Ergod. Th. & Dynam. Sys. (2001), 21, 1213–1237
Printed in the United Kingdom
c 2001 Cambridge University Press
Local rigidity of certain partially hyperbolic
actions of product type
VIOREL NIŢICÆ and ANDREI TÖRÖK‡
Institute of Mathematics of the Romanian Academy, PO Box 1–764,
RO-70700 Bucharest, Romania
(Received 8 November 1999 and accepted in revised form 29 July 2000)
Abstract. We prove certain rigidity properties of higher-rank abelian product actions of the
type α × IdN : Zκ → Diff(M × N), where α is (TNS) (i.e. is hyperbolic and has some
special structure of its stable distributions). Together with a result about product actions
of property (T ) groups, this implies the local rigidity of higher-rank lattice actions of the
form α × IdT : 0 → Diff(M × T), provided α has some rigidity properties itself, and
contains a (TNS) subaction.
1. Introduction
This paper is a contribution to the rigidity program initiated by Zimmer [Z].
The goal of the program is to classify the smooth actions of higher-rank semi-simple Lie
groups and of their (irreducible) lattices on compact manifolds. It was expected that any
such lattice action that preserves a smooth volume form and is ergodic can be reduced to
one of the following standard models: isometric actions, linear actions on nilmanifolds, and
left translations on compact homogeneous spaces. This original conjecture was disproved
by Katok and Lewis (see [KL2]): by blowing up a linear nilmanifold-action at some fixed
points they exhibit real-analytic, volume-preserving, ergodic lattice actions on manifolds
with complicated topology.
Nevertheless, imposing additional assumptions on the action, for example, some
hyperbolicity, allows for global classification results. The existence of an Anosov element
in the action is used in [KLZ] to give a global classification of SL(n, Z) actions on Tn that
preserve an absolutely continuous probability measure. Other global classification results
for Anosov actions can be found in [GS].
Much work was also done to study perturbations of higher-rank lattices and Lie groups
actions (see [H, KL1, KL2, KLZ, KS2, MQ, QY1, QY2]). In many of these papers, the
† Current address: Department of Mathematics, University of Notre Dame, Room 370, CCMB, Notre Dame,
IN 46556-5683, USA (e-mail: [email protected]).
‡ Current address: Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA
(e-mail: [email protected]).
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V. Niţică and A. Török
existence of an Anosov element, respectively of a spanning family of directions that are
hyperbolic for certain elements of the action, was essential. A first attempt to study the
deformation rigidity of partially hyperbolic product actions that contain a compact factor
was carried out in [NT1].
Notation. (1) Unless specified otherwise, we assume that all manifolds and maps are
smooth.
−
By C k we denote the class of functions that are C k−ε for any ε > 0. The C k− -topology
stands for the coarsest topology for which the inclusions C k− ⊂ C k−ε are continuous for
−
each ε > 0. However, by C 1 we mean C 1 .
(2) Throughout this paper by a C r -lamination we mean a topological foliation whose
leaves are C r -submanifolds that vary continuously in the C r -topology.
Definition. Let 0 be a finitely generated discrete group, M a compact manifold, and
e : 0 × M → M C ∞ -actions. Fix a finite set of generators {γi } of 0. We say that φ is
φ, φ
L
φ if the C ∞ -diffeomorphisms φ(γi ) and e
φ(γi ) are close in the C L -topology for
C -close to e
L
∞
all i. A C -perturbation of the action φ is a C -action C L -close to φ. A C L -deformation
of the action φ is a C L -continuous path of C ∞ -actions φt , 0 ≤ t ≤ 1, with φ0 = φ.
An action φ is said to be C L,K -locally rigid if any C L -perturbation of φ contained in
a sufficiently small C L -neighborhood of φ is conjugated to φ by a C K -diffeomorphism
which is C 0 -close to the identity. An action φ is said to be C L,K -deformation rigid if any
C L -deformation of φ contained in a sufficiently small C L -neighborhood of φ is conjugated
to φ by a continuous path of C K -diffeomorphisms C 0 -close to the identity.
We are interested in rigidity results for partially hyperbolic actions that have a trivial
factor. Deformation rigidity results were obtained in [NT1]. The main goal of this paper is
to prove the local rigidity of certain actions of this type.
Let π be the standard linear action of SL(n, Z) on the n-dimensional torus Tn . The
holonomy considerations used in this paper lead to the following improvement upon the
main result of [NT1] (see the end of §4 for the proof).
T HEOREM 1.1. Let n ≥ 3 and d ≥ 1 be integers. Let ρ be the action of SL(n, Z) on
Tn+d = Tn × Td given by ρ(A)(x, y) = (π(A)x, y), x ∈ Tn , y ∈ Td , A ∈ SL(n, Z).
−
Then, for any integer K ≥ 1, the action ρ is C 5,K -deformation rigid.
One special case of our new result is a strengthening of Theorem 1.1 when the fibers
are one-dimensional (see the proof after that of Theorem 3.3).
T HEOREM 1.2. Let 0 ⊂ SL(n, Z) be a subgroup of finite index and let ρ be the action of
0 on Tn+1 = Tn × T given by ρ(A)(x, y) = (π(A)x, y), x ∈ Tn , y ∈ T, A ∈ 0. Then,
−
for n ≥ 3 and any integer K ≥ 1, the action ρ is C 2,K -locally rigid.
More examples are given in Corollary 3.4 and the remarks following it.
The main part of Theorem 1.1, proved in [NT1], is based on three results in hyperbolic
dynamics: a generalization of Livsic’s cohomological results to cocycles with values in
diffeomorphism groups, a non-commutative version of the Anosov closing lemma, and a
version of the Hirsch–Pugh–Shub structural stability theorem.
Local rigidity of certain partially hyperbolic actions
1215
Note that Theorem 1.2 does not imply that the action ρ is C 2,∞ -locally rigid because
the size of the C 2 -neighborhood guaranteed by the theorem depends on K. The main
ingredients of the proof are two rigidity results: one concerns (TNS) abelian actions,
the other actions of property-(T ) groups. This approach obtains the rigidity of partiallyhyperbolic actions of product type from the (known) rigidity of hyperbolic actions.
It has been noticed for some time that many Anosov actions of abelian groups display an
array of rigidity properties. The vanishing of the first cohomology groups with coefficients
in Rn was noticed for the first time in [KS1]. This result is extended to cohomology
groups with coefficients in a matrix Lie group in [KNT]. Local and global rigidity results
for abelian Anosov actions are proved in [KL1] and [KS2].
As a consequence of the Hirsch–Pugh–Shub structural stability theorem (see [HPS] and
Theorem 2.1), small perturbations of abelian partially hyperbolic actions of product type
are conjugated to skew products of abelian Anosov actions by cocycles with coefficients in
diffeomorphism groups. Given the previous results, it is natural to expect certain rigidity
phenomena to appear for these actions as well.
A basic fact proved in this paper is that for certain higher-rank abelian partially
hyperbolic actions of product type, the sum of the stable and unstable distributions of
any regular element of the perturbation is integrable. In view of the situation for Lie group
valued cocycles (see [KNT]), one might expect the leaves of the integral lamination to
be closed manifolds covering the base simply, thus obtaining a conjugacy between the
perturbation and a product action. This amounts to showing that the only small solutions
of a particular system of equations in Diff (or Homeo), determined by the holonomy of
the integral lamination, are the trivial ones. We obtain the above-mentioned conclusion
under the supplementary assumption that some regular element of the perturbation has
a pointwise fixed center leaf. It is not clear to us how to remove this additional
assumption.
Namely, one has to deal with the following question.
Question. Let T : M → M be an Anosov diffeomorphism on the compact manifold M.
Let N be a compact manifold and f a partially hyperbolic diffeomorphism of M × N
which is C 1 -close to T × IdN . Assume that the stable and unstable foliations of f are
jointly integrable. Does it follow that the diffeomorphism f is conjugated to T × S, where
S ∈ Homeo(N)? That is, does the foliation integrating the stable and unstable foliations
of f have closed leaves?
Theorem 1.2 is obtained as follows. Consider first the restriction of the action ρ to a
diagonalizable abelian subgroup of rank n − 1 of 0. The existence of a pointwise fixed
center fiber is ensured by a theorem of Stowe [St1, St2], first used in this context by Hurder
[H]. Applying the rigidity result to this abelian action, we obtain a conjugacy between the
original abelian action and its perturbation. Using the property (T ), we show that this
conjugacy reduces the 0-action to a family of perturbations of hyperbolic actions.
This paper has the following structure: in §2 we recall a few facts from the theory of
partially hyperbolic diffeomorphisms and the definition of a (TNS) action, respectively of
property (T ). In §3 we present the main results, and in §§4 and 5 we give the proofs.
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V. Niţică and A. Török
2. Preliminaries
We recall first the definition of a partially hyperbolic diffeomorphism.
If T is a linear transformation between two normed linear spaces, the norm and the
conorm of T are defined by
kT k = sup{kT vk; kvk = 1} and m(T ) = inf{kT vk; kvk = 1}.
Let X be a compact, connected, boundaryless manifold. A C 1 -diffeomorphism f :
X → X is called partially hyperbolic if the derivative Tf : T X → T X leaves invariant
a continuous splitting T X = E s ⊕ E c ⊕ E u , E s 6= 0 6= E u , such that, with respect to a
fixed Riemannian metric on T X, Tf expands E u , Tf contracts E s , and the inequalities
kTps k < m(Tpc f ),
kTpc f k < m(Tpu f )
are true for all p ∈ X. If the center bundle E c = 0, then f is called an Anosov (or
hyperbolic) diffeomorphism.
Assume that the partially hyperbolic C r -diffeomorphism f leaves invariant a C 1 lamination L tangent to the central direction E c . We say that f is r-normally hyperbolic
at L if for all p ∈ X and 0 ≤ k ≤ r one has
m(Tpu f ) > kTpc f kk
and kTps f k < m(Tpc f )k .
The partially hyperbolic diffeomorphism f is said to satisfy the rth-order centerbunching conditions if for all p ∈ X and 0 ≤ ` ≤ r
kTps f kkTpc f k` < m(Tpc f ) and kTpc f k < m(Tpu f )m(Tpc f )` .
We recall the results of [HPS, Theorems 6.1, 6.8, 7.1, 7.2] and [PSW, Theorem B]
about partially hyperbolic Z-actions and their small perturbations. We describe only the
case that will be of interest in the following, and skip the description of the terms that are
self-explanatory. In the case of hyperbolic diffeomorphisms, we have the classical results
of Anosov [A]. See also the Remark following Theorem 2.1.
T HEOREM 2.1. ([HPS]) Let X be a compact manifold and f ∈ Diff r (X), r ≥ 1, a
diffeomorphism which is r-normally hyperbolic at a C r -lamination Lf having compact
leaves.
(1) Through each leaf of Lf there exists a C r center-stable manifold. The center-stable
manifold through x ∈ X consists of those points whose forward f -orbit does not
stray away from the orbit of Lf (x). Hence, since the leaves of the lamination Lf are
compact, each center-stable manifold is a union of center leaves; the center-stable
manifolds form the center-stable lamination W cs (f ). A similar statement holds for
the center-unstable lamination, W cu (f ).
(2) There exists a C r stable lamination W s (f ) whose leaves lie in those of W cs (f ).
The points of a stable leaf are characterized by sharp forward asymptoticity. If f
satisfies the (r − 1)th order center-bunching conditions, then the stable distribution
is C r−1 on each center-stable leaf. In particular, the holonomy maps determined by
the stable lamination inside the center-stable leaves are C r−1 . A similar statement
holds for the unstable lamination, W u (f ).
Local rigidity of certain partially hyperbolic actions
(3)
(4)
1217
If g ∈ Diff r (X) is C 1 -close to f , then g is r-normally hyperbolic at a unique C r lamination Lg , and the stable, unstable, and center laminations of g converge in C r
to those of f as g converges to f in the C r -topology. The stable (unstable) holonomy
maps within the center-stable (respectively, center-unstable) leaves of g converge in
C r−1 to those of f , as g converges in C r to f .
Moreover, if Lf is a C r -foliation, then in the case (3) there exists a leaf-conjugacy
H ∈ Homeo(X) between (f, Lf ) and (g, Lg ): H maps the leaves of Lf to those of
Lg and Lg (H ◦ f (x)) = Lg (g ◦ H (x)). H is a C r -diffeomorphism of each leaf of
Lf onto its image, varying continuously in C r with the leaf. For x ∈ X, Lg (H (x))
is uniquely characterized by the fact that its g-orbit does not stray away from the
f -orbit of Lf (x). Modulo the choice of a normal bundle to Lf , H is uniquely
determined. If g converges to f in the C r -topology then H converges to the identity
in the C r -topology along the leaves of Lf and to IdX in C 0 .
Here ‘never strays away’ means that g n (Lg (H (x))) stays within a tubular
neighborhood of predetermined small size of f n (Lf (x)), for each n ∈ Z.
Remark. The statement in (2) about the smoothness of the stable distribution along the
leaves of W cs follows from the C r -section theorem [HPS, Theorem 3.5] (applied in this
case for (r − 1)). The compactness of the base space can be replaced by the appropriate
uniformities. The continuous dependence of these holonomies described in (3) follows
from a straightforward generalization of the similar continuity contained in the C r -section
theorem. Theorem B of [PSW] proves that the holonomy of W s inside W cs is C r−1 under
milder conditions.
The crucial property on which the rigidity results for abelian actions are based is that of
a (TNS) action, introduced by Katok (see [KNT]).
Definition. Let α : A × X → X be an action of A = Zκ on a compact manifold X. We
say that the action α is totally non-symplectic, or (TNS), if there is a family S of partially
hyperbolic elements in A and a continuous splitting of the tangent bundle T X = ⊕`i=1 Ei
into A-invariant distributions such that:
(i) the stable and unstable distributions of any element in S are direct sums of a subfamily of the Ei ’s;
(ii) any two distributions Ei and Ej , 1 ≤ i, j ≤ `, are included in the stable distribution
of some element in S.
Remarks. (1) It is easy to see that given a (TNS) action, one can assume that S consists
only of Anosov elements.
(2) Given a (TNS) action described by S ⊂ A with all elements of S Anosov
and a splitting T X = ⊕`i=1 Ei , one can replace the distributions {Ei } by the non-zero
T
intersections a∈S E σ (a)(a), where σ (a) ∈ {u, s}. Indeed, denote the new splitting by
T X = ⊕ki=1 Fi . It obviously satisfies (i), and (ii) can be checked as follows: given Fi
and Fj , there are 1 ≤ i 0 , j 0 ≤ ` such that Ei 0 ⊂ Fi and Ej 0 ⊂ Fj and a ∈ S such that
Ei 0 , Ej 0 ⊂ E s (a); then Fi , Fj ⊂ E s (a), by the choice of the new splitting.
(3) If the action α is linear, and Lj : Zκ → R are its Lyapunov exponents (i.e. the
logarithms of the absolute values of the eigenvalues of the matrix corresponding to the
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V. Niţică and A. Török
derivative of the action; see [KS3]), then the (TNS) property can be characterized by:
if Lj = cLi for some constant c, then c > 0.
In the following we will use the splitting given by Remark (2) above. Note that it is
given by integrable distributions. We call the corresponding laminations minimal.
Several examples of (TNS) actions are presented in [KNT, §7]. We also discuss an
example in §3 of this paper, after Corollary 3.4.
Call an element a ∈ A regular if α(a) is hyperbolic.
Recall the definition of Kazhdan’s property (T ) (see Proposition 1.14 in [HV]).
Definition. A discrete group 0 has the property (T ) if there exists a finite set S ⊂ 0 and
δ > 0 such that any unitary representation π : 0 → U(H ) that has a non-zero (δ, S)invariant vector has a non-zero invariant vector as well. (A vector ξ ∈ H is (δ, S)-invariant
if kπ(a)ξ − ξ k ≤ δkξ k for all a ∈ S.)
Finally, we introduce one more piece of notation, motivated by the examples we are
going to consider.
Notation. Given a partially hyperbolic diffeomorphism, by a horizontal lamination we
mean a C 1 -lamination whose leaves are transverse to the center distribution. In particular,
by the horizontal foliation of a product action α × IdN on M × N we mean the foliation
with leaves M × {y}, y ∈ N.
3. Results
To state our results in full generality we need the following definition.
Definition. Let 0 be a discrete group, M a compact manifold, K, L ≥ 1, and α : 0 →
Diff K (M) an action. The action α is called continuously C L,K -locally rigid if it is
−
C L,K -locally rigid, and the conjugacy varies continuously in the C k topology when the
perturbation varies continuously within a compact set in the C k -topology, for 1 ≤ k ≤ K.
Examples of continuously locally rigid actions are described before Corollary 3.4.
It is not hard to see that ‘bounded’ C L,K -local rigidity implies continuous C L,K -local
rigidity. We do not know, however, whether either of them are implied by C L,K -local
rigidity.
The two main results of this paper are the following.
T HEOREM 3.1. Let A = Zκ (κ ≥ 2), M = Tm , N be a compact manifold and
α : A → Diff K+1 (M) be a (TNS) action, where K ≥ 0 is fixed. Consider ρ : A →
Diff K+1 (M × N) given by ρ(a) := α(a) × IdN .
e
e that is C 1 -close to ρ has an invariant horizontal lamination H
(a) Any C K+1 -action ρ
−
e. The closeness
which has C (K+1) leaves and C K -holonomy between the center leaves of ρ
depends on (K + 1)-normal hyperbolicity and Kth-order center-bunching. As ρ
e converges
e converges to H in C (K+1)− and the holonomy converges uniformly in C K
to ρ in C K+1 , H
to the identity on each center leaf.
(b) Assume, moreover, that for some regular element a ∈ A there is a center leaf which
is pointwise fixed by ρ
e(a). Then, for ρ
e C 1 -close enough to ρ, there is a homomorphism
Local rigidity of certain partially hyperbolic actions
1219
e is conjugated to the product action α × π by a map
π : A → Diff K (N) such that ρ
h ∈ Homeo(M × N ):
ρ
e(a 0 ) = h(α(a 0 ), π(a 0 ))h−1 ,
a 0 ∈ A.
h(x, · ) is a C K -diffeomorphism onto its image for each x ∈ M and varies continuously in
e is the image through h of the invariant horizontal foliation
C K with x. The lamination H
H of ρ. If ρ
e converges to ρ in C 1 , then h converges to IdM×N in C 0 .
−
(c) Let K ≥ 1 and assume that the action α is continuously C L ,K -locally rigid for
some K + 1 ≥ L ≥ 1. If the conditions of (b) hold, ρ
e is C L -close to ρ, and the
K
e and
corresponding homomorphism π : A → Diff (N) is trivial (i.e. h conjugates ρ
ρ), then h( · , y) : M → M × N is C K and varies continuously in C K− with y. Hence, by
−
e converges to ρ in C L+1
Journé’s theorem (see Theorem 4.5), h ∈ Diff K (M × N ). If ρ
−
then h converges to IdM×N in C L .
Remark. Given a C K+1 cocycle β : A × M → Diff K+1 (N) over the action α, one can
construct a skew-product action ρ
e : A → Diff K+1 (M × N) by
ρ
e(a)(x, y) = (α(a), β(a, x)(y)),
where x ∈ M, y ∈ N.
The conclusion of Theorem 3.1(b) is equivalent to the fact that the cocycle β is
cohomologous to a constant cocycle. Hence, under the additional assumption that the
skew-product has a regular element that pointwise fixes a center fiber, we extend to
cocycles with values in diffeomorphism groups the results obtained in [KNT] for cocycles
with values in Lie groups.
T HEOREM 3.2. Let 0 be a discrete group with property (T ) acting through a C 1 -action
ρ on M × T, where M is a compact connected Riemannian manifold. Let µ be a smooth
probability measure whose support is M × T. Assume that:
(1) ρ preserves µ;
(2) ρ preserves each set My := M × {y}, y ∈ T;
(3) there is an element a0 ∈ 0 such that ρ(a0 ) is ergodic (with respect to the measure
induced by µ) on each set My , y ∈ T.
Then any action ρ
e : 0 → Diff(M × T) which is C 1 -close to ρ and satisfies:
(i) ρ
e(a0 ) = ρ(a0 );
(ii) ρ
e(a)(My ) ∩ My 6= ∅ for all a ∈ 0 and y ∈ T;
preserves each set My .
Remark. One can replace T by R, as can be seen from the proof. However, in that case,
more care is needed in the definition of the C 1 -closeness. If the fiber is non-compact it is
enough that the measure µ be locally finite.
Using these, one obtains the following:
T HEOREM 3.3. Fix K, L ≥ 1. Let 0 be a discrete group with property (T ) and α a smooth
action of 0 on a torus M = Tm such that:
(1) α has a periodic point x0 ∈ M (i.e. the α-orbit of x0 is finite); denote its stabilizer
by 00 ⊂ 0;
1220
V. Niţică and A. Török
α 00 contains a (TNS) abelian action α0 ;
α 00 preserves an absolutely continuous probability measure ν with support M;
−
the (TNS) abelian action α0 is continuously C L ,K -locally rigid;
−
α 00 is C L ,0 -locally rigid.
H 1 (00 , Vrep ), the first cohomology group of 00 , is trivial for any finite-dimensional
representation rep : 00 → GL(V ).
−
Then the action ρ := α × IdT of 0 on M × T is C L∨2,K -locally rigid, where a ∨ b =
max{a, b}.
(2)
(3)
(4)
(5)
(6)
Remarks.
(1) By [LS] and [LMM], the α 00 -invariant absolutely continuous measure ν
introduced in (3) is automatically smooth.
(2) Conditions (4) and (5) of Theorem 3.3 can be replaced by:
−
(40 ) the (TNS) abelian action α0 is continuously C L ,1 -locally rigid;
−
(50 ) α 00 is continuously C L ,K -locally rigid.
Proof of Theorem 3.3. Let ρ
e be a C L∨2 -small perturbation of ρ.
We first prove that ρ
e00 is conjugated to ρ 00 through a diffeomorphism h0 ∈
K−
Diff (M × T) which is C 1 -close to the identity. We then show that the diffeomorphism
e to ρ on the whole group. This last argument might be well known,
h0 actually conjugates ρ
we include it here for the sake of completeness.
By (1), ρ(00 ) has a pointwise fixed center leaf. One concludes from the theorem of
Stowe [St1, St2] and assumption (6) that there is a pointwise fixed center leaf for ρ
e(00 ).
e0 ,
Denote by A ⊂ 00 the abelian group that induces the (TNS) action α0 and by ρ0 , ρ
the restriction of ρ, respectively ρ
e, to A. Applying Theorem 3.1(c) to the action ρ0 , one
−
−
e0 and ρ0 , which is C L -small on each
obtains a C 1 -small C K conjugacy h0 between ρ
horizontal leaf of ρ0 .
Since α0 contains Anosov diffeomorphisms and any Anosov diffeomorphism that
preserves a smooth measure is ergodic, it follows that we can apply Theorem 3.2 to the
e ◦ h−1
action ρ 00 and its perturbation ρ̂ 00 , where ρ̂ := h0 ◦ ρ
0 , by taking µ to be the
product of the smooth measure ν and the Lebesgue measure on T.
Therefore, ρ̂ 00 preserves each set My := M × {y}, y ∈ T. Using assumption (5), it
follows that for each y ∈ T, the restriction (ρ̂ 00 )y of ρ̂ 00 to the set My is conjugated to
the action α 00 through a homeomorphism hy which is C 0 -close to IdM . But the Anosov
actions α A and (ρ̂ A )y coincide, hence hy is in the centralizer of α A , which is discrete
e ◦ h−1
(see [PY]). Thus hy has to be the identity, and therefore ρ 00 = h0 ◦ ρ
0 00 .
−1
e ◦ h0 00 = ρ 00 on a subgroup of finite index 00 ⊂ 0,
Next, we show that if h0 ◦ ρ
ρ ◦h−1
where ρ = α×IdT and h0 ∈ Diff 1 (M ×T) is C 0 -close to the identity, then h0 ◦e
0 =ρ
on 0 as well. The only property we need is that there is an element γ0 ∈ 00 such that α(γ0 )
is Anosov; the fact that the center leaf is one-dimensional is not relevant.
Without loss of generality, we may assume that 00 is a normal subgroup. Denote
e ◦ h−1
h0 ◦ ρ
0 by ρ̂, as before.
Let g ∈ 0. Then gγ0 g −1 ∈ 00 , and thus ρ(gγ0 g −1 ) = ρ̂(gγ0 g −1 ) = ρ̂(g)ρ(γ0 )ρ̂(g −1 )
preserves each subset My , y ∈ T. Therefore, ρ(γ0 ) preserves each set ρ̂(g −1 )My , y ∈ T,
Local rigidity of certain partially hyperbolic actions
1221
g ∈ 0. However, each of these sets is a C 1 -manifold which is not far from My (at least,
on a family of generators of 0). The only such invariant manifolds have to contain the
stable and unstable manifolds of ρ(γ0 ) (see, e.g., the proof of Lemma 4.1) and therefore
must coincide with My 0 for some y 0 ∈ T. This means that there is a homomorphism
8 : 0 → Homeo(T) with the property that ρ(g)My = M8g (y), 8 00 ≡ IdT , and 8 is
C 0 -close to the identity. Then Theorem 4.7 implies that 8 ≡ IdT on 0.
Denote, as before, by ρ̂y the restriction of ρ̂ to My , y ∈ T. For any g ∈ 0,
α(g)α(γ0 )α(g)−1 = ρy (gγ0 g −1 ) = ρ̂y (gγ0 g −1 ) = ρ̂y (g)α(γ0 )ρ̂y (g)−1 ,
therefore ρ̂y (g)−1 α(g) ∈ Homeo(T) commutes with α(γ0 ), which is an Anosov
diffeomorphism. The same argument as above shows that ρ̂y (g)−1 α(g) = IdT on a family
2
of generators, hence ρ̂y (g) = α(g) for all g ∈ 0.
Proof of Theorem 1.2. An example of action satisfying the assumptions of Theorem 3.3
is the standard linear action of 0 on Tn , n ≥ 3, where 0 is a subgroup of finite index
in SL(n, Z). Indeed, by a theorem of Prasad and Raghunathan [PR], 0 intersects a
conjugate of a given R-split Cartan subgroup of SL(n, R) in a uniform lattice. Hence, the
intersection consists of commuting matrices which generate a (TNS) action on Tn . Take
L = 1. Condition (4) follows from [KL1], condition (5) from [KLZ], and condition (6) is
a corollary of Theorem 2.1 in [Mar]. Thus Theorem 1.2 is a corollary of Theorem 3.3. 2
Katok and Spatzier have shown in a recent paper [KS2] that if α0 is an algebraic Anosov
action of Zκ , κ ≥ 2, acting on an infranilmanifold with semisimple linear part such
that no non-trivial element of the group has roots of unity as eigenvalues in the induced
representation on the abelianization, then α0 is C 1,∞ -locally rigid. Continuity of this local
rigidity follows from the proof. In the same paper, they prove the C 1,∞ -local rigidity of
linear Anosov actions of irreducible lattices in linear semisimple Lie groups G all of whose
factors have real rank at least two. Moreover, for such a lattice action, there is a maximal
abelian R-split subgroup whose action on the abelianization satisfies the above conditions.
Hence, one has the following.
C OROLLARY 3.4. Let 0 be an irreducible lattice in a linear semisimple Lie group G all
of whose factors have real rank at least two. If α : 0 → Diff ∞ (Tm ) is a linear action that
−
contains a (TNS) subaction, then α × IdT is C 2,K -locally rigid, for any K ≥ 1.
By the aforementioned theorem of Prasad and Raghunathan and Remark (3) following
the definition of the (TNS) property, given a lattice 0 in a semisimple Lie group G, the
existence of a (TNS) subaction of a linear action α : 0 → Diff(Tm ) coming from a
representation π : G → SL(m, Z) can be decided by computing the weights of π on an
R-split Cartan subgroup of G. The irreducible representations of SL(n, R) are described
by Young tableaus. For example, for n ≥ 3 and 1 ≤ k ≤ n − 1, the representations
corresponding to the tableaus with one column of height k 6= n/2, or two equal columns
of height k 6= n/4, n/3, n/2, 2n/3, 3n/4, or one line of length k < n/2 contain a (TNS)
sub-representation. However, for each given semisimple Lie group, only finitely many
irreducible representations do so.
1222
V. Niţică and A. Török
4. The (TNS) property
In this section we prove Theorem 3.1.
Notice first that given a small perturbation ρ
e of the A-action ρ and a fixed regular
e, hence it is the center
element a ∈ A, the center lamination of ρ
ea is preserved by ρ
lamination of each regular partially hyperbolic diffeomorphism ρ
eb , b ∈ A. Moreover,
ea and ρa is a leaf-wise conjugacy
the leaf-wise conjugacy Ha of Theorem 2.1(4) between ρ
for each such ρ
eb .
e that of e
ρa . Then, by
Indeed, denote by L the center foliation of ρa and by L
Theorem 2.1(4) for any η, ξ ∈ M × N :
e a (ξ )) ⇐⇒ ρ
ea n (η) never strays away from ρa n (L(ξ )).
η ∈ L(H
Pick a finite family of regular generators of A, and let b be one element of this family.
ρa n (Ha (ξ ))) is close to ρb (ρa n (L(ξ ))), uniformly with respect
Then, for ξ ∈ M × N, ρ
eb (e
ρa n (e
ρb (Ha (ξ ))) never strays away from
to n ∈ Z. Since ρb preserves L, this means that e
e
e
eb (Ha (ξ )) ∈ L(Ha (ρb (ξ ))), which shows that e
ρb preserves L
ρa n (L(ρb (ξ ))). Therefore, ρ
e
ρb , L) and (ρb , L). The claim now follows from
and Ha is a leaf-wise conjugacy between (e
the uniqueness parts of (3) and (4) in Theorem 2.1.
Assume the minimal laminations of α are given by
\
W s (a), Si ⊂ S ⊂ A, i ∈ I.
Fi =
a∈Si
Hence Si ∩ Sj 6= ∅ for any i, j ∈ I , by the (TNS) property. We can also assume that each
α(a), a ∈ S, is hyperbolic.
Let ρ
e be C 1 -close to ρ = α × Id, so that Theorem 2.1 can be applied for r = K + 1.
For i ∈ I and b ∈ Si consider the lamination
\
\
cs
e(b) :=
W
(e
a
)
W s (e
b),
(4.1)
F
i
a∈Si
where e
a stands for ρ
e(a). By Theorem 2.1, the above lamination has C K+1 leaves, and its
T
a ) is C K .
holonomy within a∈Si W cs (e
e(a) = F
e(b) .
L EMMA 4.1. If a, b ∈ Si then F
i
i
Proof. By the unique integrability of these laminations, it is enough to deal with their
a) ⊕
tangent distributions. Since the action is abelian, the splittings T (M × N ) = E s (e
ec ⊕ E u (e
ec ⊕ E u (e
a ) = E s (e
b) ⊕ E
b) are A-invariant. We claim that
E
ec ⊕ [E s (e
a ) ∩ E cs (e
b) = E
a ) ∩ E s (e
b)].
(4.2)
E cs (e
This clearly implies that
a ) ∩ [E cs (e
a ) ∩ E cs (e
b)] = E s (e
b) ∩ [E cs (e
a ) ∩ E cs (e
b)] = E s (e
a ) ∩ E s (e
b),
E s (e
from which the desired conclusion follows.
a ) ∩ Eξcs (e
b). Then v = v0 + va = v00 + vb ,
To prove (4.2), let ξ ∈ M × N and v ∈ Eξcs (e
ec , va ∈ E s (e
a ), vb ∈ E s (e
b). If v0 = v00 , we are done. Otherwise, let
with v0 , v00 ∈ E
s
0
e
a∗n w)/ke
a∗n wk approaches unit
w := vb = (v0 − v0 ) + va ∈ Eξ (b). As n → ∞, wn := (e
ec , while staying in E s (e
b). However, the angle between the distributions E s (e
b)
vectors in E
c
e
2
and E is bounded away from zero, a contradiction.
Local rigidity of certain partially hyperbolic actions
1223
ei . Moreover,
e(a) , a ∈ Si by F
By Lemma 4.1, one can denote F
i
\
ei =
W s (e
a ).
F
a∈Si
Definition. Let X be a (compact) Riemannian manifold and F and G two C 1 -laminations
of X.
(a) Given δ > 0, by the local leaf of size δ at x of F , denoted F (x, δ), we mean the
connected component of x in F (x) ∩ {y ∈ X : distX (x, y) < δ}.
(b) We say that the laminations F and G are non-tangent if
inf 6 (Tx F , Tx G) > 0.
x∈X
More generally, a family {Fi }i of C 1 -laminations is jointly non-tangent if
inf 6 (Tx Fi , span Tx Fj : j 6= i ) > 0, for each i.
x∈X
(c) We say that the non-tangent laminations F and G commute locally if there are
δ, ε > 0 such that for any x ∈ X, y ∈ F (x, δ), and z ∈ G(x, δ), there is exactly one
point in F (z, ε) ∩ G(y, ε).
Remarks. Assume that F and G are two non-tangent locally commuting C 1 -laminations
of a compact manifold.
(1) The following strengthening of the commutativity property holds:



there are δ, ε, A > 0 such that for any x ∈ X, y ∈ F (x, δ)
(4.3)
and z ∈ G(x, δ), the point w = F (z, ε) ∩ G(y, ε) satisfies


dist (x, w) ≤ A max{dist (x, y), dist (x, z)}.
X
X
X
(2) Given two points x and y in a leaf of G and a path γ in the leaf connecting them, one
can define the G-holonomy along γ between neighborhoods of x in F (x) and y in F (y).
This holonomy is a local homeomorphism. (This follows from local commutation for short
γ . In general, decompose γ into short segments.)
ei and F
ej commute locally.
L EMMA 4.2. For each i, j ∈ I , the laminations F
Remark. This is the only place where it is important that M is a torus. In this case any pair
of minimal laminations of α commute locally. Using the fact that ρ
e is a small perturbation
of a product map, and the structural stability of partially hyperbolic diffeomorphisms, we
ei .
show that this property can be lifted to the laminations F
Proof of Lemma 4.2. By the remarks at the beginning of this section, there is a
homeomorphism H ∈ Homeo(M × N ), which is C K+1 in the N-direction and induces
a leaf-wise conjugacy between the center laminations of ρ and ρ
e. Since W cs (ρ(a)) =
s
W (α(a)) × N, hence the center-stable leaves of ρ are spanned by center leaves, one
concludes from the characterization of Theorem 2.1 that H also takes the center-stable
laminations of ρ into those of ρ
e.
By [F] and [M], the action α is conjugated through a Hölder homeomorphism to a
linear action α0 . Via this conjugation, stable leaves are mapped to stable leaves. Since the
1224
V. Niţică and A. Török
stable laminations of α0 are affine (i.e. given by translations of linear subspaces; recall that
M = Tm ), the laminations {Fi } correspond through this conjugacy to affine foliations, and
clearly any two affine foliations on a torus commute locally. We conclude that any two of
the laminations {Fi } commute locally as well. Let δ0 , ε0 , and A0 be constants given by
(4.3), chosen to be correct for any pair Fi and Fj (i, j ∈ I ).
fc (B), its saturation
Given a subset B ⊂ M × N, let us denote by W c (B), respectively W
with respect to the center lamination of ρ, respectively ρ
e:
W c (B) := ∪{W c (ξ ) : ξ ∈ B},
fc (B) := ∪{W
fc (ξ ) : ξ ∈ B}.
W
fc (B) = H (W c (H −1 (B))).
Note that W
ej (ξ, δ). We want to show that
ei (ξ, δ), ζ ∈ F
Let i, j ∈ I , ξ ∈ M × N , η ∈ F
e
e
Fi (ζ, ε) ∩ Fj (η, ε) has exactly one element. (The values of δ and ε will be specified
later.)
fc of the above sets. In view of the definition (4.1) of the
Consider the saturation by W
ek , there are numbers δ 0 , δ 00 , ε0 , ε00 , δ̄ 0 , δ̄ 00 , ε̄0 , ε̄00 > 0 (depending on ξ, η, ζ, δ,
laminations F
and ε) such that
ei (ξ, δ))) ⊂ Fi (x, δ̄ 0 ) × N,
fc (F
Fi (x, δ 0 ) × N ⊂ H −1 (W
ej (ξ, δ))) ⊂ Fj (x, δ̄ 00 ) × N,
fc (F
Fj (x, δ 00 ) × N ⊂ H −1 (W
ei (ζ, ε))) ⊂ Fi (z, ε̄0 ) × N,
fc (F
Fi (z, ε0 ) × N ⊂ H −1 (W
ej (η, ε))) ⊂ Fj (y, ε̄00 ) × N,
fc (F
Fj (y, ε 00 ) × N ⊂ H −1 (W
where x = prM (H −1 (ξ )), y = prM (H −1 (η)) ∈ Fi (x, δ̄ 0 ), and z = prM (H −1 (ζ )) ∈
ek }
Fj (x, δ̄ 00 ). (Here prM stands for the projection M × N → M.) Since the laminations {F
1
1
±1
e → ρ in C and H are uniformly continuous, for each ρ
e
converge in C to {Fk } as ρ
C 1 -close to ρ one can first choose ε > 0 such that
0 < ε1 :=
and then δ > 0 such that
min {ε0 , ε00 } ≤
η,ζ ∈M×N
max {ε̄0 , ε̄00 } < ε0
η,ζ ∈M×N
max {δ̄ , δ̄ } ≤ min δ0 ,
0
ξ ∈M×N
00
ε1
.
A0 + 1
See Figure 1 for what follows.
Therefore Fi (z, ε0 ) ∩ Fj (y, ε 00 ) contains exactly one element, hence
ei (ζ, ε)) ∩ W
ej (η, ε)) = H (H −1(W
ei (ζ, ε))) ∩ H −1 (W
ej (η, ε))))
fc (F
fc (F
fc (F
fc (F
W
ej (y, ε 00 ) × N))
ei (z, ε0 ) × N) ∩ (F
= H ((F
ec . Let τ 0 := F
ei (ζ, ε)∩ L
ec and τ 00 := F
ej (η, ε)∩ L
ec
fc , say L
consists of exactly one leaf of W
0
0
0
(these intersections contain at most one point, for ε small enough).
ec . Since the last intersection consists
a , δ +ε)∩ L
Pick a ∈ Si ∩Sj . Then τ 0 , τ 00 ∈ W s (ξ,e
0
2
of a single point for δ + ε small enough, we conclude that τ 0 = τ 00 .
The (TNS)-property is necessary only to obtain Lemma 4.2. One could then prove the
following independent result.
Local rigidity of certain partially hyperbolic actions
1225
F IGURE 1.
T HEOREM 4.3. Given a family {Gi } of C 1 -laminations on a compact manifold which are
jointly non-tangent and pairwise commute locally, there is a C 1 -lamination G spanned by
−
them. If the initial laminations are C k , then G is C k .
However, in our case we can take advantage of the fact that M is a torus to simplify the
proof.
e is C 1 -close to ρ, then W s (e
a ) and
L EMMA 4.4. Let a ∈ A be a regular element. If ρ
u
a ) commute locally (recall that e
a stands for ρ
e(a)).
W (e
ei : i ∈ I s (a)} and W u (e
ei : i ∈ I u (a)},
a ) = span{F
a ) = span{F
Proof. Assume W s (e
where I s (a) = {ik : 1 ≤ k ≤ S} and I u (a) = {jk : 1 ≤ k ≤ U } form a partition
of I . We claim that for any δ1 > 0 there is a δa > 0 such that, for ξ ∈ M × N ,
a , δa ), respectively ζ ∈ W u (ξ,e
a , δa ), can be reached from ξ by
each point η ∈ W s (ξ,e
juxtaposing segments of lengths at most δ1 of the laminations {Fi : i ∈ I s (a)}, respectively
{Fj : j ∈ I u (a)}. That is, there are points ηi and ζj such that
ei (ξ, δ1 ), ηi ∈ F
ei (ηi , δ1 ), . . . , η = ηi ∈ F
ei (ηi , δ1 )
ηi1 ∈ F
S
S
1
2
2
1
S−1
and
ej (ξ, δ1 ), ζj ∈ F
ej (ζj , δ1 ), . . . , ζ = ζj ∈ F
ej (ζj , δ1 ).
ζj1 ∈ F
U
U
1
2
2
1
U−1
fc , map
a , δ) by W
Indeed, proceed as in Lemma 4.2: consider the saturation of W s (ξ,e
−1
to the corresponding picture for ρ and use the fact that α is conjugated to a
it via H
linear action to obtain the decomposition of the stable and unstable laminations of α(a)
into segments of α-minimal laminations. Take the ρ-center leaves of the points obtained
this way and map them through H to center leaves of ρ
e. The intersections of these center
a , δ) yield the points ηi . By reducing δ one can obtained the desired
leaves with W s (ξ,e
a ).
conclusion. Apply the same procedure for W u (e
Pick ε > 0 small enough. One can choose δ1 > 0 (and, therefore, δa ) such
that Lemma 4.2 and (4.3) imply that the following construction is well defined: given
1226
V. Niţică and A. Török
ξ s,u
ξ 0,u =ζ
ξ 1,1
ξ 0,1
ξ s,0 =η
ξ 1,0
ξ 0,0 =ξ
F IGURE 2.
ξ ∈ M × N, η ∈ W s (ξ,e
a , δa ), and ζ ∈ W u (ξ,e
a , δa ), let ηi , ζj be as above and define
inductively the points ξk,l , 0 ≤ k ≤ S, 0 ≤ l ≤ U , by
ξk,l
ξ0,0 := ξ ; ξ0,l := ζjl ; ξk,0 := ηik ;
ei (ξk−1,l , ε) ∩ F
ej (ξk,l−1 , ε) for k, l ≥ 1.
:= F
k
l
See Figure 2.
a , U · ε) ∩ W s (ζ,e
a , S · ε). Since the stable and unstable
But then ξS,U ∈ W u (η,e
laminations of e
a are non-tangent, for ε > 0 small enough this intersection has only one
a ) and W u (e
a ).
2
element, thus proving the local commutation of W s (e
e
Proof of Theorem 3.1(a). We construct first the lamination H.
Fix a regular element a ∈ S. There is an ε̄a > 0 such that the intersection
a , ε̄a ) ∩ W u (η,e
a , ε̄a ) has at most one element for any ξ, η ∈ M × N . Choose
W s (ξ,e
a ) and W u (e
a ). For
δa > 0, ε̄a > εa > 0, Aa > 1 satisfying (4.3) applied for W s (e
ξ, η ∈ M × N , denote
a , εa ) ∩ W u (η,e
a , εa )
[ξ, η] := W s (ξ,e
(hence, either [ξ, η] is empty or it contains one point). Note that local commutativity can
be rephrased as
[ξ, η] 6= ∅ and
dist([ξ, η], ξ ) < δa ,
dist([ξ, η], η) < δa H⇒ [η, ξ ] 6= ∅.
Consider δ̄a := min{εa , δa /(Aa + 2)}. For ξ ∈ M × N , define
a , δ̄a ), W s (ξ,e
a , δ̄a )],
Q(ξ ) = Q(ξ,e
a , δ̄a ) := [W u (ξ,e
where [U, V ] := {[u, v] : u ∈ U, v ∈ V }.
a ) and W u (e
a ) are continuous, non-tangent, and commute
Since the laminations W s (e
a , δ̄a ) ×
locally, Q(ξ ) with the induced topology from M × N is homeomorphic to W s (ξ,e
u
m
a , δ̄a ), i.e. to an open set in R (m = dim M). We claim that the family
W (ξ,e
{Q(ξ ) : ξ ∈ M × N} determines plaques of a C 0 -lamination. In view of [R], this is
equivalent to showing that
Q(ξ ) ∩ Q(η) 6= ∅ H⇒ Q(ξ ) ∩ Q(η) is open in both Q(ξ ) and Q(η).
Local rigidity of certain partially hyperbolic actions
W
1227
s
η
η’
11
00
η
ζ
0
ξ
τ0
11
00
00
11
τ’
τ
Wu
F IGURE 3.
But this follows from (actually, is equivalent to) the local commutativity of the laminations
a ) and W u (e
a ).
W s (e
It is enough to show that if ζ ∈ Q(ξ ), then ζ belongs to both IntQ(ξ ) (Q(ξ ) ∩ Q(ζ ))
and IntQ(ζ ) (Q(ξ ) ∩ Q(ζ )). We will check the first statement, the other being similar. See
Figure 3.
a , δ̄a ), η0 ∈ W s (ξ,e
a , δ̄a ). Note that
Let ζ = [τ0 , η0 ], where τ0 ∈ W u (ξ,e
a , δ̄a ) and
distM×N (ζ, η0 ), distM×N (τ0 , η0 ) < δa . Since the holonomy between W s (ξ,e
a , δ̄a ) along W u (e
a ) is a local homeomorphism, there is an open neighborhood
W s (ζ,e
a , δ̄a ) such that [ζ, V s ] ⊂ W s (ζ,e
a , δ̄a ). Similarly, there exists an
V s of η0 in W s (ξ,e
u
u
u
a , δ̄a ) with [V , ζ ] ⊂ W u (ζ,e
a , δ̄a ). Then the set
open neighborhood V of τ0 in W (ξ,e
u
s
[V , V ] is an open neighborhood of ζ in Q(ξ ), and it is also included in Q(ζ ). Indeed,
for η ∈ V s , τ ∈ V u , let τ 0 = [τ, ζ ], η0 = [ζ, η]. Then [τ, η] = [τ 0 , η0 ] ∈ Q(ζ ), as claimed.
a ) and W u (e
a ),
We have proved therefore that there is a C 0 -lamination integrating W s (e
e
e
hence the family {Fi }i∈I as well. Call this lamination H. It is clearly A-invariant and
horizontal, because so are the laminations used to construct it.
−
Since it is obtained as the span of two C K+1 -laminations, its leaves will be C (K+1) , as
can be easily seen from the following theorem of Journé (the corollary is a consequence of
the proof).
T HEOREM 4.5. (Journé [J]) Assume given on a manifold two continuous transverse
laminations, Fs and Fu , with uniformly smooth leaves. If a function f is uniformly C k+δ smooth along the leaves of Fs and Fu , then f is C k+δ -smooth (1 ≤ k ≤ ∞, δ ∈ (0, 1)).
C OROLLARY 4.6. Moreover, if Fs0 → Fs , Fu0 → Fu , f 0 Fu0 → f Fu , f 0 Fs0 → f Fs
in the C k+δ -topology, then f 0 → f in the C k+δ -topology.
Indeed, given a point ξ ∈ M × N, consider a smooth coordinate system χ : U × V →
e ) is given by the
M × N such that in these coordinates a neighborhood of ξ in the leaf H(ξ
graph of a function F : U → V (here U and V are open subsets of Euclidean spaces).
This function is uniformly C K+1 along the laminations of U obtained by projecting onto
1228
V. Niţică and A. Török
e ), respectively W u (e
e ). Therefore, by Theorem 4.5, F
a ) ∩ H(ξ
a ) ∩ H(ξ
U the leaves of W s (e
−
(K+1)
.
is C
e
e is C K between center leaves because any path along H
Finally, the holonomy of H
cs
cu
a ) and W (e
a ), and,
can be realized as a sequence of steps taken alternately inside W (e
by Theorem 2.1(2), the stable and unstable holonomies are C K within the leaves of the
center-stable, respectively center-unstable, laminations.
2
Proof of Theorem 3.1(b). Pick some y0 ∈ N and consider the projection q : M × N →
e (which is transversal to
M ≡ M × {y0 } along the center foliation of the perturbed action ρ
e and the submanifold M × {y0 }). Thus M × N becomes
both the horizontal lamination H
a fiber bundle over M, with the fibers C K -diffeomorphic to N (by Theorem 2.1(4)).
Moreover, this identification between the fibers and N is C k -uniformly bounded (k ≤ K)
with respect to ρ
e in a C k -small neighborhood of ρ.
e is constructed, each leaf of it is a covering space
Due to the way the lamination H
−1
of M. Given a center leaf q (x), consider the holonomy map of H, 1 : π1 (M, x) →
Homeo(q −1 (x)), obtained as follows: a loop γ ∈ (M, x) defines the map that associates
e
Note that, by
to y ∈ q −1 (x) the endpoint of the lift beginning at y of γ to the leaf H(y).
K −1
part (a) of Theorem 3.1, the image of 1 is actually in Diff (q (x)).
The main step is to show that the holonomy map 1 is trivial.
e(a), for some non-trivial a ∈ A (this
Assume the fiber q −1 (x) is preserved by ρ
always happens, because each partially-hyperbolic map has periodic points in the quotient
e and the center laminations imply that
M∼
e(a)-invariance of both H
= (M × N )/q). The ρ
1 ◦ Aa (γ ) = φa ◦ 1(γ ) ◦ φa−1 ,
γ ∈ π1 (M, x),
(4.4)
e(a) and φa := ρ
e(a) q −1 (x) .
where Aa ∈ Aut(π1 (M, x)) is the action induced by ρ
In the case when 1 takes values in a Lie group, equation (4.4) has only the solution
1 ≡ I , provided 1 and φa are close to the identity (see [KNT, Lemma 4.5]). One would
expect the same to be true in Diff K , but we were not able to find a proof. However, if we
add the condition that φa = Id (i.e. we consider the equation 1 ◦ Aa = 1), then this is
an easy consequence of the following theorem of Newman [N], presented here in a version
due to Smith [S] (see [B, §9] for a proof).
T HEOREM 4.7. (Newman [N]) Let X be a connected topological manifold endowed with
a metric. Then there is an ε > 0 such that any non-trivial action of a finite group (or
compact Lie group) on X has an orbit of diameter larger that ε.
L EMMA 4.8. Let M = Tm and a ∈ Diff(M) be an Anosov diffeomorphism that has a fixed
point x0 ∈ M. Fix a set T of generators of π1 (M, x0 ) ∼
= Zm . Consider the automorphism
A ∈ Aut(π1 (M, x0 )) ∼
= SL(m, Z) induced by a.
Given a compact manifold X, there is a neighborhood U of the identity in Homeo(X)
with the following property: if 1 : π1 (M, x0 ) → Homeo(X) is an A-invariant
homomorphism (i.e. 1◦A = 1) which maps T into U , then 1 is the trivial homomorphism.
Remark. As in Lemma 4.5 of [KNT], this implies that the same result holds for a an
infranilmanifold Anosov diffeomorphism.
Local rigidity of certain partially hyperbolic actions
1229
F IGURE 4.
Proof of Lemma 4.8. It is enough to consider the case when T is the set {ei }1≤i≤m , given
by the canonical basis of Zm . Denote 1i := 1(ei ).
The equation 1 ◦ A = 1 implies that 1 is the identity on the image of A − Im .
Since A is induced by an Anosov diffeomorphism, 1 is not in the spectrum of A, hence
Zm / Im(A − Im ) is a finite group, of order c = det(A − Im ). Therefore, 1ci = IdX for each
1 ≤ i ≤ m. Let U be a neighborhood of the identity in Homeo(X) such that Theorem 4.7
can be applied to any cyclic group of order dividing c.
2
e is trivial, we can construct the desired
Once we know that the holonomy of H
homeomorphism h as follows. See Figure 4. Let H ∈ Homeo(M × N ) be the leafconjugacy between ρ and e
ρ given by Theorem 2.1(4). Hence, the center leaves of ρ
e are
given by the images of H (x, · ) : N → M × N , x ∈ M. Denote H (x, · ) by Hx . Since
each Hx is a diffeomorphism from N to its image, one can speak of Hx−1 .
Fix a point x∗ ∈ M and let
e x∗ (y)).
h(x, y) := Hx (N) ∩ H(H
(4.5)
e between the center leaves given by
is the holonomy along H
That is, h(x, · ) ◦ Hx−1
∗
Hx∗ and Hx . Thus, by part (a) of Theorem 3.1, h(x, · ) : N → M × N is a C K diffeomorphism onto its image, as desired. The intersection in the definition is a unique
e and the center lamination of
point by Lemma 4.8, and h is a homeomorphism because H
ρ
e are transverse and give a global ‘product structure’ on M × N . Note that the leaves of
e are the images of h( · , y) : M → M × N, y ∈ N, and that h(x, N) = Hx (N) for each
H
x ∈ M.
It remains to check that h conjugates ρ
e to a product action:
e x (y)))
ea (Hx (N)) ∩ ρ
ea (H(H
ρ
ea (h(x, y)) = ρ
∗
e
ρa (Hx∗ (y))) = Hα
= Hα (x)(N) ∩ H(e
a
= h(αa (x), Aa (y)),
a (x)
e x∗ (Aa (y)))
(N) ∩ H(H
(4.6)
1230
V. Niţică and A. Török
ea (Hx∗ (y)), hence ρ
ea (Hx∗ (y))
where Aa ∈ Diff K (N) is defined by h(αa (x∗ ), Aa (y)) = ρ
e (The second equality in (4.6) follows from the
and Hx∗ (Aa (y)) are in the same leaf of H.
e
fact that H is a leaf-conjugacy and the ρ
e-invariance of H.)
e converges to ρ in C 1 follows from Theorem 2.1
The C 0 -convergence of h to IdM×N as ρ
and (4.5).
2
ey the image of M through h( · , y), y ∈ N. This
Proof of Theorem 3.1(c). Denote by M
K
e
is a leaf of H, hence a C -submanifold of M × N , diffeomorphic to M. Denote by
e is a lamination C K prM : M × N → M the projection on the first factor. Since H
close to the horizontal foliation of ρ, there are uniquely determined C K -diffeomorphisms
ey ⊂ M × N that are right inverses of prM ; in addition, y ∈ N 7→ φy ∈ C K
φy : M → M
−
is continuous. In view of Theorem 2.1(3), and Corollary 4.6, φy converges in C L to
x ∈ M 7→ (x, y) ∈ M × N, uniformly with respect to y ∈ N, as ρ
e converges to ρ in
C L . Under the same conditions, h0 converges in C 0 to IdM×N , by Theorem 3.1(b). Hence,
eM
for each y ∈ N, φy−1 ◦ h( · , y) conjugates the A-actions α and φy−1 ◦ ρ
ey ◦ φy , and is
C 0 -small. Moreover, since the actions are hyperbolic, this is the only conjugacy C 0 -close
−
to IdM . Therefore, since α is continuously C L ,K -locally rigid, φy−1 ◦ h( · , y) is C K and
e is C L -close enough to ρ.
varies continuously with y ∈ N in the C K -topology, provided ρ
K
Thus the same is true for y ∈ N 7→ h( · , y) ∈ C (M, M × N ).
−
e converges to ρ in C L+1 follows from
The fact that h converges to IdM×N in C L when ρ
the convergence of the holonomy maps (Theorem 2.1(3)), the continuous local rigidity of
α, and Journé’s theorem.
2
Proof of Theorem 1.1. Theorem 7.1 in [NT1] proves the C K+Lip,K−3 -deformation rigidity
of ρ for K ≥ 4. In particular, ρ is C 5,1 -deformation rigid. Let K ≥ 1 and assume that
e|A with
the smooth action ρ
e is C 5 -close enough to ρ so that Theorem 2.1 applies for ρ
r = K + 1, where A is an R-split abelian subgroup of SL(n, Z) provided by the theorem
of Prasad and Raghunathan. Denote by h0 ∈ Diff 1 (Tn × Td ) the conjugacy between ρ
e the image through h0 of the horizontal foliation H of
and e
ρ given by [NT1]. Then H,
ρ, is a ρ
eA -invariant horizontal foliation which has trivial holonomy. Since it must be
−
spanned by the stable and unstable laminations of ρ
eA , it is a C (K+1) -lamination. Let
h be the conjugacy constructed as at the end of the proof of Theorem 3.1(b). Note that
the homomorphism π is trivial, because ρ and ρ
e are conjugated. Then h is C K along the
center direction of ρ; along H one uses the result of [KL1], as was done in [NT1]. Journé’s
−
e they
theorem implies that h belongs to C K . Finally, since both h and h0 carry H to H,
e. 2
differ by a constant y-factor: h(x, y) = h0 (x, 8(y)). Thus h also conjugates ρ and ρ
5. Property (T )
In this section we prove Theorem 3.2. Although some of the following properties are
true in a more general setting, we will state them only for the situation considered by the
theorem.
Recall the following consequence of the property (T ) (see [HV, Proposition 1.16]):
Assume the group 0 has the property (T ) relative to the finite set S ⊂ 0.
Then, for any ε > 0 there is a δT = δT (ε) > 0 such that given a unitary
Local rigidity of certain partially hyperbolic actions
1231
representation π : 0 → U(H ) and a (δT , S)-invariant vector ξ ∈ H , ξ 6= 0,
there exists an invariant vector ξ 0 6= 0 with kξ − ξ 0 k ≤ εkξ k.
Denote by 1A : M × T → R the characteristic function of the set A ⊂ M × T.
ρ (a)∗ µ)/∂µ, the Radon–Nikodym derivative, where
For a ∈ 0, let 1a := ∂(e
∗
−1
(8 µ)(A) := µ(8 (A)) for A ⊂ M × T and 8 : M × T → M × T. One obtains a
1/2
e(a)−1 ) · 1a . In
unitary representation U : 0 → U(L2 (M × T, µ)) by Ua (f ) = (f ◦ ρ
the following we use the notation Ua , a ∈ 0, for the unitary action described above and Va ,
a ∈ 0, for the unitary action induced by ρ on L2 (M × T, µ) (note that since ρ preserves
µ, Va (f ) = f ◦ ρ(a)−1 ).
Let µT be the measure induced on T from µ via the projection prT : M × T → T:
µT (A) := µ(M
R × A) for AR ⊂RT. Denote by µy the conditional measure induced by µ on
My , y ∈ T: M×T g dµ = T [ My g dµy ] dµT (y). Then µy (My ) = 1 for all y ∈ T.
Note that if f ∈ L2 (M × T, µ) is invariant under the 0-action U , then |f |2 dµ is a
ρ
e-invariant measure on M × T. Conditions (2), (3) and (i) of Theorem 3.2 imply that f
has to be µy -a.e. constant on My , for µT -a.e. y ∈ T.
e) := max{distC 1 (ρ(a), e
ρ (a)) : a ∈ S}. Write ρ
e(a)−1 =
Notation. Denote distC 1 (ρ, ρ
(ha , va ), where ha : M × T → M, va : M × T → T.
If two functions differ only on a set of ν-measure zero we will say that they are equal
mod ν. We use the same notation for sets that are equal up to a ν-null set. When there is
no possibility of confusion, we will abbreviate the notation for Lp -spaces.
Remark. Denote by L2 (M × T)ρ , respectively L2 (M × T)ρe, the vectors of L2 (M × T)
that are fixed by the actions V , respectively U . The hypotheses of Theorem 3.2 imply that
L2 (M × T)ρe ⊂ L2 (M × T)ρ ∼
= L2 (T, µT ).
The conclusion of the Theorem is that L2 (M × T)ρe = L2 (M × T)ρ .
If L2 (T, µT ) were finite dimensional, the result would follow easily from the fact that 0
has property (T ). In the general case, one needs a generating family of L2 (M × T)ρ which
is uniformly almost-invariant for U . Note, however, that even if two different measure
preserving transformations are close to each other, the unitaries they induce are far apart
in the operator norm. Hence, in order to take advantage of the fact that ρ
e is close to ρ, one
has to choose these vectors in a special way. Our approach is, roughly speaking, to select
them recursively.
Remark. The next two lemmas hold if we replace T by any compact manifold. Only in
Lemma 5.3 is it important that the center direction is one-dimensional.
We begin with some simple estimates.
L EMMA 5.1.
(a) If f ∈ L2 (M × T, µ), then
e(a)−1 − f ◦ ρ(a)−1 kL2 · k1a kL∞
kUa (f ) − Va (f )kL2 ≤ kf ◦ ρ
1/2
1/2
+ kf kL2 · k1a
− 1kL∞ .
1232
(b)
V. Niţică and A. Török
If φ ∈ C 0 (T) and f = φ ◦ prT (i.e. f depends only on the second variable, hence is
V -invariant), then
kf ◦ ρ
e(a)−1 − f kL2 ≤ kf kLip · distC 0 , (va , prT ) · µ()1/2,
where  = supp f ∪ supp(f ◦ ρ
e(a)−1 ) ⊂ M × T.
Proof. Both inequalities are straightforward. The first follows from
e(a)−1 · 1a
kUa (f ) − Va (f )kL2 = kf ◦ ρ
1/2
− f ◦ ρ(a)−1 kL2
≤ k(f ◦ ρ
e(a)−1 − f ◦ ρ(a)−1 ) · 1a kL2
1/2
+ kf ◦ ρ(a)−1 · (1a
1/2
− 1)kL2 ,
using the fact that kf ◦ ρ(a)−1 kL2 = kf kL2 , by the unitarity of V .
For the second, notice that both f ◦ ρ
e(a)−1 and f are zero outside , while for
(x, y) ∈  ⊂ M × T
|f ◦ ρ
e(a)−1 (x, y) − f (x, y)| = |f (ha (x, y), va (x, y)) − f (ha (x, y), y)|
≤ kf kLip · sup{distT (va (x, y), y) : (x, y) ∈ }.
Integrating over  yields the desired conclusion.
2
The next lemma is probably well known.
completeness.
We include it here for the sake of
e) < ω then: given
L EMMA 5.2. For any ε > 0 there is a ω > 0 such that if distC 1 (ρ, ρ
aρ
e-invariant set  ⊂ M × T of positive (finite) measure, there is a U -invariant function
ξ ∈ L2 (M × T, µ) which is positive µ-a.e. on  and
kξ − 1 kL2 ≤ εk1 kL2 .
Proof. Let ε 0 be such that ε0 /(1 − ε0 ) < ε2 . Since 1a → 1 uniformly for any a ∈ 0 as
e) → 0, one can find ω > 0 such that
distC 1 (ρ, ρ
Z
1/2
1/2
1/2
2
|1a − 1| dµ
≤ k1a − 1kL∞ · k1A kL2 ≤ δT (ε 0 )k1A kL2
kUa (1A ) − 1A kL2 =
A
for any ρ
e-invariant set A ⊂ M × T and all a ∈ S, provided distC 1 (ρ, ρ
e) < ω. Then, by
e) < ω: given a ρ
e-invariant set A
property (T ), the following holds whenever distC 1 (ρ, ρ
of positive (finite) measure, there is a U -invariant function ξA0 such that supp(ξA0 ) ⊂ A and
e-invariant set and
kξA0 − 1A kL2 ≤ ε0 k1A kL2 . In particular, A \ supp(ξA0 ) is a ρ
µ(A \ supp(ξA0 ))1/2 ≤ kξA0 − 1A kL2 ≤ ε0 k1A kL2 = ε0 µ(A)1/2
(5.1)
(the condition on the support can be satisfied by restricting to A the U -invariant function
provided by the property (T )).
Let 0 := , ξ0 := ξ0 and define recursively n := n−1 \ supp(ξn−1 ), ξn := ξ0 n as
long as µ(n ) > 0.
Local rigidity of certain partially hyperbolic actions
1233
P
Indeed, k1n kL2 ≤ ε0 k1n−1 kL2
Then ξ :=
n≥0 ξn has the desired properties.
P
0
by (5.1) and kξn kL2 ≤ (1 + ε )k1n kL2 , hence n≥0 ξn converges in L2 ; the limit is
U -invariant, and its support is equal to , up to a measure-zero set. Finally,
kξ − 1 kL2 =
X
1/2
kξn − 1n \n+1 k2L2
n≥0
≤
X
ε0 k1n k2L2
2
1/2
n≥0
≤
X
1/2
kξn − 1n k2L2
n≥0
X
1/2
≤
(ε 0 )n+1
k1 kL2 .
2
n≥0
The main step in the proof is the following.
L EMMA 5.3. Assume the hypotheses of Theorem 3.2 hold. Fix p ≥ 1. There is ε∗ > 0
e) ≤ ε∗ then the following holds.
such that if distC 1 (ρ, ρ
Let I ⊂ T be a connected set with a non-empty interior such that ρ
e preserves M × I .
Then there is a measurable function ψ : M × I → [0, ∞) such that ψ ◦ ρ
ea = ψ µ-a.e.
for each a ∈ 0 and µ(ψ −1 ({λ})) ≤ µ(M × I )/p for any λ ∈ R.
Proof. We can choose a new Riemannian structure on T such that T coincides with T
endowed with the standard metric of volume one, and volT = µT . Note that this affects
distC 0 and distC 1 by at most a constant factor. We will assume from now on that this is the
original set-up of the lemma.
Replacing I by its closure we may assume without loss of generality that I is closed.
Denote M × I by . Let ` = volT (I ) = µ(M × I ).
Consider first the case I 6= T. Identify  and M × [0, `], but, for simplicity, keep all
the other notation unchanged.
Let f¯ : [0, `] → [0, 1] be given by
(
2y/`,
y ∈ [0, `/2],
f¯(y) =
2 − (2y/`), y ∈ [`/2, `],
and define f : M × [0, `] → R by f (x, y) = f¯(y).
In the following we compute some L2 -norms explicitly, but it would be enough to
notice to which power of ` these are proportional; this follows from the behavior of these
quantities under rescaling.
Let a ∈ S. Since M × [0, `] is ρ
e-invariant, for each x ∈ M the function va (x, · ) maps
[0, `] into itself, hence it has a fixed point. (Actually, condition (ii) of Theorem 3.2 implies
that both endpoints of the interval are fixed.) Therefore,
sup{distT (va (x, y), y) : (x, y) ∈ M × [0, `]} ≤ kva − pr[0,`] kLip · diam ([0, `])
e) · `,
≤ C distC 1 (ρ, ρ
where C does not depend on ρ
e or I . Since supp f ∪ supp(f ◦ e
ρ (a)−1 ) ⊂ M × [0, `],
Lemma 5.1(b) implies that
e) · `] · µ(M × [0, `])1/2.
kf ◦ ρ
e(a)−1 − f kL2 ≤ Ckf kLip · [distC 1 (ρ, ρ
1234
V. Niţică and A. Török
But
kf kL2
hence
Z
= 2
0
`/2 2
`
2
t
r
1/2
dt
=
`
,
3
e),
kf ◦ ρ
e(a)−1 − f kL2 ≤ C 0 kf kL2 distC 1 (ρ, ρ
(5.2)
e or I .
where the constant C 0 does not depend on ρ
In view of Lemma 5.1(a), the inequality (5.2) shows that if I 6= T then
sup
a∈S
kUa (f ) − f kL2
→ 0 as distC 1 (ρ, ρ
e) → 0,
kf kL2
independently of I.
(5.3)
If I = T then ` = 1 and we consider the function f :  → [0, 1] as above by
e(a)−1 −
identifying T \ {y0 } with (0, 1) for some arbitrary point y0 ∈ T. Since kf ◦ ρ
00
e) for a ∈ S, the property (5.3) is still valid.
f kL∞ ≤ C distC 1 (ρ, ρ
e) is small
Assume fixed small values ε1 , ε2 > 0, to be specified later. If distC 1 (ρ, ρ
enough (independently of `), then by Lemma 5.2, respectively (5.3) and the property (T ),
there are U -invariant functions ξ , φ ∈ L2 (M × T, µ) such that ξ > 0 µ-a.e. on  and
kξ − 1 kL2 ≤ ε2 k1 kL2 ,
(5.4)
kf − φkL2 ≤ ε1 kf kL2 .
(5.5)
We can assume that φ ≥ 0, because |φ| has the same property.
e-invariant mod µ. Since ψ is µ-a.e. preserved by
Then ψ := (φ · 1 )/ξ is ρ
ρ
e(a0 ) = ρ(a0 ), it follows from (3) and Fubini’s theorem that ψ essentially depends only
on the second variable: ψ(x, y) = ψ̄(y), (x, y) ∈ M × [0, `] µ-a.e. We have to show that
ψ̄ is not constant on a subset of I of too large measure.
Let λ ∈ R be such that ∗ := ψ −1 ({λ}) has positive µ-measure. Since ∗ is ρ(a0 )
invariant mod µ, there is a measurable set I∗ ⊂ I such that ∗ = M × I∗ , again mod µ.
From (5.5) and the triangle inequality one obtains that
ε1 kf kL2 ≥ kφ ∗ − f ∗ kL2 = k(ψξ ) ∗ −f ∗ kL2
= kλξ ∗ − f ∗ kL2 = kλ(ξ ∗ − 1∗ ) − (λ · 1∗ − f ∗ )kL2
≥ kλ · 1∗ − f ∗ kL2 − |λ| · kξ ∗ − 1∗ kL2 ,
hence, using (5.4),
ε1 k1 kL2 ≥ ε1 kf kL2 ≥ kλ · 1∗ − f ∗ kL2 − |λ| · ε2 k1 kL2 ,
that is
kλ · 1∗ − f ∗ kL2 ≤ (ε1 + |λ| · ε2 ) k1 kL2 .
But
kλ · 1∗ − f ∗ kL2
 
1 µ(∗ )) 3/2


k1 kL2 ,

 6 µ()
≥


µ(∗ )) 1/2
1


 |λ|
k1 kL2 ,
2
µ()
(5.6)
|λ| ≤ 2,
(5.7)
|λ| ≥ 2.
Local rigidity of certain partially hyperbolic actions
Indeed,
1235
volT ({y ∈ I : |λ − f¯(y)| ≤ δ}) ≤ 2δ`,
hence
kλ · 1∗ − f ∗ kL2 = kλ · 1I∗ − f¯ I∗ k2L2 (I ) ≥ δ 2 volT (I∗ \ {|λ − f¯| ≤ δ})
≥ δ 2 (volT (I∗ ) − 2δ`),
and the choice δ = volT (I∗ )/(4`) = µ(∗ )/(4µ()) yields the estimate for the case
|λ| ≤ 2. The other case is straightforward.
It is clear that for ε1 and ε2 chosen small enough, the inequalities (5.6) and (5.7) cannot
2
be satisfied simultaneously if µ(∗ )/µ() ≥ 1/p.
e) ≤ ε∗ , where ε∗ is given by
Proof of Theorem 3.2. Choose ρ
e such that distC 1 (ρ, ρ
Lemma 5.3 for p = 2.
e acts through continuous
Let B := {y ∈ T : ρ
ea (My ) 6= My for some a ∈ 0}. Since ρ
maps, B is an open set. If B = ∅, we are done. Assume B is non-void.
The complement of M × B is ρ
e-invariant, hence so is M × B. Let I ⊂ T be one of
the connected components of B. Then M × I is ρ
e-invariant because, by condition (ii) of
Theorem 3.2, ∪{e
ρa (M × I ) : a ∈ 0} ⊂ M × B is connected and contains M × I . Let
ψ : M × I → R be the function provided by Lemma 5.3. We will show that there is a
connected open non-void set J0 ⊂ I such that ψ is µ-a.e. constant on M × J0 . Taking
the union of all the connected open sets J ⊂ I that have this property and contain J0 , we
deduce that there is a maximal open connected set J∗ ⊂ I such that ψ is µ-a.e. constant
on M × J∗ .
e-invariant mod µ. Indeed, consider the open ρ
e-invariant set
But then M × J∗ is ρ
 = ∪{e
ρa (M × J∗ ) : a ∈ 0} ⊃ M × J∗ . By (ii),  is connected. Then conditions
(3) and (i) imply that  = M × T0 mod µ for some open and connected set T0 ⊂ T. Since
ψ is ρ
e-invariant, ψ  is constant µ-a.e. Thus, by the maximality of J∗ , we conclude that
ea (M × J∗ ) = M × J∗ mod µ for all a ∈ 0, as claimed.
T0 = J∗ , i.e. that ρ
e-invariant. Since µT (J∗ ) ≤ µT (I )/2 by
In particular, the boundary of M × J∗ is ρ
e-invariant,
Lemma 5.3, there is a point y 0 ∈ I ∩ ∂J∗ . But then (ii) implies that My 0 is ρ
contradicting the connectedness of I .
It remains to show that given a measurable function ψ supported on a ρ
e-invariant set
M × I where ∅ 6= I ⊂ B is open, connected and such that ψ ◦ ρ
ea = ψ µ-a.e. for all a ∈ 0,
there is an open connected set J0 ⊂ I such that ψ is µ-a.e. constant on M × J0 .
Notice first that in view of the conditions (3) and (i) of Theorem 3.2, by changing ψ on
a µ-null set we may assume that ψ(x, y) = ψ̄(y) for any (x, y) ∈ M × I . Moreover, there
is a set Z ⊂ M × I of full measure such that for any a ∈ 0:
ρ
ea (Z) = Z,
ψ(e
ρa (z)) = ψ(z),
for all z ∈ Z
ρb (z)} and take Z := M × I \ ∪{e
ρa (Ab ) : a, b ∈ 0}).
(let Ab := {z ∈ M × I : ψ(z) 6= ψ(e
Pick y ∈ I such that Zy := My ∩ Z has full µy -measure in My and a ∈ 0 such that
ea (Zy ), hence ψ̄ is constant on J00 := prT (e
ρa (Zy )),
ρ
ea (My ) 6= My . Then ψ is constant on ρ
1236
V. Niţică and A. Török
which implies that ψ is constant on M × J00 . We will show that J00 coincides mod µT with
ρa (My )).
the connected set J¯0 := prT (e
Indeed, let γ : [0, 1] → My be a C 1 curve such that prT ρa (γ ([0, 1])) = J¯0 . By Sard’s
theorem, a.e. point of J¯0 is a regular value for the mapping prT ◦ρa ◦ γ : [0, 1] → J¯0 .
Assume by contradiction that J¯0 \ J00 has positive measure in T and let y∗ ∈ J¯0 \ J00 be a
ρa ◦ γ . Let t∗ ∈ [0, 1] be a preimage
density point which is also a regular value for prT ◦e
of y∗ and x∗ := γ (t∗ ) ∈ My . Then t∗ is a density point of A := γ −1 (My \ Zy ) ⊂ [0, 1].
ρa , there is a neighborhood O of x∗ in My such that
Since x∗ is a regular point for prT ◦e
ρa (x) = y∗ } is a codimension-one submanifold of My . Moreover, by
O1 := {x ∈ O : prT ◦e
the inverse mapping theorem, for some δ > 0 and possibly after shrinking O and O1 , one
can choose around x∗ local coordinates given by a C 1 -map 8 : O1 × (t∗ − δ, t∗ + δ) → O
such that if x ∈ O and |t − t∗ | < δ then
prT (ρa (x)) = prT (ρa (γ (t))) ⇐⇒ x ∈ 8(O1 × {t}).
Therefore, 8(O1 × {t ∈ A : |t − t∗ | < δ}) ⊂ My \ Zy . Since 8 is a local diffeomorphism,
2
the left-hand side has a positive µy -measure, which contradicts our choice of y.
Acknowledgements. We would like to thank F. Almgren, G. Forni and A. Katok for
helpful discussions.
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