intermediate measure,and is essentially saturated by stable and unstable manifolds. Let Abe the set of J density points of A0 and let B be the set of J densitypoints of M\A0. According to Theorem 8.4, almost every point of A0

is a J density point, so A has positive measure. Likewise the set Bhas positive measure. The set P of all us-paths originating at pointsof A is, by definition, us-saturated and it contains A, so it forms a us-saturated set of positive measure. Essential accessibility implies thatP has full measure, and therefore it meets B. The upshot is that thereare points p ∈ A, q ∈ B and a us-path from p to q. Since p is a Jdensity point of A0, Theorem 8.7 implies that the entire first leg of thepath consists of J density points of A0. The same applies to the secondleg, and so forth. We conclude that q is also a J density point of A0

and this is incompatible with it also being a J density point of M\A0.


10. Affine diffeomorphisms

In this section we prove Theorems B, C, and Corollaries b, c fromSection 1. For readers unfamiliar with Lie groups, we include somestandard material, see also [22] where some of this is worked out for thecase of left translation. We use the standard notation from differentialtopology that TxM is the tangent space to M at x and that Txf is thetangent to a map f at x.

Recall what we assumed in Section 1.(a) G is a connected Lie group and B ⊂ G is a proper, closed sub-

group.(b) G/B is compact.(c) Haar measure projects to a finite measure m on G/B, invariant

under left translation, xB 7→ gxB(d) f : G/B → G/B is an affine diffeomorphism. That is, it is

part of a commutative diagram

Gf = Lg ◦ A−−−−−−−−−−−−−−−→ G

πy yπ

G/Bf−−−−−−−−−−−−−−−→ G/B

where A is an automorphism of G, Lg is left multiplication bysome fixed element g ∈ G, and, to make the projection welldefined, A(B) = B.

Further, f induces a Lie algebra automorphism defined by

a(f) = Ad(g) ◦ TeA = TgRg−1 ◦ TeLg ◦ TeA.Given a linear subspace E of the Lie algebra g = TeG, we extend it

to a right invariant subbundle R(E) ⊂ TG whose fiber at x ∈ G is

Ex = TeRx(E).

Right invariance of R(E) means that for all x, y ∈ G, TxRy carries Exto Exy.

Proposition 10.1. a(f)-invariance of E is equivalent to T f -invarianceof R(E).

Proof. Observe first that

Tef = TeRg ◦ a(f).(32)

For f(x) = gA(x) = Lg ◦ A(x) implies that

TeRg ◦ a(f) = TeRg ◦ TgRg−1 ◦ TeLg ◦ TeA = TeLg ◦ TeA,


which is Tef .From (32) and right invariance of R(E) we infer that E is a(f)-

invariant if and only if Tef sends E = Ee to Eg. It remains to show thatthis T f -invariance of R(E) from e to f(e) = g implies T f -invarianceat the general point y ∈ G.

Commutativity of the diagram

Gf−−−−−−−−→ G

Ry

y yRA(y)

Gf−−−−−−−−→ G

is simple to check: if x ∈ G then

RA(y) ◦ f(x) = gA(x)A(y) = gA(xy) = f(xy) = f(Ry(x)).

Taking the tangent maps of the diagram at x = e gives

TeGTef−−−−−−−−−→ TgG

TeRy

y yTgRA(y)

TyGTyf−−−−−−−−−→ Tf(y)G

Hence

Tyf = TgRA(y) ◦ Tef ◦ (TeRy)−1.

SinceR(E) is TR-invariant, it is also (TR)−1-invariant, and T f -invarianceof R(E) from e to f(e) propagates to T f -invariance everywhere.

Because B acts on the right, the bundle R(E) projects naturally toa quotient bundle, Tπ(R(E)) = R(E) ⊂ T (G/B), where E = Tπ(E),

R(E)T f−−−−−−−−−→ R(E)

Tπ

y yTπR(E)

Tf−−−−−−−−−→ R(E).

The fiber of R(E) at xB ∈ G/B is Txπ(Ex), which is isomorphic toEx/(Tx(xB)∩ Ex). Here, we think of xB both as a subset of G and asa point in G/B.

Corollary 10.2. a(f)-invariance of E implies Tf -invariance of R(E).

Proof. From commutativity of the diagrams this is clear.


Proposition 10.3. Let gρ be the sum of the generalized eigenspacesof a(f) whose eigenvalues have modulus ≤ ρ where 0 < ρ ≤ 1. Thengρ is a Lie subalgebra of g, not merely a linear subspace of TeG.

Proof. The assertion is valid not only for a(f), but for any Lie algebraautomorphism a : g → g of a complex, finite dimensional Lie algebra,and is a consequence of the following general fact: the factors S and Min the Jordan Decomposition a = SM , are automatically Lie algebraautomorphisms of g, not merely linear transformations TeG → TeG.As usual S is semi-simple and M = I + N with N nilpotent. SeeChapter I.4 of Borel’s book about linear algebraic groups [3] wherethis is discussed. Note that the automorphism group of g is a linearalgebraic group.

Suppose first that a itself is semi-simple, a = S. Then there is a com-plete eigenbasis {v1, . . . , vn} corresponding to eigenvalues {λ1, . . . , λn}.Since a is a Lie algebra automorphism,

S[vi, vj] = [Svi, Svj] = λiλj[vi, vj],

and [vi, vj] is seen to be either the zero vector or an S-eigenvector witheigenvalue λiλj. In particular, gρ is closed under Lie bracket since|λi|, |λj| ≤ ρ ≤ 1 implies that |λiλj| ≤ ρ2 ≤ ρ.

Next assume that a = SM with S semi-simple, M = I+N , and N 6=0 nilpotent. Since MS = SM , the S-eigenvalues are identical with thea-eigenvalues and the S-eigenspaces are identical with the generalizeda-eigenspaces. The Lie bracket operation is defined independently froma, S,M , so the conclusions for S hold also for a.

The Lie algebra automorphism a(f) in the statement of the proposi-tion is real, not complex, but since eigenvalues, generalized eigenspaces,and brackets act naturally under complexification, the complex caseimplies the real one.

As in Section 1 we split g and a(f) according to the generalizedeigenspaces of with eigenvalues of modulus > 1,= 1, and < 1 respec-tively,

a(f) = au(f) ⊕ ac(f) ⊕ as(f)

g = gu ⊕ gc ⊕ gs.

By Proposition 10.3 these are Lie subalgebras of g. We extend themto right invariant bundles Eu, Ec, Es over G. According to Propo-sition 10.1, these bundles are T f -invariant. By Corollary 10.2 their


projections to G/B, Eu, Ec, and Es are Tf -invariant,

Eu ⊕ Ec ⊕ Es T f−−−−−−−−−→ Eu ⊕ Ec ⊕ Es

Tπ

y yTπEu ⊕ Ec ⊕ Es Tf−−−−−−−−−→ Eu ⊕ Ec ⊕ Es.

A priori, some of these bundles may be zero.

Proposition 10.4. There is a Riemann structure on T (G/B) adaptedto Tf with respect to which Tf expands Eu sharply, Tf is nearly iso-metric on Ec, and Tf contracts Es sharply.

Proof. Choose ρ, 0 < ρ < 1, such that the moduli of the eigenvaluesof (au(f))−1 and as(f) lie in the interval (0, ρ). For any ε > 0 thereis an inner product on TeG which is adapted to a(f) in the sensethat the norms of (au(f))−1 and as(f) are < ρ, while the norms of(ac(f))−1 and ac(f) are < 1 + ε. Right translate this inner product toa right invariant Riemann structure on TG. Right translations becomeisometries. By (32) and the proof of Proposition 10.1, Tef is isometricto Txf for all x ∈ G. Hence, the Riemann structure is adapted to T f .Vectors in Eu are expanded more sharply than the factor 1/ρ, vectorsin Es are contracted more sharply than the factor ρ, and vectors in Ec

are affected isometrically, up to factors 1− ε and 1 + ε.Since B acts on the right, and since the T f -adapted Riemann struc-

ture is right invariant, it projects by Tπ : TG → T (G/B) to a Rie-mann structure on T (G/B). Under quotients, norms become no larger.Hence, ‖T sf‖ ≤ ‖T sf‖ < ρ. Similarly, ‖T csf‖ ≤ ‖T csf‖, and if thebundles are non-zero so the inverses exist, ‖T uf−1‖ ≤ ‖T uf−1‖ and‖T cuf−1‖ ≤ ‖T cuf−1‖. Hence, T uf expands by at least the factor 1/ρ,T sf contracts by at most the factor ρ, and since Ec = Ecu ∩ Ecs, T cfis an isometry up to a factor between 1− ε and 1 + ε.

Recall that h is the hyperbolic Lie subalgebra of f and b is the Liealgebra of B. Both are subalgebras of g.

Corollary 10.5. The affine diffeomorphism f is partially hyperbolicif and only if h /⊂ b.

Proof. We prove the equivalence of the opposites. If h ⊂ b thenT (G/B) = Ec and f is completely central; it is not partially hyperbolic.On the other hand, if f is not partially hyperbolic then at least one ofEu, Es is zero, say

T (G/B) = Ec ⊕ Es.


Since f preserves the projected Haar measure, the determinant of Tfis ±1 everywhere, which implies that Es = 0 also. Thus Eu, Es ⊂ b,and h ⊂ b.

Proposition 10.6. The right invariant bundles Eu, Ecu, Ec, Ecs, Es

and their projections to T (G/B) are tangent to smooth dynamicallycoherent foliations; i.e., f and f are dynamically coherent.

Proof. Since gu, gcu, gc, gcs, gs are Lie subalgebras of g they are tan-gent to unique connected, locally closed subgroups, Gu, Gcu, Gc, Gcs, Gs

respectively. The orbits Gux of the left Gu-action are integral manifoldsof the right invariant subbundle Eu. For if we consider the general pointhx ∈ Gux then the right multiplication diffeomorphism Rhx : G → Gsends Gu onto Gux, sends e to hx, and hence

Thx(Gux) = (TeRhx)(TeG

u) = (TeRhx)(gu) = (TeRhx)(E

ue ) = Eu

hx

by right invariance of Eu. The corresponding facts are true for theother subgroups. Also, these subgroups act on G/B and their orbitsintegrate the projected bundles there.

Recall that the smallest Lie subalgebra of g containing gu and gs, h,is an ideal. It is therefore the tangent space at e of a unique connected,locally closed, normal, Lie subgroup H of G.

Proposition 10.7. An element y ∈ G/B is us-accessible from x ∈G/B if and only is there is an h ∈ H such that y = hx.

Proof. Suppose that y is accessible from x. From the proof of Propo-sition 10.6 it follows that there are elements u1, . . . , uk ∈ Gu ands1, . . . , sk ∈ Gs such that

y = ukskuk−1sk−1 . . . u1s1x.

Since H contains Gu and Gs, the product uk . . . s1 lies in H. Theconverse can be proved similarly since h is generated by gu and gs.

Proof of Theorem B. Recall that for an affine diffeomorphism f ,Theorem B asserts

(a) Partial hyperbolicity is equivalent to h /⊂ b.(b) Center bunching and dynamical coherence are automatic.(c) Accessibility in G/B is equivalent to g = b + h.Corollary 10.5 gives (a); Proposition 10.4 gives center bunching;

Proposition 10.6 gives dynamical coherence.It remains to prove (c). Consider the connected Lie subgroup H of

G that is tangent at e to the hyperbolic Lie subalgebra h. According


to Proposition 10.7, accessibility in G/B is equivalent to

HB = G,

which, since G is connected, is true if and only if h + b = g.

Proof of Theorem C. Recall that Theorem C asserts the stable er-godicity of the affine diffeomorphism f if g = h + b. According toTheorem B, f satisfies all the hypotheses of Theorem A (which impliesit is stably ergodic) except one: stable us-accessibility. However, weknow that the unstable and stable foliations are smooth (since they areorbits of subgroups), and, by Theorem B, they have the (a priori notstable) us-accessibility property. But, as we showed in [22], smoothaccessibility implies stable accessibility, and Theorem A applies: f isstably ergodic.

Proof of Corollary b. Recall that Corollary b gives a necessaryand sufficient condition for stable ergodicity of a special type of affinediffeomorphism f . The assumptions are:

(a) G is simple.(b) B is a uniform discrete subgroup of G.(c) f = Lg is left translation by some fixed g ∈ G. (In the previous

notation, this means that A is the identity automorphism.)Under these hypotheses it is asserted that

f is stably ergodic if and only if Ad(g) has at least one eigenvaluewith modulus 6= 1

By Proposition 5.3 of [22], h is an ideal in g. If Ad(g) has an eigen-value with modulus 6= 1, h is non-trivial. Simplicity of G implies thath = g. Theorems B and C show that Lg is stably ergodic among C2

m-preserving diffeomorphisms of G/B. For the converse, we know from[4], that stable ergodicity of Lg, even among left translations, impliesthat h = g. Hence h is non-trivial and Ad(g) has an eigenvalue withmodulus 6= 1.

Proof of Corollary c. Recall that Corollary c concerns the speciallinear group SL(n,R). One assumes that

(a) B is a uniform discrete subgroup of G = SL(n,R).(b) f : G/B → G/B is left multiplication by some fixed matrix

M ∈ G.The assertion is that f is stably ergodic among C2 volume preservingdiffeomorphisms of G/B if and only if M has at least one eigenvaluewith modulus 6= 1.

To deduce Corollary c from Corollary b we note that SL(n,R) is


simple, and as in Proposition 5.6 of [22], Ad(M) has an eigenvaluewith modulus 6= 1 if and only if M does.

11. An accessibility example

Let f be a partially hyperbolic diffeomorphism of M with splittingEu ⊕ Ec ⊕ Es. It has the accessibility property if every pair of pointsp, q ∈ M can be joined by a us-path. If this remains true for all per-turbations of f then f (or the pair Eu, Es) has the stable accessibilityproperty.

Question 1. Does accessibility imply stable accessibility?

When Eu, Es are of class C1 the answer is “yes”, a fact we madeuse of in the proof of Theorem C in Section 10. In general, however,Eu, Es have only the following regularity: they are Holder and uniquelyintegrable. (Unique integrability of a plane field E means that E istangent to a foliation and any curve everywhere tangent to E lies in aleaf of the foliation. See [24] for more discussion of this.)

In this section we give an example indicating that the answer toQuestion 1 may be “no”. Although not definitive, it shows the limitsof too naive an approach.

Let Eu, Es ⊂ TM be fixed subbundles. We say that a point q ∈ Mis strongly accessible from p ∈ M if the following non-zero degreecondition holds. There are continuous, uniquely integrable vector fieldsX1, . . . , X2k such that

(a) The vector fields with odd index are subordinate to Eu, and theones with even index are subordinate to Es.

(b) A concatenation of the flows ϕi = ϕit(x) generated by the vec-tor fields X i joins p to q. That is, for some time vector t∗ =(t∗1, . . . , t

∗2k),

Φ(t∗) = ϕ2kt∗2k◦ ϕ2k−1

t∗2k−1◦ · · · ◦ ϕ1

t∗1(p) = q.

(c) For some neighborhood N of t∗ in R2k the map Φ : N →M ,

Φ : t 7→ Φ(t),

is topologically essential. That is, there exists a disc D ⊂ N ofthe same dimension as M , and

Degree(Φ|∂D, q) 6= 0.

In particular, q /∈ Φ(∂D).


It is an easy application of topological degree theory to show thatstrong accessibility implies stable accessibility. The converse is

Question 2. Does stable accessibility imply strong accessibility?

Proposition 11.1. Local 4-legged accessibility does not imply local 4-legged stable accessibility.

Proof. Specifically, there is a pair of uniquely integrable vector fieldsX, Y on a 4-manifold M such that for some p ∈ M , and some non-empty open set U ⊂M , every q ∈ U is accessible from p by a 4-leggedXY -path,

Φ(t1, t2, t3, t4) = ϕYt4 ◦ ϕXt3◦ ϕYt2 ◦ ϕ

Xt1

(p) = q,

where t = (t1, t2, t3, t4) ranges in some open T ⊂ R4, but there exist

small, uniquely integrable perturbations X, Y of X, Y for which Φ(T )is merely a 3-dimensional subset of U .

The construction of X, Y is local. It takes place in one coordinatechart, say with coordinates x, y, z, w. The point p is the origin, p =(0, 0, 0, 0), and the vector field Y is everywhere constant,

Y =∂

∂y.

The time set T is a neighborhood of t∗ = (1, 1, 1, 1). On the unit cube,X is also constant,

X =∂

∂x.

Thus, the first two legs of the XY -path with time vector t∗ are the unitsegments along the x-axis and parallel to the y-axis,

[0, 1]× 0× 0× 0 and 1× [0, 1]× 0× 0.

Let β : R → R+ be a smooth function with support in [4/3, 5/3],and integral 1, ∫

Rβ(x)dx = 1.

Let σ : R3 → R+ be a smooth function with compact support suchthat σ(y, z, w) = 1 whenever |y − 1| ≤ 1, |z| ≤ 1, and |w| ≤ 1.

Let γ : R → [−1, 1]2 be a Peano curve with compact support [0, 2].That is, γ is continuous, onto, and γ(y) = 0 for y /∈ [0, 2]. Necessarilyγ is not differentiable. Write γ in components as

γ(y) = (ζ(y), ω(y)).


We then define

X =∂

∂x+ a

∂

∂z+ b

∂

∂w,

where

a = β(x)σ(y, z, w) ζ(y) and b = β(x)σ(y, z, w)ω(y).

Although X is not smooth with respect to y, its ∂/∂y componentis zero. Thus, X is uniquely integrable and the X-flow curves lie inplanes y = const. Specifically, if t = (t1, t2, t3, t4) and

|t1 − 1| < 1

3|t2 − 1| < 1 |t3 − 1| < 1

3,(33)

then

Φ(t) = (t1 + t3, t2 + t4, ζ(t2), ω(t2)).(34)

Consider the time set T defined by times t satisfying (33) and

|t4 − 1| < 2.

Suppose that q = (x, y, z, w) satisfies

|x− 2| < 1

3|y − 2| < 1 |z| < 1 |w| < 1.(35)

Since γ is a Peano curve, there exists t2 ∈ [0, 2] such that

γ(t2) = (z, w).

We then set

t1 = 1

t3 = x− 1

t4 = y − t2,which makes t = (t1, t2, t3, t4) ∈ T . Therefore Φ(t) is given by (34) and

Φ(t) = q.

That is, each q in the set U defined by (35) is the Φ image of somet ∈ T , or to put it in terms of accessibility, each q ∈ U is accessiblefrom p.

Now let X, Y be smooth approximations to X, Y of a special type:

X =∂

∂x+ a

∂

∂z+ b

∂

∂wand Y = Y =

∂

∂y,

where

a = β(x)σ(y, z, w) ζ(y) b = β(x)σ(y, z, w) ω(y)


and γ = (ζ , ω) is a smooth approximation to γ. The expression (34)becomes

Φ(t) = (t1 + t3, t2 + t4, ζ(t2), ω(t2)).

The Φ image of T is bracketted as

[4/3, 8/3]× [1, 3]× γ(R) ⊂ Φ(T ) ⊂ R2 × γ(R),

and hence has dimension 3.

Besides the fact that perturbations destroy accessibility in the pre-ceding example, one can see that although all points of U are accessiblefrom p, no point is strongly accessible by 4-legged paths. Increasingthe number of legs in the path, well beyond the dimension of M , maybe the way to answer Questions 1 and 2 affirmatively.

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naire Bourbaki, 1991-92, Asterisque, # 206, 311-345

Mathematics Department, University of California, Berkeley Cal-ifornia, 94720

E-mail address: [email protected]

IBM, Thomas J. Watson Research Center, Yorktown Heights NewYork, 10598

E-mail address: [email protected]

michael shub...abstract. in this paper we eliminate some of the most nettle-some hypotheses of our...

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