Proceedings of the International Congress of Mathematicians

Hyderabad, India, 2010

Conservative partially hyperbolic

dynamics ∗
Amie Wilkinson

Abstract. We discuss recent progress in understanding the dynamical properties of par-

tially hyperbolic diffeomorphisms that preserve volume. The main topics addressed are
density of stable ergodicity and stable accessibility, center Lyapunov exponents, patho-
logical foliations, rigidity, and the surprising interrelationships between these notions.

Mathematics Subject Classification (2000). Primary 37D30; Secondary 37C40.

Introduction
Here is a story, told at least in part through the exploits of one of its main char-
acters. This character, like many a Hollywood (or Bollywood) star, has played
a leading role in quite a few compelling tales; this one ultimately concerns the
dynamics of partially hyperbolic diffeomorphisms.
We begin with a connected, compact, smooth surface S without boundary, of
genus at least 2. The Gauss-Bonnet theorem tells us that the average curvature of
any Riemannian metric on S must be negative, equal to 2πχ(S), where χ(S) is the
Euler characteristic of S. We restrict our attention to the metrics on S of every-
where negative curvature; among such metrics, there is a finite-dimensional moduli
space of hyperbolic metrics, which have constant curvature. Up to a normalization
of the curvature, each hyperbolic surface may be represented by a quotient H/Γ,
where H is the complex upper half plane with the metric y −2 (dx2 + dy 2 ), and Γ is
a discrete subgroup of PSL(2, R), isomorphic to the fundamental group of S. More
generally, any negatively curved metric on S lies in the conformal class of some
hyperbolic metric, and the space of all such metrics is path connected. Throughout
this story, S will be equipped with a negatively curved metric.
This negatively curved muse first caught the fancy of Jacques Hadamard in the
late 1890’s [39]. Among other things, Hadamard studied the properties of geodesics
on S and a flow ϕt : T 1 S → T 1 S on the unit tangent bundle to S called the geodesic
flow. The image of a unit vector v under the time-t map of this flow is obtained
by following the unique unit-speed geodesic γv : R → S satisfying γ̇v (0) = v for a
∗ Thanks to Christian Bonatti, Keith Burns, Jordan Ellenberg, Andy Hammerlindl, François

Ledrappier, Charles Pugh, Mike Shub and Lee Wilkinson for reading earlier versions of this text
and making several helpful suggestions. This work was supported by the NSF.
2 Amie Wilkinson

distance t and taking the tangent vector at that point:

ϕt (v) := γ̇v (t).

This geodesic flow, together with its close relatives, plays the starring role in the
story told here.

ϕt (v)

Figure 1. The geodesic flow.

A theorem of Liouville implies that ϕt preserves a natural probability measure

m on T 1 S, known as Liouville measure, which locally is just the product of nor-
malized area on S with Lebesgue measure on the circle fibers. Poincaré recurrence
then implies that almost every orbit of the geodesic flow comes back close to itself
infinitely often.
In the special case where S = H/Γ is a hyperbolic surface, the unit tangent
bundle T 1 S is naturally identified with P SL(2, R)/Γ, and the action of the geodesic
flow ϕt is realized by left multiplication by the diagonal matrix
 t/2 
e 0
gt = .
0 e−t/2

Liouville measure is normalized Haar measure.

In his study of ϕt , Hadamard introduced the notion of the stable manifold of a
vector v ∈ T 1 S:

W s (v) := {w ∈ T 1 S | lim dist(ϕt (v), ϕt (w)) = 0}.

t→∞

The proof that such sets are manifolds is a nontrivial consequence of negative cur-
vature and a noted accomplishment of Hadamard’s. Indeed, each stable manifold
W s (v) is an injectively immersed, smooth copy of the real line, and taken together,
the stable manifolds form a foliation W s of T 1 M . Similarly, one defines an unstable
manifold by:

W u (v) := {w ∈ T 1 S | lim dist(ϕt (v), ϕt (w)) = 0}

t→−∞

and denotes the corresponding unstable foliation W u . The foliations W s and W u

are key supporting players in this story.
 3

In the case where S = H/Γ, the stable manifolds are orbits of the positive
horocyclic flow on PSL(2, R)/Γ defined by left-multiplication by
 
s 1 t
ht = ,
0 1
and the unstable manifolds are orbits of the negative horocyclic flow, defined by
left-multiplication by  
1 0
hut = .
t 1
This fact can be deduced from the explicit relations:
g−t hsr gt = hsre−t and g−t hur gt = huret . (1)
The stable and unstable foliations stratify the future and past, respectively,
of the geodesic flow. It might come as no surprise that their features dictate the
asymptotic behavior of the geodesic flow. For example, Hadamard obtained from
the existence of these foliations and Poincaré recurrence that periodic orbits for ϕt
are dense in T 1 S.
Some 40 years after Hadamard received the Prix Poncelet for his work on
surfaces, Eberhard Hopf introduced a simple argument that proved the ergodicity
(with respect to Liouville measure) of the geodesic flow on T 1 S, for any closed
negatively curved surface S [44]. In particular, Hopf proved that almost every
infinite geodesic in S is dense (and uniformly distributed), not only in S, but in
T 1 S. It was another thirty years before Hopf’s result was extended by Anosov to
geodesic flows for negatively curved compact manifolds in arbitrary dimension.
Up to this point the discussion is quite well-known and classical, and from here
the story can take many turns. For example, for arithmetic hyperbolic surfaces,
the distribution of closed orbits of the flow and associated dynamical zeta functions
quickly leads us into deep questions in analytic number theory. Another path leads
to the study the spectral theory of negatively curved surfaces, inverse problems and
quantum unique ergodicity. The path we shall take here leads to the definition of
partial hyperbolicity.
Let us fix a unit of time t0 > 0 and discretize the system ϕt in these units; that
is, we study the dynamics of the time-t0 map ϕt0 of the geodesic flow. From a
digital age perspective this is a natural thing to do; for example, to plot the orbits
of a flow, a computer evaluates the flow at discrete, usually equal, time intervals.
If we carry this computer-based analogy one step further, we discover an in-
teresting question. Namely, a computer does not “evaluate the flow” precisely,
but rather uses an approximation to the time-t0 map (such as an ODE solver or
symplectic integrator) to compute its orbits. To what extent does iterating this
approximation retain the actual dynamical features of the flow ϕt , such as ergod-
icity?
To formalize this question, we consider a diffeomorphism f : T 1 S → T 1 S such
that the C 1 distance dC 1 (f, ϕt0 ) is small. Note that f in general will no longer
embed in a flow. While we assume that the distance from f to ϕt0 is small, this is
no longer the case for the distance from f n to ϕnt0 , when n is large.
4 Amie Wilkinson

f n (x)

ϕnt0 (x)

f (x)

ϕt0 (x)

Figure 2. f n (x) is not a good approximation to ϕnt0 (x).

The earliest description of the dynamics of such a perturbation f comes from

a type of structural stability theorem proved by Hirsch, Pugh, and Shub [43].
The results there imply in particular that if dC 1 (f, ϕt0 ) is sufficiently small, then
there exists an f -invariant center foliation W c = W c (f ) that is homeomorphic
to the orbit foliation O of ϕt . The leaves of W c are smooth. Moreover, the
homeomorphism h : T 1 S → T 1 S sending W c to O is close to the identity and W c
is the unique such foliation.
The rest of this paper is about f and, in places, the foliation W c (f ).
What is known about f is now substantial, but far from complete. For example,
the following basic problem is open.
Problem. Determine whether f has a dense orbit. More precisely, does there exist
a neighborhood U of ϕt0 in the space Diff r (T 1 S) of C r diffeomorphisms of T 1 S
(for some r ≥ 1) such that every f ∈ U is topologically transitive?
Note that ϕt0 is ergodic with respect to volume m, and hence is itself topo-
logically transitive. In what follows, we will explain results from the last 15 years
implying that any perturbation of ϕt0 that preserves volume is ergodic, and hence
has a dense orbit. For perturbations that do not preserve volume, a seminal re-
sult of Bonatti and Dı́az shows that ϕt0 can be approximated arbitrarily well by
C 1 -open sets of transitive diffeomorphisms [9]. But the fundamental question of
whether ϕt0 lives in such an open set remains unanswered.
In most of the discussion here, we will work in the conservative setting, in
which the diffeomorphism f preserves a volume probability measure. To fix no-
tation, M will always denote a connected, compact Riemannian manifold without
boundary, and m will denote a probability volume on M . For r ≥ 1, we denote by
Diff rm (M ) the space of C r diffeomorphisms of M preserving m, equipped with the
C r topology.
 5

1. Partial hyperbolicity

The map ϕt0 and its perturbation f are concrete examples of partially hyperbolic
diffeomorphisms. A diffeomorphism f : M → M of a compact Riemannian man-
ifold M is partially hyperbolic if there exists an integer k ≥ 1 and a nontrivial,
Df -invariant, continuous splitting of the tangent bundle

T M = Es ⊕ Ec ⊕ Eu

such that, for any p ∈ M and unit vectors v s ∈ E s (p), v c ∈ E c (p), and v u ∈ E u (p):

Dp f k v s  < 1 < Dp f k v u , and

Dp f k v s  < Dp f k v c  < Dp f k v u .

Up to a change in the Riemannian metric, one can always take k = 1 in this

definition [37]. In the case where E c is the trivial bundle, the map f is said to be
Anosov. The central example ϕt0 is partially hyperbolic: in that case, the bundle
E c = Rϕ̇ is tangent to the orbits of the flow, and E s and E u are tangent to the
leaves of W s and W u , respectively.
Partial hyperbolicity is a C 1 -open condition: any diffeomorphism sufficiently
1
C -close to a partially hyperbolic diffeomorphism is itself partially hyperbolic.
Hence the perturbations of ϕt0 we consider are also partially hyperbolic. For an
extensive discussion of examples of partially hyperbolic dynamical systems, see
the survey articles [20, 41, 62] and the book [55]. Among these examples are: the
frame flow for a compact manifold of negative sectional curvature and most affine
transformations of compact homogeneous spaces.
As is the case with the example ϕt0 , the stable and unstable bundles E s and E u
of an arbitrary partially hyperbolic diffeomorphism are always tangent to foliations,
which we will again denote by W s and W u respectively; this is a consequence of
partial hyperbolicity and a generalization of Hadamard’s argument. By contrast,
the center bundle E c need not be tangent to a foliation, and can even be nowhere
integrable. In many cases of interest, however, there is also a center foliation W c
tangent to E c : the content of the Hirsch-Pugh-Shub work in [43] is the properties
of systems that admit such foliations, known as “normally hyperbolic foliations.”
There is a natural and slightly less general notion than integrability of E c that
appears frequently in the literature. We say that a partially hyperbolic diffeomor-
phism f : M → M is dynamically coherent if the subbundles E c ⊕ E s and E c ⊕ E u
are tangent to foliations W cs and W cu , respectively, of M . If f is dynamically
coherent, then the center bundle E c is also integrable: one obtains the center
foliation W c by intersecting the leaves of W cs and W cu . The examples ϕt0 are
dynamically coherent, as are their perturbations (by [43]: see [24] for a discussion).
6 Amie Wilkinson

2. Stable ergodicity and the Pugh-Shub Conjec-

tures
Brin and Pesin [16] and independently Pugh and Shub [57] first examined the
ergodic properties of partially hyperbolic systems in the early 1970’s. The methods
they developed give an ergodicity criterion for partially hyperbolic f ∈ Diff 2m (M )
satisfying the following additional hypotheses:

(a) the bundle E c is tangent to a C 1 foliation W c , and

(b) f acts isometrically (or nearly isometrically) on the leaves of W c .

In [16] it is shown that such an f is ergodic with respect to m if it satisfies a

condition called accessibility.
Definition 2.1. A partially hyperbolic diffeomorphism f : M → M is accessible
if any point in M can be reached from any other along an su-path, which is a
concatenation of finitely many subpaths, each of which lies entirely in a single leaf
of W s or a single leaf of W u .
This ergodicity criterion applies to the discretized geodesic flow ϕt0 : its center
bundle is tangent to the orbit foliation for ϕt , which is smooth, giving (a). The
action of ϕt0 preserves the nonsingular vector field ϕ̇, which implies (b). It is
straightforward to see that if S is a hyperbolic surface, then ϕt0 is accessible: the
stable and unstable foliations are orbits of the smooth horocyclic flows hst and hut ,
respectively, and matrix multiplication on the level of the Lie algebra sl2 shows
that locally these flows generate all directions in PSL(2, R):
     1

1 0 1 0 0 2 0
, = ; (2)
2 0 0 1 0 0 − 12

the matrices appearing on the left are infinitesimal generators of the horocyclic
flows, and the matrix on the right generates the geodesic flow. Since ϕt0 is acces-
sible, it is ergodic.
Now what of a small perturbation of ϕt0 ? As mentioned above, any f ∈
Diff 2m (T 1 S) sufficiently C 1 close to ϕt0 also has a center foliation W c , and the
action of f on the leaves is nearly isometric. With some work, one can also show
that f is accessible (this was carried out in [16]). There is one serious reason why
the ergodicity criterion of [16] cannot be applied to f : the foliation W c is not C 1 .
The leaves of W c are C 1 , and the tangent spaces to the leaves vary continuously,
but they do not vary smoothly. We will explore in later sections the extent to which
W c fails to be smooth, but for now suffice it to say that W c is pathologically bad,
not only from a smooth perspective but also from a measure-theoretic one.
The extent to which W c is bad was not known at the time, but there was little
hope of applying the existing techniques to perturbations of ϕt0 . The first major
breakthrough in understanding the ergodicity of perturbations of ϕt0 came in the
1990’s:
 7

Theorem A (Grayson-Pugh-Shub [38]). Let S be a hyperbolic surface, and let ϕt

be the geodesic flow on T 1 S. Then ϕt0 is stably ergodic: there is a neighborhood
U of ϕt0 in Diff 2m (T 1 S) such that every f ∈ U is ergodic with respect to m.

The new technique introduced in [38] was a dynamical approach to understand-

ing Lebesgue density points which they called juliennes. The results in [38] were
soon generalized to the case where S has variable negative curvature [74] and to
more general classes of partially hyperbolic diffeomorphisms [59, 60]. Not long after
[38] appeared, Pugh and Shub had formulated an influential circle of conjectures
concerning the ergodicity of partially hyperbolic systems.

Conjecture 1 (Pugh-Shub [58]). On any compact manifold, ergodicity holds for an

open and dense set of C 2 volume preserving partially hyperbolic diffeomorphisms.

This conjecture can be split into two parts using the concept of accessibility.

Conjecture 2 (Pugh-Shub [58]). Accessibility holds for an open and dense subset
of C 2 partially hyperbolic diffeomorphisms, volume preserving or not.

Conjecture 3 (Pugh-Shub [58]). A partially hyperbolic C 2 volume preserving dif-

feomorphism with the essential accessibility property is ergodic.

Essential accessibility is a measure-theoretic version of accessibility that is im-

plied by accessibility: f is essentially accessible if for any two positive volume sets
A and B, there exists an su-path in M connecting some point in A to some point
in B – see [20] for a discussion.
In the next two sections, I will report on progress to date on these conjectures.
Further remarks.

1. Volume-preserving Anosov diffeomorphisms (where dim E c = 0) are always

ergodic. This was proved by Anosov in his thesis [1]. Note that Anosov
diffeomorphisms are also accessible, since in that case the foliations W s and
W u are transverse. Hence all three conjectures hold true for Anosov diffeo-
morphisms.

2. It is natural to ask whether partial hyperbolicity is a necessary condition for

stable ergodicity. This is true when M is 3-dimensional [27] and also in the
space of symplectomorphisms [45, 68], but not in general [71]. What is true
is that the related condition of having a dominated splitting is necessary for
stable ergodicity (see [27]).

3. One can also ask whether for partially hyperbolic systems, stable ergodicity
implies accessibility. If one works in a sufficiently high smoothness class,
then this is not the case, as was shown in the groundbreaking paper of F.
Rodrı́guez Hertz [61], who will also speak at this congress. Hertz used meth-
ods from KAM theory to find an alternate route to stable ergodicity for
certain essentially accessible systems.
8 Amie Wilkinson

4. On the other hand, it is reasonable to expect that some form of accessibility

is a necessary hypothesis for a general stable ergodicity criterion for par-
tially hyperbolic maps (see the discussion at the beginning of [24]). Unlike
Anosov diffeomorphisms, which are always ergodic, partially hyperbolic dif-
feomorphisms need not be ergodic. For example, the product of an Anosov
diffeomorphism with the identity map on any manifold is partially hyper-
bolic, but certainly not ergodic. See also Theorem 11.16 in [10].

3. Accessibility
In general, the stable and unstable foliations of a partially hyperbolic diffeomor-
phism are not smooth (though they are not pathological, either – see below). Hence
it is not possible in general to use infinitesimal techniques to establish accessibility
the way we did in equation (2) for the discretized hyperbolic geodesic flow. The
C 1 topology allows for enough flexibility in perturbations that Conjecture 2 has
been completely verified in this context:

Theorem B (Dolgopyat-Wilkinson [31]). For any r ≥ 1, accessibility holds for a

C 1 open and dense subset of the partially hyperbolic diffeomorphisms in Diff r (M ),
volume-preserving or not.

Theorem B also applies inside the space of partially hyperbolic symplectomor-

phisms.
More recently, the complete version of Conjecture 3 has been verified for sys-
tems with 1-dimensional center bundle.

Theorem C (Rodrı́guez Hertz-Rodrı́guez Hertz-Ures [63]). For any r ≥ 1, acces-

sibility is C 1 open and C r dense among the partially hyperbolic diffeomorphisms
in Diff rm (M ) with one-dimensional center bundle.

This theorem was proved earlier in a much more restricted context by Niţică-
Török [54]. The C 1 openness of accesssibility was shown in [28]. A version of
Theorem C for non-volume preserving diffeomorphisms was later proved in [19].
The reason that it is possible to improve Theorem B from C 1 density to C r
density in this context is that the global structure of accessibility classes is well-
understood. By accessibility class we mean an equivalence class with respect to the
relation generated by su-paths. When the dimension of E c is 1, accessibility classes
are (C 1 immersed) submanifolds. Whether this is always true when dim(E c ) > 1 is
unknown and is an important obstacle to attacking the general case of Conjecture 2.

Further remarks.

1. More precise criteria for accessibility have been established for special classes
of partially hyperbolic systems such as discretized Anosov flows, skew prod-
ucts, and low-dimensional systems [23, 21, 64].
 9

2. Refined formulations of accessibility have been used to study higher-order

statistical properties of certain partially hyperbolic systems, in particular
the discretized geodesic flow [29, 52]. The precise relationship between ac-
cessibility and rate of mixing (in the absence of other hypotheses) remains a
challenging problem to understand.
3. Accessibility also plays a key role in a recently developed Livsič theory for
partially hyperbolic diffeomorphisms, whose conclusions closely mirror those
in the Anosov setting [47, 73].

4. Ergodicity
Conjecture 1 has been verified under one additional, reasonably mild hypothesis:
Theorem D (Burns-Wilkinson [22]). Let f be C 2 , volume-preserving, partially
hyperbolic and center bunched. If f is essentially accessible, then f is ergodic, and
in fact has the Kolmogorov property.
The additional hypothesis is “center bunched.” A partially hyperbolic diffeo-
morphism f is center bunched if there exists an integer k ≥ 1 such that for any
p ∈ M and any unit vectors v s ∈ E s (p), v c , wc ∈ E c (p), and v u ∈ E u (p):

Dp f k v s  · Dp f k wc  < Dp f k v c  < Dp f k v u  · Dp f k wc . (3)

As with partial hyperbolicity, the definition of center bunching depends only on

the smooth structure on M and not the Riemannian structure; if (3) holds for a
given metric and k ≥ 1, one can always find another metric for which (3) holds
with k = 1 [37]. In words, center bunching requires that the non-conformality of
Df | E c be dominated by the hyperbolicity of Df | E u ⊕ E s . Center bunching
holds automatically if the restriction of Df to E c is conformal in some metric (for
this metric, one can choose k = 1). In particular, if E c is one-dimensional, then f
is center bunched. In the context where dim(E c ) = 1, Theorem D was also shown
in [63].
Combining Theorems C and D we obtain:
Corollary 1. The Pugh-Shub conjectures hold true among the partially hyperbolic
diffeomorphisms with 1-dimensional center bundle.

Further remarks.
1. The proof of Theorem D builds on the original argument of Hopf for er-
godicity of geodesic flows and incorporates a refined theory of the juliennes
originally introduced in [38].
2. It appears that the center bunching hypothesis in Theorem D cannot be
removed without a significantly new approach. On the other hand, it is
possible that Conjecture 1 will yield to other methods.
10 Amie Wilkinson

3. Formulations of Conjecture 1 in the C 1 topology have been proved for low-

dimensional center bundle [12, 61] and for symplectomorphisms [3]. These
formulations state that ergodicity holds for a residual subset in the C 1 topol-
ogy.

5. Exponents
By definition, a partially hyperbolic diffeomorphism produces uniform contraction
and expansion in the directions tangent to E s and E u , respectively. In none
of the results stated so far do we make any precise assumption on the growth
of vectors in E c beyond the coarse bounds that come from partial hyperbolicity
and center bunching. In particular, an ergodic diffeomorphism in Theorem D can
have periodic points of different indices, corresponding to places in M where E c
is uniformly expanded, contracted, or neither. The power of the julienne-based
theory is that the hyperbolicity in E u ⊕ E s , when combined with center bunching
and accessibility, is enough to cause substantial mixing in the system, regardless
of the precise features of the dynamics on E c .
On the other hand, the asymptotic expansion/contraction rates in E c can give
additional information about the dynamics of the diffeomorphism, and is a poten-
tially important tool for understanding partially hyperbolic diffeomorphisms that
are not center bunched.
A real number λ is a center Lyapunov exponent of the partially hyperbolic
diffeomorphism f : M → M if there exists a nonzero vector v ∈ E c such that

1
lim sup log Df n (v) = λ. (4)
n→∞ n

If f preserves m, then Oseledec’s theorem implies that the limit in (4) exists for
each v ∈ E c (x), for m-almost every x. When the dimension of E c is 1, the limit
in (4) depends only on x, and if in addition f is ergodic with respect to m, then
the limit takes a single value m-almost everywhere.

Theorem E (Shub-Wilkinson [70]). There is an open set U ⊂ Diff ∞ 3

m (T ) of par-
tially hyperbolic, dynamically coherent diffeomorphisms of the 3-torus T3 = R3 /Z3
for which:

• the elements of U approximate arbitrarily well (in the C ∞ topology) the linear
automorphism of T3 induced by the matrix:
⎛ ⎞
2 1 0
A=⎝ 1 1 0 ⎠
0 0 1

• the elements of U are ergodic and have positive center exponents, m-almost
everywhere.
 11

Note that the original automorphism A has vanishing center exponents, every-
where on T3 , since A is the identity map on the third factor. Yet Theorem E says
that a small perturbation mixing the unstable and center directions of A creates
expansion in the center direction, almost everywhere on T3 .
The systems in U enjoy the feature of being non-uniformly hyperbolic: the
Lyapunov exponents in every direction (not just center ones) are nonzero, m-
almost everywhere. The well-developed machinery of Pesin theory guarantees a
certain level of chaotic behavior from nonuniform hyperbolicity alone. For example,
a nonuniformly hyperbolic diffeomorphism has at most countably many ergodic
components, and a mixing partially hyperbolic diffeomorphism is Bernoulli (i.e.
abstractly isomorphic to a Bernoulli process). A corollary of Theorem E is that
the elements of U are Bernoulli systems.
The constructions in [70] raise the question of whether it might be possible
to “remove zero exponents” from any partially hyperbolic diffeomorphism via a
small perturbation. If so, then one might be able to bypass the julienne based
theory entirely and use techniques from Pesin theory instead as an approach to
Conjecture 1. More generally, and wildly optimistically, one might ask whether any
f ∈ Diff 2m (M ) with at least one nonzero Lyapunov exponent on a positive measure
set might be perturbed to produce nonuniform hyperbolicity on a positive measure
set (such possibilities are discussed in [70]).
There is a partial answer to these questions in the C 1 topology, due to Baraveira
and Bonatti [7]. The results there imply in particular that if f ∈ Diff rm (M ) is
partially hyperbolic, then there exists g ∈ Diff rm (M ), C 1 -close to f so that the
sum of the center Lyapunov exponents is nonzero.
Further remarks.

1. Dolgopyat proved that the same type of construction as in [70] can be applied
to the discretized geodesic flow ϕt0 for a negatively curved surface S to
produce perturbations with nonzero center exponents [30]. See also [66] for
further generalizations of [70].

2. An alternate approach to proving Conjecture 1 has been proposed, taking into

account the center Lyapunov exponents [18]. For systems with dim(E c ) = 2,
this program has very recently been carried out in the C 1 topology in [65],
using a novel application of the technique of blenders, a concept introduced
in [9].

6. Pathology
There is a curious by-product of nonvanishing Lyapunov exponents for the open
set U of examples in Theorem E. By [43], there is a center foliation W c for each
f ∈ U, homeomorphic to the trivial R/Z fibration of T3 = T2 × R/Z; in particular,
the center leaves are all compact. The almost everywhere exponential growth
associated with nonzero center exponents is incompatible with the compactness of
12 Amie Wilkinson

the center foliation, and so the full volume set with positive center exponent must
meet almost every leaf in a zero set (in fact a finite set [67]).
The same type of phenomenon occurs in perturbations of the discretized geodesic
flow ϕt0 . While in that case the leaves of W c are mostly noncompact, they are in a
sense “dynamically compact.” An adaptation of the arguments in [67] shows that
any perturbation of ϕt0 with nonvanishing center exponents, such as those con-
structed by Dolgopyat in [30], have atomic disintegration of volume along center
leaves.
Definition 6.1. A foliation F of M with smooth leaves has atomic disintegration
of volume along its leaves if there exists A ⊂ M such that
• m(M \ A) = 0, and
• A meets each leaf of F in a discrete set of points (in the leaf topology).

Figure 3. A pathological foliation

At the opposite end of the spectrum from atomic disintegration of volume is

a property called absolute continuity. A foliation F is absolutely continuous if
holonomy maps between smooth transversals send zero volume sets to zero volume
sets. If F has smooth leaves and is absolutely continuous, then for every set A ⊂ M
satisfying m(M \ A) = 0, the intersection of A with the leaf F through m-almost
every point in M has full leafwise Riemannian volume. In this sense Fubini’s
theorem holds for absolutely continuous foliations. If F is a C 1 foliation, then it
is absolutely continuous, but absolute continuity is a strictly weaker property.
Absolute continuity has long played a central role in smooth ergodic theory.
Anosov and Sinai [1, 2] proved in the 60’s that the stable and unstable foliations
 13

of globally hyperbolic (or Anosov) systems are absolutely continuous, even though
they fail to be C 1 in general. Absolute continuity was a key ingredient in Anosov’s
celebrated proof [1] that the geodesic flow for any compact, negatively curved man-
ifold is ergodic. When the center foliation for f fails to be absolutely continuous,
this means that one cannot “quotient out by the center direction” to study ergodic
properties f .
The existence of such pathological center foliations was first demonstrated by
A. Katok (whose construction was written up by Milnor in [53]). Theorem E shows
that this type of pathology can occur in open sets of diffeomorphisms and so is
inescapable in general. In the next section, we discuss the extent to which this
pathology is the norm.
Further remarks.

1. An unpublished letter of Mañé to Shub examines the consequences of non-

vanishing Lyapunov center exponents on the disintegration of volume along
center foliations. Some of the ideas there are pursued in greater depth in
[42].

2. The examples of Katok in [53] in fact have center exponents almost ev-
erywhere equal to 0, showing that nonvanishing center exponents is not a
necessary condition for atomic disintegration of volume.

3. Systems for which the center leaves are not compact (or even dynamically
compact) also exhibit non-absolutely continuous center foliations, but the
disintegration appears to be potentially much more complicated than just
atomic disintegration [69, 36].

7. Rigidity
Examining in greater depth the potential pathologies of center foliations, we dis-
cover a rigidity phenomenon. To be concrete, let us consider the case of a per-
turbation f ∈ Diff ∞ m (M ) of the discretized geodesic flow on a negatively-curved
surface. If the perturbation f happens to be the time-one map of a smooth flow,
then W c is the orbit foliation for that flow. In this case the center foliation for f
is absolutely continuous – in fact, C ∞ . In general, however, a perturbation f of
ϕt0 has no reason to embed in a smooth flow, and one can ask how the volume m
disintegrates along the leaves of W c .
There is a complete answer to this question:

Theorem F (Avila-Viana-Wilkinson [6]). Let S be a closed negatively curved sur-

face, and let ϕt : T 1 S → T 1 S be the geodesic flow.
For each t0 > 0, there is a neighborhood U of ϕt0 in Diff ∞ 1
m (T S) such that for
each f ∈ U:

1. either m has atomic disintegration along the center foliation W c , or

14 Amie Wilkinson

2. f is the time-one map of a C ∞ , m-preserving flow.

What Theorem F says is that, in this context, nothing lies between C ∞ and
absolute singularity of W c – pathology is all that can happen. The geometric
measure-theoretic properties of W c determine completely a key dynamical property
of f – whether it embeds in a flow.
The heart of the proof of Theorem F is to understand what happens when the
center Lyapunov exponents vanish. For this, we use tools that originate in the
study of random matrix products. The general theme of this work, summarized
by Ledrappier in [49] is that “entropy is smaller than exponents, and entropy
zero implies deterministic.” Original results concerning the Lyapunov exponents of
random matrix products, due to Furstenberg, Kesten [34, 33], Ledrappier [49], and
others, have been extended in the past decade to deterministic products of linear
cocycles over hyperbolic systems by Bonatti, Gomez-Mont, Viana [11, 13, 72]. The
Bernoulli and Markov measures associated with random products in those earlier
works are replaced in the newer results by invariant measures for the hyperbolic
system carrying a suitable product structure.
Recent work of Avila, Viana [5] extends this hyperbolic theory from linear to
diffeomorphism cocycles, and these results are used in a central way. For cocycles
over volume preserving partially hyperbolic systems, Avila, Santamaria, and Viana
[4] have also recently produced related results, for both linear and diffeomorphism
cocycles, which also play an important role in the proof. The proof in [4] employs
julienne based techniques, generalizing the arguments in [24].

Further remarks.

1. The only properties of ϕt0 that are used in the proof of Theorem F are ac-
cessibility, dynamical coherence, one-dimensionality of E c , the fact that ϕt0
fixes the leaves of W c , and 3-dimensionality of M . There are also more gen-
eral formulations of Theorem F in [6] that relax these hypotheses in various
directions. For example, a similar result holds for systems in dimension 3 for
whom all center manifolds are compact.

2. Deep connections between Lyapunov exponents and geometric properties of

invariant measures have long been understood [48, 50, 51, 46, 8]. Theorem F
establishes new connections in the partially hyperbolic context.

3. Theorem F gives conditions under which one can recover the action of a Lie
group (in this case R) from that of a discrete subgroup (in this case Z).
These themes have arisen in the related context of measure-rigidity for alge-
braic partially hyperbolic actions by Einsiedler, Katok, Lindenstrauss [32].
It would be interesting to understand more deeply the connections between
these works.
 15

8. Summary, questions.
We leave this tale open-ended, with a few questions that have arisen naturally in
its course.
New criteria for ergodicity. Conjecture 1 remains open. As discussed in Sec-
tion 4, the julienne based techniques using the Hopf argument might have reached
their limits in this problem (at least this is the case in the absence of a significantly
new idea). One alternate approach which seems promising employs Lyapunov ex-
ponents and blenders [65]. Perhaps a new approach will find a satisfying conclusion
to this part of the story.
Classification problem. A basic question is to understand which manifolds
support partially hyperbolic diffeomorphisms. As the problem remains open in
the classical Anosov case (in which E c is zero-dimensional), it is surely exremely
difficult in general. There has been significant progress in dimension 3, however;
for example, using techniques in the theory of codimension-1 foliations, Burago
and Ivanov proved that there are no partially hyperbolic diffeomorphisms of the
3-sphere [17].
Modifying this question slightly, one can ask whether the partially hyperbolic
diffeomorphisms in low dimension must belong to certain “classes” (up to homo-
topy, for example) – such as time-t maps of flows, skew products, algebraic systems,
and so on. Pujals has proposed such a program in dimension 3, which has spurred
several papers on the subject [15, 14, 64, 40].
It is possible that if one adds the hypotheses of dynamical coherence and abso-
lute continuity of the center foliation, then there is such a classification. Evidence
in this direction can be found in [6].
Nonuniform and singular partial hyperbolicity. Unless all of its Lyapunov
exponents vanish almost everywhere, any volume-preserving diffeomorphism is in
some sense “nonuniformly partially hyperbolic.” Clearly such a general class of
systems will not yield to a single approach. Nonetheless, the techniques developed
recently are quite powerful and should shed some light on certain systems that are
close to being partially hyperbolic. Some extensions beyond the uniform setting
have been explored in [3], in which the center bunching hypotheses in [24] has been
replaced by a pointwise, nonuniform center bunching condition. This gives new
classes of stably ergodic diffeomorphisms that are not center bunched.
It is conceivable that the methods in [3] may be further extended to apply in
certain “singular partially hyperbolic” contexts where partial hyperbolicity holds
on an open, noncompact subset of the manifold M but decays in strength near the
boundary. Such conditions hold, for example, for geodesic flows on certain non-
positively curved manifolds. Under suitable accessibility hypotheses, these systems
should be ergodic with respect to volume.
Rigidity of partially hyperbolic actions. The rigidity phenomenon described
in Section 7 has only begun to be understood. To phrase those results in a more
general context, we consider a smooth, nonsingular action of an abelian Lie group
G on a manifold M . Let H be a discrete group acting on M , commuting with
16 Amie Wilkinson

the action of G, and whose elements are partially hyperbolic diffeomorphisms in

Diff ∞
m (M ). Can such an action be perturbed, preserving the absolute continuity
of the center foliation? How about the elements of the action? When absolute
continuity fails, what happens?
The role of accessibility and accessibility classes has been exploited in a serious
way in the important work of Damjanović and A. Katok on rigidity of abelian
actions on quotients of SL(n, R) [26]. It seems reasonable that these explorations
can be pushed further, using some of the techniques mentioned here, to prove
rigidity results for other partially hyperbolic actions. A simple case currently
beyond the reach of existing methods is to understand perturbations of the action
of a Z2 lattice in the diagonal subgroup on SL(2, R) × SL(2, R)/Γ, where Γ is an
irreducible lattice.
Our final question takes us further afield, but back once again to the geodesic
flow. Fix a closed hyperbolic surface S, and consider the standard action on T 1 S by
the upper triangular subgroup T < PSL(2, R), which contains both the geodesic
and positive horocyclic flows. Ghys proved that this action is highly rigid and
admits no m-preserving C ∞ deformations [35]. Does the same hold true for some
countable subgroup of T ? For example, consider the solvable Baumslag Solitar
subgroup BS(1, 2) generated by the elements
 √   
2 0 1 1
a= √1
and b = ,
0 2
0 1

which has the presentation BS(1, 2) = a, b | aba−1 = b2 . Can the standard

action be perturbed inside of Diff ∞ 1
m (T S)? More generally, can one classify all
faithful representations
ρ : BS(1, 2) → Diff ∞
m (M ),

where M is a 3-manifold? For results of a similar nature in lower dimensions, see

[25, 56].

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Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208-2730, USA
E-mail: wilkinso@math.northwestern.edu