Table Of ContentLOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS
∗ ∗∗ ∗∗∗
ANDREY GOGOLEV , BORIS KALININ , VICTORIA SADOVSKAYA
2
1
WITH AN APPENDIX BY RAFAEL DE LA LLAVE
0
2
n Abstract. We consider an irreducible Anosov automorphism L of a torus Td
a suchthat no three eigenvalues havethe same modulus. We show that L is locally
J
rigid, that is, L is C1+H¨older conjugate to any C1-small perturbation f such that
7
the derivative D fn is conjugate to Ln whenever fnp = p. We also prove that
1 p
toral automorphisms satisfying these assumptions are generic in SL(d,Z). Ex-
] amples constructed in the Appendix show importance of the assumption on the
S
eigenvalues.
D
.
h
t
a
m 1. Introduction
[
Hyperbolic dynamical systems have been one of the main objects of study in
2 smooth dynamics. Basic examples of such systems are given by Anosov automor-
v
phisms of tori: for a hyperbolic matrix F in SL(d,Z) the map F : Rd → Rd projects
4
9 toanautomorphism ofthe torusTd = Rd/Zd. Moregenerally, adiffeomorphism f of
9
a compact Riemannian manifold M is called Anosov if there exist a decomposition
2
. of the tangent bundle TM into two f-invariant continuous distributions Es,f and
9
Eu,f, and constants C > 0, λ > 0, such that for all n ∈ N,
0
0
kDfn(v)k ≤ Ce−λnkvk for all v ∈ Es,f,
1
:
v kDf−n(v)k ≤ Ce−λnkvk for all v ∈ Eu,f.
i
X
The distributions Es,f and Eu,f are called the stable and unstable distributions of f.
r
a Structural stability is a fundamental property of hyperbolic systems. If g is an
Anosov diffeomorphism and f is sufficiently C1 close to g, then f is also Anosov and
is topologically conjugate to g, i.e. there is a homeomorphism h of M such that
g = h−1 ◦f ◦h.
In this paper we study regularity of the conjugacy h. It is well known that in
general h is only H¨older continuous. A necessary condition for it to be C1 is that
the derivatives of the return maps of f and g at the corresponding periodic points
Date: January 18, 2012.
∗
Supported in part by NSF grant DMS-1001610.
∗∗
Supported in part by NSF grant DMS-0701292.
∗∗∗
Supported in part by NSF grant DMS-0901842.
1
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 2
are conjugate. Indeed, differentiating gn = h−1◦fn◦h at a periodic point p = fn(p)
gives
D gn = (D h)−1 ◦D fn ◦D h.
p p h(p) p
A diffeomorphism g is said to be locally rigid if for any C1-small perturbation f
this condition is also sufficient for the conjugacy to be a C1 diffeomorphism. The
problem of local rigidity has been extensively studied and Anosov systems with
one-dimensional stable and unstable distributions were shown to be locally rigid
[dlL87, dlLM88, dlL92, P90].
Local rigidity problem in higher dimensions is much less understood. Examples
where the periodic condition is not sufficient were constructed by R. de la Llave
[dlL92, dlL02]. However, the one-dimensional results were extended in two direc-
tions. In the case when g is conformal on the full stable and unstable distributions,
local rigidity was established for some classes ofsystems [dlL02, KS03, dlL04, KS09].
Inadifferent direction, localrigiditywasproved in[G08]foranirreducible Anosov
toral automorphism L : Td → Td with real eigenvalues of distinct moduli, as well
as for some nonlinear systems with similar structure. We recall that L is said
to be irreducible if it has no rational invariant subspaces, or equivalently if its
characteristic polynomial is irreducible over Q. It follows that all eigenvalues of L
are simple. An important feature of this case is that Rd splits into a direct sum of
one-dimensional L-invariantsubspaces. This splittinggivesrisetothecorresponding
linear foliations on Td which are expanded or contracted by L at different rates.
Such a splitting persist for C1-small perturbations of L and provides a framework
for studying regularity of the conjugacy.
Examples in [G08] show that irreducibility of L is a necessary assumption for local
rigidity except when L is conformal on the stable and unstable distributions. The
main result of this paper is the following theorem which establishes local rigidity for
a broad class of irreducible toral automorphisms. We give a concise proof that uses
techniques from [G08, dlL02, KS09] along with some new results on conformality of
cocycles from [KS10].
Theorem 1.1. Let L : Td → Td be an irreducible Anosov automorphism such that
no three of its eigenvalues have the same modulus. Let f be a C1-small perturbation
of L such that the derivative D fn is conjugate to Ln whenever p = fnp. Then f is
p
C1+H¨older conjugate to L.
We note that irreducibility of L implies that it is diagonalizable over C. Hence
assuming that D fn is conjugate to Ln is equivalent to assuming that D fn is
p p
also diagonalizable over C and has the same eigenvalues as Ln. The only extra
assumption in the theorem ensures that the dimensions of the subspaces in the
splitting by rates of expansion/contraction are not higher than two. It allows L to
have pairs of complex conjugate eigenvalues as well as pairs λ,−λ. We prove this
theorem in Section 2.
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 3
In Section 3, we show that toral automorphisms satisfying the assumptions of the
theoremaregenericinthefollowingsense. ConsiderthesetofmatricesinSL(d,Z)of
norm at most T. Then the proportion of matrices corresponding to automorphisms
that do not satisfy our assumptions goes to zero as T → ∞. Moreover, it can be
estimated by cT−δ for some δ > 0.
Example A.3 in the Appendix yields an Anosov toral automorphism conformal on
a three-dimensional invariant subspace and a perturbation with conjugate periodic
data whose derivative is not uniformly quasi-conformal on the corresponding three-
dimensional invariant distribution. This, in particular, precludes smoothness of the
conjugacy. The automorphism is reducible, so the example does not prove that the
extra assumptionisindeed necessary forour theorem. However, itclearlyshows that
currentmethodscannotbepushedfurthertogivetheresultwithoutthisassumption.
2. Proof of Theorem 1.1
2.1. Notation and outline of the proof. We denote by Es,L and Eu,L the stable
and unstable distributions of L. Since f is C1 close to L, f is also Anosov and we
denote its stable and unstable distributions by Es,f and Eu,f. They are tangent to
the stable and unstable foliations Ws,f and Wu,f respectively (see, e.g. [KH95]).
The leaves of these foliations are C∞ smooth, but in general the distributions Es,f
and Eu,f are only H¨older continuous transversally to the corresponding foliations.
Let 1 < ρ < ρ < ··· < ρ be the distinct moduli of the unstable eigenvalues of
1 2 l
L, and let
Eu,L = EL ⊕EL ⊕···⊕EL
1 2 l
be the corresponding splitting of the unstable distribution.
By the assumption, the distributions EL, k = 1,...,l, are either one- or two-
k
dimensional. As f is C1-close to L, the unstable distribution Eu,f splits into a
direct sum of l invariant H¨older continuous distributions close to the corresponding
distributions for L:
Eu,f = Ef ⊕Ef ⊕···⊕Ef
1 2 l
(see, e.g. [Pes04, Section 3.3]). We also consider the distributions
Ef = Ef ⊕Ef ⊕...⊕Ef.
(i,j) i i+1 j
For any 1 < k ≤ l, the distribution Ef is a fast part of the unstable distribution
(k,l)
and thus it integrates to a H¨older foliation Wf with C∞ smooth leaves (see,
(k,l)
e.g. [Pes04, Section 3.3]). Moreover, the leaves Wf (x) depend C∞ smoothly on x
(k,l)
along the unstable leaves Wu,f (see, e.g. [KS07, Proposition 3.9]).
Notation. We say that an object is C1+ if it is C1 and its differential is H¨older
continuous with some positive exponent. We say that a homeomorphism h is C1+
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 4
along a foliationF if the restrictions of h to the leaves of F is C1+ and the derivative
Dh| is H¨older continuous on the manifold.
F
For any 1 ≤ k < l the distribution Ef is a slow part of the unstable distribu-
(1,k)
tion. It also integrates to an f-invariant foliation Wf with C1+ smooth leaves.
(1,k)
One way to see this is to view L as a partially hyperbolic automorphism with the
splitting Es,L ⊕EL ⊕El . It follows from the structural stability of partially
(1,k) (k+1,l)
hyperbolic systems [HPS77, Theorem 7.1] that for a C1-small perturbation f the
“central” foliation survives; that is, Ef integrates to a foliation Wf . For an
(1,k) (1,k)
alternative simple and short proof that uses specifics of our setup and also gives
unique integrability, (as opposed to existence of some foliation tangent to Ef )
(1,k)
see [G08, Lemma 6.1].
Thus within the unstable distribution Eu,f there are flags of weak and strong
distributions
Ef = Ef ⊂ Ef ⊂ ... ⊂ Ef = Eu,f,
1 (1,1) (1,2) (1,l)
Ef = Ef ⊂ Ef ⊂ ... ⊂ Ef = Eu,f.
l (l,l) (l−1,l) (1,l)
Sincebothflagsareuniquely integrableandtheleaves ofthecorresponding foliations
areatleast C1+, forany1 ≤ k ≤ l thedistributionEf = Ef ∩Ef alsointegrates
k (1,k) (k,l)
uniquely to a H¨older foliation
Vf = Wf ∩Wf
k (1,k) (k,l)
with C1+ smooth leaves. Similarly, the distributions Ef = Ef ∩Ef , 1 ≤ i ≤
(i,j) (1,j) (i,l)
j ≤ l, integrate to H¨older foliations
Wf = Wf ∩Wf .
(i,j) (1,j) (i,l)
We use similar notation for the automorphism L: EL = EL ⊕ ... ⊕ EL, and
(i,j) i j
WL and VL are the linear foliations tangent to EL and EL respectively.
(i,j) i (i,j) i
Since L is Anosov and f is C1 close to L, there exists a bi-Ho¨lder continuous
homeomorphism h : Td → Td close to the identity in C0 topology such that
h◦L = f ◦h.
The conjugacy h takes the flag of weak foliations for L into the corresponding weak
flag for f:
Lemma 2.1. For any 1 ≤ k ≤ l, h(WL ) = Wf .
(1,k) (1,k)
The proof is the same as that of Lemma 6.3 in [G08]. We give the argument for
the reader’s convenience.
Proof. Let h˜, f˜ and L˜ be the lifts of h, f and L to Rd. Similarly we use the tilde
sign to denote lifts of various foliations.
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 5
Since h˜(W˜ u,L) = W˜ u,f we have that h˜(W˜ L ) ⊂ W˜ u,f. Let y ∈ W˜ u,L(x), then
(1,k)
y ∈ W˜ L (x) if and only if dist(L˜n(x),L˜n(y)) ≤ (ρ +ǫ)ndist(x,y) for all n > 0,
(1,k) k
where dist is the standard metric on Rd. Since ˜h is C0 close to Id we further obtain
that y ∈ W˜ L (x) if and only if for all n > 0
(1,k)
dist(f˜n(h˜(x)),f˜n(h˜(y))) = dist(h˜(L˜n(x)),˜h(L˜n(y))) ≤ (ρ +ǫ)ndist(x,y)+c.
k
The latter condition in turn is equivalent to h˜(y) ∈ W˜ f (h˜(x)). (cid:3)
(1,k)
We note that Lemma 2.1 holds for any sufficiently C1-small perturbation of an
Anosov automorphism of Td.
The coarse strategy of the proof of Theorem 1.1 is showing inductively that h is
C1+ along WL for any k and thus along WL = Wu(L). By the same argument,
(1,k) (1,l)
h is C1+ along Ws(L) and hence h is C1+ by Journ´e Lemma:
Lemma 2.2 (Journ´e [J88]). Let M be a manifold and Fs, Fu be continuous
j j j
transverse foliations on M with uniformly smooth leaves, j = 1,2. Suppose that
j
h : M → M is a homeomorphism that maps Fs into Fs and Fu into Fu. More-
1 2 1 2 1 2
over, assume that the restrictions of h to the leaves of these foliations are uniformly
Cr+ν, r ∈ N, 0 < ν < 1. Then h is Cr+ν.
The main steps of the proof of the Theorem are the following statements:
• h(VL) = Vf;
i i
• h is a C1+ diffeomorphism along VL.
i
Their proofs are interdependent and organized into an inductive process given by
Propositions 2.3 and 2.4.
Proposition 2.3. If h(VL) = Vf, then h is a C1+ diffeomorphism along VL.
i i i
The proof of this proposition is given in Section 2.2 below. Since VL = WL ,
1 (1,1)
Lemma 2.1 implies that h(VL) = Vf, and then Proposition 2.3 yields that h is C1+
1 1
along VL. This provides the base of the induction. The inductive step is given by
1
the following proposition.
Proposition 2.4. Suppose that h(VL) = Vf, 1 ≤ i ≤ k − 1, and h is a C1+
i i
diffeomorphism along WL . Then h(VL) = Vf and h is a C1+ diffeomorphism
(1,k−1) k k
along WL .
(1,k)
The proof of this proposition is given in Section 2.3 (and also uses an inductive
argument). In the proof we only need to establish that h(VL) = Vf. Then Propo-
k k
sition 2.3 implies the smoothness of h along VL, and the smoothness along WL
k (1,k)
follows from the Journ´e Lemma 2.2.
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 6
2.2. Proof of Proposition 2.3. In this subsection we write
VL d=ef VL, Vf d=ef Vf, EL d=ef EL = TVL, Ef d=ef Ef = TVf.
i i i i
The proof is an adaptation of arguments of de la Llave [dlL02]. First we show
that h is Lipschitz along VL as a limit of smooth maps with uniformly bounded
derivatives. Then we prove that the measurable derivative of h along VL is actually
H¨older continuous. Both steps use Livˇsic Theorem for commutative and noncom-
mutative cocycles and rely on conformality of L and f along VL and Vf respectively.
Conformality of f along Vf is crucial and to establish it we use a result from [KS10].
First we construct a map h close to h and satisfying the following conditions:
0
(1) h (VL) = Vf, moreover, h (VL(x)) = Vf(h(x)) for all x in Td;
0 0
(2) sup d (h (x),h(x)) < +∞, where d is the distance along the leaves;
x∈Td Vf 0 Vf
(3) h is C1+ diffeomorphism along the leaves of VL.
0
Let V¯L be the linear integral foliation of Es,L ⊕EL ⊕...⊕EL ⊕EL ⊕...⊕EL.
1 i−1 i+1 l
We define the map h by intersecting local leaves:
0
h (x) = Vf,loc(h(x))∩V¯L,loc(x).
0
The map is well-defined and satisfies (2) since h is close to the identity. Condition
(1) holds since h(VL(x)) = Vf(h(x)) by the assumption, and (3) is satisfied since
for any x the leaf Vf(h(x)) is C1+ and C1 close to VL(x).
It follows easily as in [dlL02, Theorem 2.1] that
h = lim h , where h = f−n ◦h ◦Ln.
n n 0
n→∞
Indeed, let us endow the space of maps satisfying (1) and (2) with the metric
d(k ,k ) = sup d (k (x),k (x)). Then, since f−1 contracts the leaves of Vf, it
1 2 x Vf 1 2
follows that the map k 7→ f−1 ◦k ◦L is a contraction with the fixed point h.
Now we prove that h is Lipschitz along Vf. For this it suffices to show that the
derivatives of the maps h along VL are uniformly bounded. We estimate
n
kD h k ≤ kDf−n| k·kD h k·kLn| k
VL(x) n Ef(h0(Lnx)) VL(Lnx) 0 EL(x)
= k Dfn|Ef(f−n(h0(Lnx))) −1k·kDVL(Lnx)h0k·kLn|ELk
≤ k(cid:0)Dfn|Ef(hn(x)) −1k·(cid:1)kLn|ELk·supkDVL(z)h0k.
z
(cid:0) (cid:1)
Since D h is continuous on Td, the supremum on the right is finite. Now we show
VL 0
that the product k(Dfn| )−1k·kLn| k is uniformly bounded in y and n.
Ef(y) EL
We concentrate on the case when Vf is two-dimensional. The one-dimensional
case is similar except for conformality of L along VL and of f along Vf is trivial.
Since L is irreducible it is diagonalizable over C. Therefore, as the eigenvalues of
L| have the same modulus, L| is conformal with respect to some norm on EL.
EL EL
We can assume that our background norm k·k is chosen so that L| is conformal.
EL
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 7
By the assumption of the theorem, D fn is conjugate to Ln whenever fnp = p. It
p
follows that D fn| is also diagonalizable over C and has eigenvalues of the same
p Ef(p)
modulus. To obtain conformality of Df| , we apply the following result to vector
Ef
bundle E = Ef and cocycle F = Df| .
Ef
[KS10,Theorem1.3]LetE be a H¨oldercontinuouslinearbundlewith two-dimensional
fibers over a compact Riemannian manifold M. Let F : E → E be a H¨older continu-
ous linear cocycle over a transitive Anosov diffeomorphism f : M → M. If for each
periodic point p ∈ M, the return map Fn : E → E is diagonalizable over C and
p p p
its eigenvalues are equal in modulus, then F is conformal with respect to a H¨older
continuous Riemannian metric on E.
Wedenoteby k·kf thenorminduced bythemetriconEf(x)given bythetheorem.
x
The conformality of Df| with respect to this norm means that
Ef
kDf(v)kf = c(x)·kvkf for any x ∈ Td and v ∈ Ef(x).
f(x) x
Clearly, c(x) = kDf| kf, the norm of Df : (Ef(x),k·kf) → (Ef(f(x)),k·kf ).
Ef(x) x f(x)
We set
a(x) = kL| k = kL| k and b(x) = c(h(x)) = kDf| kf.
EL(x) EL Ef(h(x))
The function a(x) is constant in our context, however we will keep the variable for
consistency with b(x). Since L is conformal on EL, a(x) satisfies
a (x) d=ef a(x)a(Lx)···a(Ln−1x) = kLn| k.
n EL
The function b(x) is H¨older continuous, and using the relation fm◦h = h◦Lm and
the conformality of Df| we obtain
Ef
b (x) d=ef b(x)b(Lx)···b(Ln−1x) =
n
= kDf|Ef(h(x))kf ·kDf|Ef(h(Lx))kf ···kDf|Ef(h(Ln−1x))kf = kDfn|Ef(h(x))kf.
We claim that the functions a and b are cohomologous, i.e. the exists a continuous
function φ : Td → R such that
+
a(x)/b(x) = φ(Lx)/φ(x).
This follows from the Livˇsic Theorem [Liv71], [KH95, Theorem 19.2.1] once we
show that a (p) = b (p) for any periodic point p = Lnp. We note that b (p) =
n n n
kDfn| kf is the modulus of the eigenvalues of Dfn| since this linear
Ef(h(p)) Ef(h(p))
map is conformal with respect to norm k·kf . A similar statement holds for a (p)
h(p) n
and Ln| . The coincidence of the periodic data for f and L implies that indeed
EL
a (p) = b (p)andhence thefunctions aandbarecohomologous. Using conformality
n n
we obtain that
kLn| k·k(Dfn| )−1kf = kLn| k· kDfn| kf −1 =
EL Ef(h(x)) EL Ef(h(x))
= a (x)/b (x) = φ(Lnx)/φ(x)
n n (cid:0) (cid:1)
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 8
is uniformly bounded since φ is continuous on Td. Since the norm k·kf is equivalent
to k·k we obtain that k(Dfn| )−1k·kLn| k is uniformly bounded in y and n. We
Ef(y) EL
conclude that kD h k is uniformly bounded in x and n, and hence h is Lipschitz
VL(x) n
along Vf.
A similar argument shows that k(D h )−1k is uniformly bounded and hence
VL(x) n
h is bi-Lipschitz along Vf. In particular, D h exists and is invertible almost ev-
VL
erywhere.
Differentiating f◦h = h◦L along VL on a set of full Lebesgue measure we obtain
Df| ◦D h = D h◦L| ,
Ef(h(x)) VL(x) VL(Lx) EL(x)
i.e., the cocycles Df| and L| are cohomologous with transfer function
Ef(h(x)) EL(x)
D h. The bundle Ef is trivial since it is close to the trivial bundle EL. Therefore,
VL(x)
Df| and L| can be viewed as H¨older continuous GL(2,R)-valued cocycles
Ef(h(x)) EL(x)
over the automorphism L. Moreover, the existence of conformal metrics implies
that they are cohomologous to cocycles with values in the conformal subgroup. We
remark that in general measurable transfer functions are not necessarily continuous
[PW01, Section9]. However, forconformalcocyclesthemeasurable transferfunction
coincidesalmost everywhere withaH¨oldercontinuousone. Thisfollowsfrom[Sch99,
Theorem 6.1] or from [PP97, Theorem 1] after reducing cocycles to orthogonal ones
by factoring out the norms. See also [S10] for stronger results on GL(2,R)-valued
cocycles. We conclude that D h is H¨older continuous, and hence h is a C1+
VL(x)
diffeomorphism along VL. (cid:3)
2.3. Proof of Proposition 2.4. The proof is based on the following proposition.
Proposition 2.5. Assume that h(WL ) = Wf , h(VL) = Vf and h is a C1+
(i,k) (i,k) i i
diffeomorphism along VL. Then h(WL ) = WL .
i (i+1,k) (i+1,k)
We apply Proposition 2.5 inductively with i = 1,...,k − 1. At every step the
assumption of the proposition is fulfilled due to the assumptions in the Proposi-
tion 2.4 and the conclusion of Proposition 2.5 at the previous step. We obtain the
conclusion of Proposition 2.4 at the final step when WL = WL = VL.
(i+1,k) (k,k) k
It remains to prove Proposition 2.5. We will use the following simplified notation:
(WL,VL,UL) = (WL ,VL,WL ),
(i,k) i (i+1,k)
(Wf,Vf,Uf) = (Wf ,Vf,Wf ).
(i,k) i (i+1,k)
We note that VL and UL are slow and fast sub-foliations of WL respectively. Sim-
ilarly, Vf and Uf are slow and fast sub-foliations in Wf. We also note that Uf =
Wf∩Wf . The foliation Wf is a fast part of the unstable foliation and hence
(i+1,l) (i+1,l)
is C∞ inside the unstable leaves, see for example [KS07, Proposition 3.9]. There-
fore, the foliation Uf is C1+ inside the leaves of Wf and the holonomies between
the leaves of Vf along Uf are uniformly C1+.
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 9
Let F = h−1(Uf). Then F is a continuous foliation with continuous leaves that
subfoliates WL. We need to show that F = UL. Since VL = h−1(Vf), F is topologi-
cally transverse to VL, that is, any leaf of F and and any leaf of VL in the same leaf
of WL intersect at exactly one point. First we prove an auxiliary statement that
gives some insight into relative structure of F and VL.
For any point a ∈ Td and any b ∈ F(a) we denote by H : VL(a) → VL(b) the
a,b
holonomy along the foliation F, i.e. for every x ∈ VL(a) we define H (x) to be the
a,b
unique point of intersection F(x)∩VL(b).
Lemma 2.6. For any point a ∈ Td and any b ∈ F(a) the holonomy map H is a
a,b
restriction to VL(a) of a parallel translation inside WL.
Proof. For any point c ∈ Td and any d ∈ Uf(c) we denote by H˜ : Vf(c) → Vf(d)
c,d
the holonomy along the foliation Uf, which is C1+ as we noted above. Since F =
h−1(Uf) and h(VL) = Vf we have
H = h−1 ◦H˜ ◦h.
a,b h(a),h(b)
Since h is a C1+ diffeomorphism along VL we conclude that H is C1+.
a,b
UL(a) Uf(h(a))
b Ha,b(x) h−1 H˜h(a),h(b)(x)
VL(b) h(b) Vf(h(b))
F(a)
h
a x VL(a) h(a) h(x) Vf(h(a))
L−n
Figure 1.
To show that H is the restriction of a parallel translation, we prove that the
a,b
differential DH = Id. We apply L−n and denote a = L−n(a), b = L−n(b). Since
a,b n n
F = h−1(Uf) and f preserves the foliation Uf, L preserves F and we can write
H = Ln ◦H ◦L−n.
a,b an,bn
LOCAL RIGIDITY FOR ANOSOV AUTOMORPHISMS 10
Differentiating and denoting D H = Id+∆ we obtain
an an,bn n
D H = Ln| ◦D H ◦L−n| = Id+Ln| ◦∆ ◦L−n| .
a a,b VL an an,bn VL VL n VL
Since L is conformal on VL with respect to some inner product,
kLn| k·kL−n| k ≤ C and hence kLn| ◦∆ ◦L−n| k ≤ Ck∆ k for all n.
VL VL VL n VL n
Itremainstoshowthatk∆ k → 0. Thisfollowseasilybydifferentiatingtheequation
n
H = h−1 ◦H˜ ◦h.
an,bn h(an),h(bn)
Indeed, we obtain
D H = (D h)−1 ◦D H˜ ◦D h.
an an,bn bn h(an) h(an),h(bn) an
Since dist(a ,b ) → 0 as n → ∞ we obtain that D H˜ → Id. Also, by
n n h(an) h(an),h(bn)
choosing a subsequence if necessary, we can assume that lima = limb = c ∈ Td.
n n
ThusD h → D h,D h → D handhence∆ = D H −Id → 0asn → ∞. (cid:3)
an c bn c n an an,bn
Now let a be a fixed point of L and let B be the unit ball in UL(a) centered at a.
If B ⊂ F(a), then UL(a) = F(a) by invariance under L. Since L is irreducible and
UL is invariant, the leaf UL(a) is dense in Td. It follows that the set of points x such
that UL(x) = F(x) is dense in Td and hence UL = F. Therefore, it suffices to show
that B ⊂ F(a).
We argue by contradiction. Assume that there is z ∈ B such that z ∈/ F(a).
1 1
Let
x = VL(z )∩F(a).
1 1
Since VL has dense leaves we can choose a sequence {b ,n ≥ 1} ⊂ VL(a) so that
n
b → x as n → ∞. Let
n 1
y = H (b ),
n a,x1 n
where H is the holonomy map along F from VL(a) to VL(x ). Continuity of F
a,x1 1
implies that the sequence y converges to a point x ∈ F(a). Moreover, Lemma 2.6
n 2
implies that {x ,x } is a parallel translation of {a,x }.
1 2 1
We continue this procedure inductively to construct the sequence {x ,n ≥ 1} ⊂
n
F(a). Let
z = VL(x )∩UL(a).
n n
Then according to the construction
(2.1) d (z ,a) = n·d (z ,a) and d (x ,z ) = n·d (x ,z ).
UL n UL 1 VL n n VL 1 1
For every n we take N(n) to be the smallest integer such that L−N(n)(z ) ∈ B.
n
Since L−1 contracts UL exponentially faster than VL, equation (2.1) implies that
d (L−N(n)(x ),L−N(n)(z )) → ∞ as n → ∞.
VL n n
This contradicts an obvious bound due to compactness of B:
maxd (z,VL(z)∩F(a)) < ∞.
VL
z∈B
