LYAPUNOV EXPONENTS
AND INVARIANT CONFORMAL STRUCTURES,
WITH APPLICATIONS TO RIGIDITY PROBLEMS IN GEOMETRY
– PART I –
VICTORIA SADOVSKAYA
Abstract. In the first part of the minicourse we will concentrate on properties of linear
cocycles. A linear cocycle over a diffeomorphism f of a manifold M is an automorphism
of a vector bundle over M that projects to f . Important examples are the differential
of f and its restrictions to invariant sub-bundles of T M. We will discuss the notions
of conformality and uniform quasiconformality of the cocycle, and the related notion of
an invariant conformal structure. We will prove existence and continuity of an invariant
conformal structure for a uniformly quasiconformal cocycle A and give applications to
rigidity of Anosov systems. Further, we will obtain conformality of a cocycle A over a
hyperbolic diffeomorphism f from its periodic data, i.e. the values of A at the periodic
points of f . An important role in these arguments is played by fiber bunching and
holonomies of the cocycle. Finally, we will consider cocycles with one Lyapunov exponent
over hyperbolic and partially hyperbolic systems and obtain a structural theorem for
them, which can be viewed as a continuous version of the Amenable Reduction.
1
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
2
1. Lecture 1
1.1. Linear cocycles.
Let f be a diffeomorphism of a compact connected manifold M and let P : E → M be
a finite dimensional vector bundle over M. A linear cocycle over f is an automorphism
A of E which projects to f , that is satisfies P ◦ A = f ◦ P .
We denote by Ax : Ex → Ef x the linear map between the fibers and use the following
notations for the iterates of A: A0x = Id, and for n ∈ N.
Anx = A(f n−1 x) ◦ · · · ◦ A(x) : Ex → Ef n x
and
−1
n
: Ex → Ef −n x .
A−n
x = (Af −n x )
Clearly, A satisfies the cocycle equation Axn+k = Anfk x ◦ Akx .
In the case of a trivial vector bundle E = M×Rd , any linear cocycle A can be identified
with a matrix-valued function A : M → GL(d, R) given by A(x) = Ax . We call such A a
GL(d, R)-valued cocycle.
For us, the primary example of a linear cocycle is the differential Df viewed as an
automorphism of the tangent bundle T M or its restriction to a Df -invariant sub-bundle
E 0 ⊂ T M, such as the stable or unstable sub-bundle of a hyperbolic or partially hyperbolic
system. In these examples,
Ax = Dx f or Ax = Df |E 0 (x)
and
Anx = Dx f n or Anx = Df n |E 0 (x) .
Since the stable and unstable sub-bundles are Hölder continuous but usually not more
regular, we will assume that the fiber bundle E is Hölder continuous and we will consider
linear cocycles in the Hölder category.
We fix a Hölder continuous Riemannian metric on E. A linear cocycle A is called
Hölder continuous if the fiber maps Ax depend Hölder continuously on x, i.e. there exist
K, β > 0 such that for all nearby x, y ∈ M
(1.1)
d(Ax , Ay ) = kAx − Ay k + k(Ax )−1 − (Ay )−1 k ≤ K · dist(x, y)β
where Ax and Ay are viewed as matrices using local coordinates and k.k is the operator
norm. In fact, the second term in the sum is not needed for continuous A and compact M
as (Ax )−1 is then automatically continuous in x and bounded on M, so we can estimate
k(Ax )−1 − (Ay )−1 k = k(Ax )−1 (Ay − Ax ) (Ay )−1 k ≤ K 0 · kAx − Ay k.
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
3
1.2. Conformal structures.
A conformal structure on Rd , d ≥ 2, is a class of proportional inner products. The space Cd
of conformal structures on Rd can be identified with the space of real symmetric positive
definite d × d matrices with determinant 1, which is isomorphic to SL(d, R)/SO(d, R).
The group GL(d, R) acts transitively on Cd via
X(C) = (det X T X)−1/d · X T C X,
C ∈ Cd , X ∈ GL(d, R).
The space Cd carries a GL(d, R)-invariant Riemannian metric of non-positive curvature.
The distance to the identity in this metric is
√
1/2
(1.2)
dist(Id, C) = d/2 · (log λ1 )2 + · · · + (log λd )2
,
where λ1 , . . . , λd are the eigenvalues of C.
If C ∈ Cd and X ∈ GL(d, R) is sufficiently close to the identity, then
dist (C, X(C)) ≤ k(C) · kX − Id k,
where k(C) is bounded on compact sets in Cd [KS10].
For a vector bundle E → M we can consider a bundle C over M whose fiber Cx is
the space of conformal structures on Ex . Using a background Riemannian metric on E,
the space Cx can be identified with the space of symmetric positive linear operators on
Ex with determinant 1. We equip the fibers of C with the Riemannian metric as above.
A continuous (measurable) section σ of C is called a continuous (measurable) conformal
structure on E. A measurable conformal structure σ is called bounded if the distance
between σ(x) and τ (x) is bounded on M for a continuous conformal structure τ on M.
An invertible linear map A : Ex → Ey induces an isometry from Cx to Cy via
A(C) = (det(A∗ A))1/d · (A−1 )∗ C A−1 ,
where C ∈ Cx is a conformal structure viewed as an operator, and A∗ is the adjoint of
A. If A : E → E is a linear cocycle over f , we say that a conformal structure σ on E is
A-invariant if
Ax (σ(x)) = σ(f x) for all x ∈ M.
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
4
1.3. Conformality and uniform quasiconformality.
The quasiconformal distortion is a measure of non-conformality of the cocycle.
Definition 1.1. The quasiconformal distortion of a cocycle A on E is the function
max { k Anx (v) k : v ∈ Ex , kvk = 1 }
, x ∈ X and n ∈ Z.
QA (x, n) = kAnx k · k(Anx )−1 k =
min { k Anx (v) k : v ∈ Ex , kvk = 1 }
If QA (x, n) ≤ K for all x and n, the cocycle is called uniformly quasiconformal, and
if QA (x, n) = 1 for all x and n, it is called conformal.
Clearly, a cocycle A is conformal with respect to a Riemannian metric on E if and
only if it preserves the conformal structure associated with this metric. We note that
the notion of uniform quasiconformality does not depend on the choice of a continuous
metric on E. So if A preserves a continuous conformal structure on E then A is uniformly
quasiconformal on E with respect to any continuous metric on E. Theorem 1.3 below
shows that the converse is also true if f is a transitive Anosov diffeomorphism.
In the next proposition we apply observations made by D. Sullivan and P. Tukia for
quasiconformal group actions to our case. Note that the converse statement is also true.
Proposition 1.2. Let f be a diffeomorphism of a compact manifold M and let A : E → E
be a continuous linear cocycle over f . If A is uniformly quasiconformal then it preserves
a bounded Borel measurable conformal structure σ on E.
Proof. Let τ be a continuous conformal structure on E. We denote by τ (x) the conformal
structure on Ex , x ∈ M. We consider the set
n
S(x) = { (A−n
f n x )(τ (f x)) : n ∈ Z }
in C(x), the space of conformal structures on Ex . Since A is uniformly quasiconformal,
the sets S(x) have uniformly bounded diameters. Since the space Cx has non-positive
curvature, for every x there exists a uniquely determined closed ball of the smallest radius
containing S(x). We denote its center by σ(x).
It follows from the construction that the conformal structure σ is A-invariant and its
distance from τ is bounded. For any k ≥ 0 the set
n
Sk (x) = { (A−n
f n x )(τ (f x)) : |n| ≤ k}
depends continuously on x in Hausdorff distance, and so does the center σk (x) of the
smallest ball containing Sk (x). Since Sk (x) → S(x) as k → ∞ for any x, the conformal
structure σ is the pointwise limit of continuous conformal structures σk (x). Hence σ is
Borel measurable.
Theorem 1.3. [S02, KS10] Let f be a transitive Anosov diffeomorphism and let A : E → E
be a Hölder continuous linear cocycle over f . If A is uniformly quasiconformal then it
preserves a Hölder continuous conformal structure on E, equivalently, A is conformal with
respect to a Hölder continuous Riemannian metric on E.
Proof of Hölder continuity will be given in the next lecture.
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
5
1.4. An application: smoothness of stable/unstable sub-bundles and rigidity
for uniformly quasiconformal Anosov diffeomorphisms. (Definition in Sec. 2.1.)
Theorem 1.4 ([S02]). Let f be a transitive C ∞ Anosov diffeomorphism of M.
Suppose that f is uniformly quasiconformal on E u with dim E u ≥ 2. Then
(i) f is conformal with respect to a Riemannian metric on E u which is Hölder
continuous on M and C ∞ along the leaves of the unstable foliation,
(ii) The holonomy maps of the stable foliation are conformal, and E s is C ∞ .
Proof. (i) By the previous theorem, A preserves a Hölder continuous conformal structure
on E. We show its smoothness along the leaves using non-stationary linearization.
Proposition 1.5. (Non-stationary linearization for 12 pinched contractions) [S02]
Let f be a diffeomorphism of a compact Riemannian manifold M, and let
W be a continuous invariant foliation with C ∞ leaves. Suppose that kDf |T W k < 1,
and there exist K > 0 and θ < 1 such that for all x ∈ M and n ∈ N,
(1.3)
k (Df n |Tx W )−1 k · k Df n |Tx W k2 ≤ K θn .
Then for every x ∈ M there exists a C ∞ diffeomorphism hx : W (x) → Tx W such that
(i) hf x ◦ f = Dx f ◦ hx ,
(ii) hx (x) = 0 and Dx hx is the identity map,
(iii) hx depends continuously on x in C ∞ topology.
Suppose f is preserves a continuous conformal structure τ on E u . Then (1.3) is satisfied
for f −1 and we get a non-stationary linearization hx along W u . For each x ∈ M we extend
τ (x) to a constant conformal structure σ on T Exu : for any t ∈ Exu we denote by σ(t) the
push forward of τ (x) by translation from 0 to t.
Proposition 1.6 ([S02]). Each hx is conformal, i.e. it pushes τ on W u (x) to σ on E u (x)
and hence τ is C ∞ along the leaves of W u .
Proof. We need to show that for any y ∈ W u (x), hx (τ (y)) = σ(hx (y)). We iterate by f −1 .
By continuity of τ and properties (ii,iii) of hx , for each > 0 there exists n > 0 such that
dist(hf −n x (τ (f −n y)), σ(hf −n x (f −n (y))) < .
To obtain the following equalities, we note that Df n induces an isometry between
the spaces of conformal structures, τ is f -invariant, σ is Df -invariant, and hx (y) =
Df n (hf −n x (f −n y)) by Proposition 1.5 (i). Thus,
> dist ( hf −n x (τ (f −n y)), σ(hf −n x (f −n y) )
= dist ( Df n (hf −n x (τ (f −n y))), Df n (σ(hf −n x (f −n y)) )
= dist ( Df n (hf −n x (f −n (τ (y))), σ(Df n (hf −n x (f −n y)) )
= dist ( hx (τ (y)), σ(hx (y)) ).
As the above holds for any > 0, it follows that hx (τ (y)) = σ(hx (y)).
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
6
(ii) Let x and y be two nearby points in M. We consider the holonomy map of the
u
u
stable foliation Hx,y : Wloc
(x) → Wloc
(y):
u
u
s
z ∈ Wloc
(x) 7→ Hx,y (z) = Wloc
(y) ∩ Wloc
(z).
First we show that the holonomy maps Hx,y are conformal, i.e. Hx,y (τ (z)) = τ (Hx,y (z)).
By the C r -section Theorem [HPS], pinching given by conformality of f on E u implies that
the stable distribution is C 1 , and hence the holonomy maps are uniformly C 1 [PSW]. It
suffices to consider the case when z = x and y ∈ W s (x) so that y = Hx,y (z). We iterate
by f and note that Hx,y = f −n ◦ Hf n x,f n y ◦ f n and that Df n x Hf n x,f n y is close to identity
for large n since f n y is close to f n x. Since τ is continuous, τ (f n x) is close to τ (f n y).
Thus, Hf n x,f n y (τ (f n x)) is close to τ (f n y). Since f −n induces an isometry between the
spaces of conformal structures on E u (f n y) and on E u (y), we conclude that Hx,y (τ (x)) is
close to τ (y), so by letting n → ∞, we get Hx,y (τ (z)) = τ (Hx,y (z)).
Now we show that Hx,y are C ∞ . Using the non-stationary linearization coordinates we
view it as the map
u
u
Gx,y = hy ◦ Hx,y ◦ h−1
x : Ex → Ey ,
which is a conformal C 1 diffeomorphism defined on a neighborhood of 0, and hence is
C ∞ . Indeed, if n > 2, it is Möbius, and if n = 2 it is complex analytic.
It follows that E s is C ∞ [PSW].
Theorem 1.7 ([KS03]). (Global rigidity of UQC Anosov diffeomorphisms)
Let f be a transitive C ∞ Anosov diffeomorphism of M which is uniformly quasiconformal
on the stable and unstable sub-bundles. Suppose either that both sub-bundles have dimension at least three, or that they have dimension at least two and M is an infranilmanifold.
Then f is C ∞ conjugate to an affine Anosov automorphism of a finite factor of a torus.
Ideas of the proof. Theorem 1.4 implies that both E u and E s are C ∞ and there is a
C ∞ Riemannian metric on M for which f is conformal on both E u and E s . In general,
smoothness of both E u and E s is not known to imply smooth conjugacy to algebraic
model (although this is conjectured). Results of Benoist and Labourie [BL93] give such a
conjugacy if f preserves a smooth affine connection.
Under the assumptions of the theorem, we showed that the holonomy maps Hx,y are
globally defined affine conformal maps between the leaves and then that the non-stationary
linearization coordinates hsx and hux depend C ∞ on x. This allowed us to construct a
smooth invariant affine connection.
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
7
2. Lecture 2
2.1. Anosov and partially hyperbolic diffeomorphisms - definitions.
A diffeomorphism f of a compact manifold M is called partially hyperbolic if there exist
a nontrivial Df -invariant splitting of the tangent bundle T M = E s ⊕ E c ⊕ E u , and a
Riemannian metric on M for which one can choose continuous functions ν < 1, ν̂ < 1,
γ, γ̂ such that for all x ∈ M and unit vectors v s ∈ E s (x), v c ∈ E c (x), and v u ∈ E u (x)
(2.1)
kDf (v s )k < ν(x) < γ(x) < kDf (v c )k < γ̂(x)−1 < ν̂(x)−1 < kDf (v u )k.
The sub-bundles E s , E u , and E c are called, respectively, stable, unstable, and center.
E s and E u are tangent to the stable and unstable foliations W s and W u respectively. If
E c is trivial, f is called hyperbolic or Anosov. Most of the results for Anosov case below
also have direct analogs for more general hyperbolic systems such as locally maximal
hyperbolic sets and subshifts of finite type.
s
We define the local stable manifold of x, Wloc
(x), as a ball centered at x of radius ρ in
s
the intrinsic metric of W (x). We choose ρ sufficiently small so that for every x ∈ M we
s
(x). Local unstable manifolds are defined similarly.
have kDfy k < ν(x) for all y in Wloc
s
u
For hyperbolic f we can choose ρ so that Wloc
(x) ∩ Wloc
(z) consists of a single point for
any sufficiently close x and z in M. This property is called the local product structure
of the stable and unstable foliations. We say that a measure µ has local product structure
u
s
(x)
(x) and Wloc
if it is locally equivalent to the product of measures on Wloc
For partially hyperbolic f , an su-path in M is defined as a concatenation of finitely
many subpaths which lie entirely in a single leaf of W s or W u and f is called accessible if
any two points in M can be connected by an su-path.
The diffeomorphism f is called center bunched if the functions ν, ν̂, γ, γ̂ can be chosen
to satisfy
ν < γγ̂ and ν̂ < γγ̂.
It follows that kDf |E c k · k(Df |E c )−1 k · ν < γ̂ −1 γ −1 ν < 1 and similarly with ν̂.
We say that f is volume-preserving if it has an invariant probability measure µ in the
measure class of a smooth volume. If f is essentially accessible, C 2 and center bunched
then µ is ergodic [BW10].
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
8
2.2. Fiber bunching and holonomies. Next we define fiber bunching of a cocycle.
This condition means that non-conformality of the cocycle is dominated, in a sense, by
the contraction and expansion in the base given by the functions ν and ν̂ in (2.1). In
particular, bounded, conformal, and uniformly quasiconformal cocycles are fiber bunched.
We use the weakest, “pointwise”, version of fiber bunching with non-constant estimates
of expansion and contraction in the base.
Definition 2.1. A β-Hölder cocycle A over a (partially) hyperbolic diffeomorphism f is
called fiber bunched if there exist θ < 1 and L such that for all x ∈ M and n ∈ N,
(2.2)
QA (x, n) · (νxn )β = kAnx k k(Anx )−1 k · (νxn )β < L θn
and
QA (x, −n) · (ν̂x−n )β < L θn ,
where νxn = ν(f n−1 x) · · · ν(x) and ν̂x−n = (ν̂(f −n x))−1 · · · (ν̂(f −1 x))−1 .
This condition plays an important role in the study of non-commutative cocycles and
in particular ensures existence of holonomies, which we define next. This terminology
for linear cocycles was introduced in [V08] and the holonomies were further studied in
[ASV13, KS13, S15]. The result below gives existence of holonomies under the weakest
fiber bunching assumption (2.2).
Proposition 2.2. (Existence of holonomies) Suppose that A is a β-Hölder
fiber bunched linear cocycle over f . Then for every x ∈ M and y ∈ W s (x) the limit
A,s
Hx,y
= lim (Any )−1 ◦ Anx ,
(2.3)
n→∞
exists and satisfies
A,s
(H1) Hx,
y is a linear map from Ex to Ey which depends continuously on (x, y);
A,s
A,s
A,s
A,s
A,s −1
(H2) Hx,
= Hy,A,sx ;
x = Id and Hy, z ◦ Hx, y = Hx, z , which implies (Hx, y )
A,s
n −1
n
(H3) Hx,
◦ HfA,s
n x, f n y ◦ Ax for all n ∈ N;
y = (Ay )
β
s
(H4) kH A,s
x,y − Id k ≤ c dist(x, y) , where c is independent of x and y ∈ Wloc (x).
A,s
The family of maps Hx,y
, x ∈ M, y ∈ W s (x), defined by (2.3) is called the (standard)
stable holonomy for A. We note that for a β-Hölder fiber bunched cocycle (2.3) is the only
family of maps satisfying (H1,2,3,4) [KS13], but there may be other families satisfying
(H1,2,3) and (H4) with exponent β 0 < β [KS16].
The unstable holonomy H A,u is defined similarly:
A,u
−n −1
n
n
−1
Hx,
◦ (A−n
, where y ∈ W u (x).
y = lim (Ay )
x ) = lim Af −n y ◦ (Af −n x )
n→∞
n→∞
It satisfies (H1, 2, 4) and
A,u
−n −1
−n
(H30 ) Hx,
◦ HfA,u
for all n ∈ N.
−n x, f −n y ◦ Ax
y = (Ay )
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
9
2.3. Continuity of a measurable invariant conformal structure.
Theorem 1.3 follows from Proposition 1.2 and the next theorem since UQC implies fiber
bunching and we can take µ to be, for example, the measure of maximal entropy, which
satisfies the assumptions on µ in Theorem 2.3.
Theorem 2.3. [KS13] Let f be a transitive Anosov diffeomorphism, A : E → E be a
β-Hölder fiber bunched linear cocycle over f , and µ be an ergodic f -invariant probability
measure with full support and local product structure. Then any A-invariant µ-measurable
conformal structure on E is β-Hölder.
We also have a similar result for volume preserving partially hyperbolic case. Note that
only continuity is obtained in this case.
Theorem 2.4. [KS13] Let f be a center bunched accessible partially hyperbolic diffeomorphism, A : E → E be a β-Hölder fiber bunched cocycle over f , and µ be an f -invariant
volume. Then any A-invariant µ-measurable conformal structure τ on E is continuous.
Let τ be an A-invariant µ-measurable conformal structure on E. Both results follow
from the next proposition which shows that τ is essentially invariant under the stable
and unstable holonomies of A. It applies to both hyperbolic and partially hyperbolic case
since it uses only contraction of W s and ergodicity of µ.
Proposition 2.5. Let H s be the stable holonomy for a linear cocycle A. If τ is a µmeasurable A-invariant conformal structure then τ is essentially H s -invariant, i.e. there
is a set G ⊂ M of full measure such that
s
s
τ (y) = Hxy
(τ (x)) for all x, y ∈ G such that y ∈ Wloc
(x).
Proof. We denote by Anx (σ) the push forward of a conformal structure σ from Ex to Ef n x
induced by Anx , and similar notations for push forwards by H s . We also let xi = f i x.
Since Any induces an isometry and τ is A-invariant, we obtain
s
s
dist (τ (y), Hxy
(τ (x))) = dist(Any (τ (y)), Any Hxy
(τ (x))) =
= dist(τ (yn ), Hxsn yn Anx (τ (x))) = dist(τ (yn ), Hxsn yn (τ (xn ))) ≤
≤ dist(τ (yn ), τ (xn )) + dist(τ (xn ), Hxsn yn (τ (xn ))).
Since τ is µ-measurable, by Lusin’s Theorem there exists a compact set S ⊂ M with
µ(S) > 1/2 on which τ is uniformly continuous and hence bounded. Let G be the set of
points in M for which the frequency of visiting S equals µ(S) > 1/2. By Birkhoff Ergodic
Theorem, µ(G) = 1.
Suppose that both x and y are in G. Then there exists a sequence {ni } such that xni ∈ S
s
and yni ∈ S. Since y ∈ Wloc
(x), dist(xni , yni ) → 0 and hence dist(τ (xni ), τ (yni )) → 0 by
s
uniform continuity of τ on S. Since H s is continuous and satisfies Hxx
= Id, we have
s
kHxni yni − Idk → 0. Since τ is bounded on S,
dist(τ (xni ), Hxsni yni (τ (xni ))) → 0.
s
We conclude that dist (τ (y), Hxy
(τ (x))) = 0 and thus τ is essentially H s -invariant.
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
10
Similarly, τ is essentially H u -invariant. By Hölder continuity of holonomies (H4) this
implies essential Hölder continuity of τ along stable and unstable leaves:
dist (τ (x), τ (y)) ≤ C dist(x, y)β
s/u
if y ∈ Wloc (x) and both x, y ∈ G.
To complete the proof of Theorem 2.3 we use the local product structure of µ and
the local product structure of the stable and unstable foliations to show that τ coincides
µ-a.e. with a Hölder continuous conformal structure on supp µ = M.
In the proof of Theorem 2.4 the local product structure is replaced by accessibility.
In this case the global continuity of τ on M follows from [ASV13, Theorem E] or [W13,
Theorem 4.2]. If one makes a stronger accessibility assumption, a simpler argument yields
Hölder continuity of τ .
2.4. Obtaining fiber bunching and conformality from periodic data.
Abundance of periodic points is one of the key features of hyperbolic systems. We consider
the periodic data of A, i.e. {Anp : p = f n p}.
In the previous results we needed fiber bunching to get the holonomies. The next
proposition allows us to obtain fiber bunching of a cocycle A from fiber bunching of its
periodic data.
Proposition 2.6. [KS10, S17] Let A be a β-Hölder cocycle over an Anosov diffeomorphism f : M → M. Suppose that there exist θ̃ < 1 and L̃ such that whenever f n p = p,
(2.4)
QA (p, n) · (νpn )β < L̃ θ̃n
and
QA (p, −n) · (ν̂p−n )β < L̃ θ̃n .
Then A is fiber bunched.
The proof uses results on subadditive sequences of continuous functions and on periodic
approximation of Lyapunov exponents of measures [K11].
If a cocycle A over an Anosov diffeomorphism f is conformal or uniformly quasiconformal then, clearly, there exists a constant Cper such that QA (p, n) ≤ Cper for every periodic
point p and n such that p = f n p. The next theorem shows that the converse is also true.
A similar result was also obtained in [LW10] under an extra bunching-type assumption.
Theorem 2.7. [KS10] Let A : E → E be a Hölder continuous linear cocycle over a
transitive C 2 Anosov diffeomorphism f . If there exists a constant Cper such that
(2.5)
QA (p, n) = kAnp k · k(Anp )−1 k ≤ Cper
whenever f n p = p,
then A is conformal with respect to a Hölder continuous Riemannian metric on E.
Proof. First, the assumption (2.5) and Proposition 2.6 give that A is fiber bunched. By
Theorem 1.3 it suffices to show that A is uniformly quasiconformal on E. We do this
using a dense orbit argument. Since f is transitive, there exists a point z ∈ M with dense
orbit O = {f k z : k ∈ Z}. We will show that the quasiconformal distortion QA (z, k) is
uniformly bounded in k ∈ Z. Since O is dense and QA (x, k) is continuous on M for each
k, this implies that QA (x, k) is uniformly bounded in x ∈ X and k ∈ Z.
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
11
We consider any two points of O with dist(f k1 z, f k2 z) < δ0 , where δ0 is sufficiently small
to apply the Anosov Closing Lemma [KtH, Theorem 6.4.15]. We assume that k1 < k2 and
denote x = f k1 z and n = k2 − k1 , so that δ = dist(x, f n x) < δ0 . By the Anosov Closing
Lemma there exists p ∈ M with f n p = p such that dist(f i x, f i p) ≤ cδ for i = 0, . . . , n.
s
u
Let y be a point in Wloc
(p) ∩ Wloc
(x). Similarly to Proposition 2.2 (H4), fiber bunching
implies
−1
β
k(Anp )−1 ◦ Any − Id k ≤ Cδ β and k(A−n
◦ A−n
y )
x − Id k ≤ Cδ .
It follows that if δ0 is sufficiently small,
QA (y, n)/QA (p, n) ≤ (1 + Cδ β )/(1 − Cδ β ) ≤ 2 and QA (x, n)/QA (y, n) ≤ 2
Thus QA (x, n) ≤ 4 QA (p, n) ≤ 4Cper .
We take m > 0 such that the set {f j z; |j| ≤ m} is δ0 -dense in M. Let
Qm = max{QA (z, j) : |j| ≤ m}.
Then for any k > m there exists j, |j| ≤ m, such that dist(f k z, f j z) ≤ δ0 and hence
QA (z, k) ≤ QA (z, j) · QA (f j z, k − j) ≤ Qm · 4Cper .
The case of k < −m is considered similarly. Thus QA (z, k) is uniformly bounded.
2.5. Weaker assumptions on periodic data.
One may try to make weaker assumptions on periodic data, for example that for each p,
there is a uniform bound C(p) on QA (p, n) for all periods n. This is equivalent to each of
the following three statements:
Anp is diagonalizable over C with its eigenvalues equal in modulus;
Anp is conjugate to a conformal linear map;
There exists a Anp -invariant conformal structure on Ep .
In fact, the periodic assumption in the first part of Theorem 2.7 is equivalent to having
such conformal stuctures for all periodic points uniformly bounded. Our next result for
two-dimensional bundles does not require any extra assumptions.
Theorem 2.8. [KS10] Let A : E → E be a Hölder continuous linear cocycle over a
transitive C 2 Anosov diffeomorphism f . Suppose that the fibers of E are two-dimensional.
If for each periodic point p ∈ M, the return map Anp : Ep → Ep is diagonalizable over C
and its eigenvalues are equal in modulus, then A is conformal with respect to a Hölder
continuous Riemannian metric on E.
We note that while Theorem 2.7 holds in any dimension, Theorem 2.8 does not hold in
dimension higher than 2, as the following example shows.
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
12
Proposition 2.9. [KS10] Let f be an Anosov diffeomorphism of M and let E = M×Rd ,
d ≥ 3. For any > 0 there exists a Lipschitz continuous linear cocycle A : E → E,
which is -close to the identity, such that for all periodic points p ∈ M the return maps
Anp : Ep → Ep are conjugate to orthogonal maps, but A is not conformal with respect to
any continuous Riemannian metric on E.

cos α(x) − sin α(x)
cos α(x)
Outline of the construction. Let E = M × R3 and Ax = sin α(x)
0
0

0 .
1
Let S be a closed f -invariant set in M without periodic points and let α : M → R
be a Lipschitz continuous function satisfying
α(x) = 0 for x ∈ S and 0 < α(x) ≤ for x ∈
/S


1 0 n
Then for x ∈ S, Anx =  0 1 0  and so QA (x, n) → ∞ as n → ∞.
0 0 1
Thus A cannot be conformal.
At p = f n p, the map Anp has eigenvalues of modulus 1 and is diagonalizable if the 2 × 2
rotation block has complex eigenvalues, that is the rotation angle α(p) + · · · + α(f n−1 p)
does not equal πk. This can be ensured for all periodic points by slightly modifying the
function α, if necessary.
Finally, we note that Theorem 2.8 and 2.7 were motivated by the study of rigidity for
conformal Anosov systems. In particular, Theorem 2.8 as well as some other results in
this lecture, were obtained for the restrictions of Df to an invariant sub-bundle of T M
for Anosov f [KS09]. Theorem 2.8 and 2.7, together with rigidity results for conformal
Anosov systems like Theorem 1.7, allow to obtain global rigidity from periodic data.
More generally, they can be applied to Df -invariant sub-bundles inside E u and E s . For
example, Theorem 2.8 was used to obtain a local rigidity result for a broad class of
irreducible Anosov automorphisms of Td in [GKS11].
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
13
3. Lecture 3
3.1. Lyapunov exponents and their periodic approximation.
For any measure µ on M, any vector bundle E over M is trivial on a set of full measure
and hence any linear cocycle A can be viewed as a GL(d, R)-valued cocycle on a set of full
measure. By Oseledets Multiplicative Ergodic Theorem, for any continuous linear cocycle
A over f , the Lyapunov exponents of A and Lyapunov decomposition of E are defined
almost everywhere for every ergodic f -invariant measure µ. We note that the exponents
and the decomposition depend on the choice of µ.
Theorem 3.1. (Oseledets Multiplicative Ergodic Theorem) Let f be an invertible
ergodic measure-preserving transformation of a Lebesgue probability space (X, µ).
Let A be a measurable GL(d, R)-valued cocycle over f satisfying log kAx k ∈ L1 (X, µ)
1
and log kA−1
x k ∈ L (X, µ).
Then there exist numbers λ1 < · · · < λ` , an f -invariant set Λ with µ(Λ) = 1,
and an A-invariant Lyapunov decomposition
Ex = Ex1 ⊕ · · · ⊕ Ex` for x ∈ Λ
such that
(i) lim n−1 log kAnx vk = λi for any i = 1, . . . , ` and any 0 6= v ∈ Exi , and
n→±∞
P
(ii) lim n−1 log | det Anx | = `i=1 mi λi , where mi = dim Exi .
n→±∞
The numbers λ1 , . . . , λ` are called the Lyapunov exponents of A with respect to µ.
By Lyapunov exponents of A at a periodic point p we mean the Lyapunov exponents
of A with respect to the invariant measure on the orbit of p. They equal (1/n) of the
logarithms of the absolute values of the eigenvalues of Anp , where n is a period of p.
Theorem 3.2 (Periodic approximation of Lyapunov exponents). [K11]
Let (M, f ) be a hyperbolic dynamical system, let A be a Hölder continuous linear cocycle
over f , and let µ be an ergodic invariant measure for f .
Then the Lyapunov exponents λ1 ≤ · · · ≤ λd of A with respect to µ, listed with
multiplicities, can be approximated by the Lyapunov exponents of A at periodic points.
More precisely, for any > 0 there exists a periodic point p ∈ M for which the Lyapunov
(p)
(p)
(p)
exponents λ1 ≤ · · · ≤ λd of A satisfy |λi − λi | < for i = 1, . . . , d.
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
14
3.2. The largest and smallest Lyapunov exponents. We are primarily interested in
the largest and the smallest Lyapunov exponents of A with respect to µ, which can be
expressed as follows:
1
λ+ (A, µ) = λ` = lim log kAnx k for µ-a.e. x ∈ M,
n→∞ n
(3.1)
1
λ− (A, µ) = λ1 = lim log k(Anx )−1 k−1 for µ-a.e. x ∈ M.
n→∞ n
These limits exist and are constant almost everywhere since the sequence of functions
an (x) = n1 log kAnx k is subadditive, i.e. it satisfies
an+k (x) ≤ ak (x) + an (f k x) for all x ∈ M and n, k ∈ N.
Hence by the Subaddititive Ergodic Theorem for µ-a.e. x,
Z
an (x)
an (µ)
an (µ)
lim
an (x) dµ.
= lim
= inf
: n ∈ N , where an (µ) =
n→∞
n→∞
n
n
n
X
Now we briefly consider the infinite-dimensional case, where each fiber Ex is a Banach space V . In this case, there is no Oseledets MET in general, but upper and lower
Lyapunov exponents λ+ (A, µ) and λ− (A, µ) can still be defined as in (3.1).
In infinite-dimensional setting, it is not always possible to approximate λ+ (A, µ) and
λ− (A, µ) by Lyapunov exponents at periodic points. The following proposition is based
on an example by L. Gurvits of a pair of operators whose joint spectral radius is greater
than the generalized spectral radius [Gu95].
Proposition 3.3. There exists a locally constant cocycle A over a full shift on two
symbols and an ergodic invariant measure µ such that λ(A, µ) > supµp λ(A, µp ),
where the supremum is taken over all uniform measures µp on periodic orbits.
In the setup of Theorem 3.2, the largest and smallest exponents of A with respect to µ
can also be approximated by n1 log kAnp k and n1 log k(Anp )−1 k−1 , respectively [K11]. We
note that for periodic measures µp ,
λ+ (A, µp ) = (1/n) log (spectral radius of Anp ) ≤ (1/n) log kAnp k,
and the inequality can be strict. The approximation of λ+ (A, µ) and λ− (A, µ) in terms
of the norms extends to the infinite-dimensional case.
Theorem 3.4. [KS17] Let (M, f ) be a hyperbolic system, let µ be an ergodic f -invariant
Borel probability measure on M, and let A be a Hölder continuous GL(V )-valued cocycle
over f , where V is a Banach space.
Then for any > 0 there exists a periodic point p = f n p such that
λ+ (A, µ) − 1 log kAnp k < and λ− (A, µ) − 1 log k(Anp )−1 k−1 < .
n
n
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
15
3.3. Cocycles with one exponent and Continuous Amenable Reduction.
Now we return to the finite-dimensional case and consider cocycles with one Lyapunov
exponent, i.e. satisfying λ+ (A, µ) = λ− (A, µ). Clearly, this is a broader class than
conformal or uniformly quasiconformal cocycles.
When A has more than one Lyapunov exponent, the invariant sub-bundles E i given
the Oseledets MET are measurable but not necessarily continuous. The next theorem
establishes continuity of measurable invariant sub-bundles for fiber bunched cocycles with
one Lyapunov exponent. The following theorem is a corollary of results by A. Avila and
M. Viana in [AV10].
Theorem 3.5. Let f be a transitive Anosov diffeomorphism, A : E → E be a β-Hölder
fiber bunched linear cocycle over f , and µ be an ergodic f -invariant probability measure
with full support and local product structure. If λ+ (A, µ) = λ− (A, µ) then any µ-measurable
A-invariant sub-bundle of E is β-Hölder.
Now we obtain a structure theorem for cocycles with one exponent.
Theorem 3.6. [KS13] Let f be a transitive Anosov diffeomorphism and A : E → E
be a β-Hölder linear cocycle over f . Suppose that for every periodic point p = f n p the
invariant measure µp on its orbit satisfies λ+ (A, µp ) = λ− (A, µp ), i.e. all eigenvalues of
Anp are equal in modulus.
Then there exist a flag of β-Hölder A-invariant sub-bundles
(3.2)
{0} = E 0 ⊂ E 1 ⊂ ... ⊂ E j−1 ⊂ E j = E
and β-Hölder Riemannian metrics on the factor bundles E i /E i−1 , i = 1, ..., j, so that the
factor-cocycles induced by A on E i /E i−1 are conformal. Moreover, there exists a positive
β-Hölder function φ : M → R s.t. the factor-cocycles of φA on E i /E i−1 are isometries.
In the case when E1 = E, the cocycle A itself is conformal on E with respect to some
continuous Riemannian metric.
Outline of the proof. We note that the periodic assumptions imply fiber bunching and
one exponent for each ergodic measure. We take µ to be the measure of maximal entropy,
or the invariant volume if it exists, and trivialize the bundle E on a set of full measure,
i.e. we measurably identify E with M × Rd and view A as a function M → GL(d, R).
Thus we can use Zimmer’s Amenable Reduction:
Theorem 3.7. (Zimmer’s Amenable Reduction) Let f be an ergodic transformation
of a measure space (X, µ) and let A : X → GL(d, R) be a measurable function.
Then there exists a measurable function C : X → GL(d, R) such that the function
B(x) = C −1 (f x) A(x) C(x) takes values in an amenable subgroup of GL(d, R).
Thus we obtain a measurable coordinate change function C : M → GL(d, R) such that
B(x) = C −1 (f x) Ax C(x) is µ-a.e. in an amenable subgroup G ⊂ GL(d, R).
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
16
Amenable subgroups of Lie groups were studied in [M79]. There are 2d−1 standard
maximal amenable subgroups of GL(d, R). They correspond to the distinct compositions
of d, d1 + · · · + dk = d, and each group consists of all block-triangular matrices of the form


B1 ∗ . . . ∗
. 
.

 0 B2 . . .. 
(3.3)
 . .

.. ... ∗ 
 ..
0 . . . 0 Bk
where each diagonal block Bi is a di × di conformal matrix, i.e. a scalar multiple of an
orthogonal matrix. Any amenable subgroup of GL(d, R) has a finite index subgroup which
is contained in a conjugate of one of these standard subgroups. Thus we may assume that
G has a finite index subgroup G0 which is contained in one of the standard subgroups.
We concentrate on the simplest case when G0 = G. Then the sub-bundle V i spanned
by the first d1 + · · · + di coordinate vectors in Rd is B-invariant for i = 1, . . . , k. Denoting
Exi = C(x)V i we obtain the corresponding flag of measurable A-invariant sub-bundles
E1 ⊂ E2 ⊂ · · · ⊂ Ek = E
with dim E i = d1 + · · · + di .
By Theorem 3.5 we may assume that the sub-bundles E i are Hölder continuous. Since
B1 (x) is a conformal matrix for µ-a.e. x, the push forward by C of the standard conformal
structure on V 1 is invariant under the restriction of A to E 1 and hence Hölder continuous.
Similarly, we consider the factor-bundles E i /E i−1 over M with the natural induced
cocycle A(i) . Since the matrix of the map induced by B on V i /V i−1 = Rdi is Bi , it
preserves the standard conformal structure on Rdi . Pushing it forward by C we obtain a
measurable conformal structure τi on E i /E i−1 invariant under A(i) . The holonomies H s
and H u induce holonomies for A(i) on E i /E i−1 . As before, we conclude that τi is essentially
invariant under these holonomies and hence is Hölder continuous on M.
Now we outline the argument for the case when G0 6= G. An example illustrating this
case is when G0 is the full diagonal subgroup and G is its normalizer, which is the finite
extension of G0 that contains all permutations of the coordinate axes.
In the general case, G0 still has the invariant flag V 1 ⊂ V 2 ⊂ · · · ⊂ V k = Rd with
conformal structures on the factors. The elements of G may not preserve this flag, instead
there are several images of it under G:
1
2
k
V(1)
⊂ V(1)
⊂ · · · ⊂ V(1)
...
1
2
k
V(`) ⊂ V(`) ⊂ · · · ⊂ V(`)
These flags are again mapped by C to the corresponding measurable flags on E as before.
The sub-bundles in the flags are not A-invariant individually, but A preserves the union
i
i
of subspaces on each level of the flag C(V(1)
) ∪ · · · ∪ C(V(`)
), i = 1, ..., k − 1. For each
level of the flag we show that the union is essentially holonomy invariant and hence is
Hölder continuous. The union may not split into individual invariant sub-bundles since it
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
17
may “twist” along non-trival loops in M. Thus to obtain actual continuous sub-bundles
one needs in general to pass to a special finite cover of M. These sub-bundles are not
necessarily invariant as A may permute them, but they are invariant under an iterate of
A. In this way we can obtain the general result below, Theorem 3.9
To complete the proof of Theorem 3.6, we claim that the assumption on the periodic
1
data implies that A is conformal on the sub-bundle spanned by the union C(V(1)
)∪
1
· · · ∪ C(V(`) ), and then similarly for the induced cocycles on the factors. Indeed, the scalar
cocycles that give expansion/contraction of A (more precisely, of an iterate of the lift of A
as above) are continuously cohomologous by the Livsic Periodic Point Theorem, since the
exponents at each periodic point are the same. This yields uniform quasiconformality and
hence conformality of A on the span. The same reasoning shows that the scalar cocycles
of expansion/contraction for the different conformal factors of A are all cohomologous to
one scalar function, whose inverse gives the function φ in the theorem.
Corollary 3.8. (Polynomial growth of quasiconformal distortion and norm)
Let f be a transitive Anosov diffeomorphism. Suppose that for every f -periodic point p
the invariant measure µp on its orbit satisfies λ+ (A, µp ) = λ− (A, µp ). Then there exists
m < dim Ex and C such that
QA (x, n) ≤ Cn2m
for all x ∈ M and n ∈ Z.
Moreover, if λ+ (A, µp ) = λ− (A, µp ) = 0 for every µp , then there exists m < dim Ex and
C such that
kAnx k ≤ C|n|m for all x ∈ M and n ∈ Z.
One can take m = j − 1, which is the number of non-trivial sub-bundles in (3.2).
3.4. The theorem for partially hyperbolic diffeomorphisms. As we indicated above,
the argument for the proof of Theorem 3.6 yields a general result, which holds also for
partially hyperbolic f . In the next theorem we assume one exponent for the volume and
fiber bunching of A. The assumptions of Theorem 3.6 imply one exponent for each ergodic
measure as well as fiber bunching. For hyperbolic f , the invariant volume µ can be replaced by any ergodic f -invariant probability measure with full support and local product
structure, and the resulting conformal structures and sub-bundles are Hölder continuous.
Theorem 3.9. [KS13] Let f be an accessible partially hyperbolic center bunched diffeomorphism preserving a volume µ and let A : E → E be a Hölder continuous linear cocycle
over f . Suppose that A is fiber bunched and λ+ (A, µ) = λ− (A, µ).
Then there exists a finite cover à : Ẽ → Ẽ of A and N ∈ N such that ÃN satisfies the
following property. There exist a flag of continuous ÃN -invariant sub-bundles
(3.4)
{0} = Ẽ 0 ⊂ Ẽ 1 ⊂ ... ⊂ Ẽ k−1 ⊂ Ẽ k = Ẽ
and continuous conformal structures on the factor bundles Ẽ i /Ẽ i−1 , i = 1, ..., k, invariant
under the factor-cocycles induced by ÃN .
LYAPUNOV EXPONENTS AND INVARIANT CONFORMAL STRUCTURES
18
The proof shows that when it is necessary to pass to a cover, the resulting cocycle ÃN
preserves more than one flag as in (3.4). Their union is preserved by à and is the lift of
an invariant object for A. To illustrate this we construct a cocycle A on E = T2 × R2
with no invariant µ-measurable sub-bundles or conformal structures. Its lift à to a double
cover preserves two continuous line bundles, while A preserves a continuous field of pairs
of lines.
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Department of Mathematics, The Pennsylvania State University, University Park, PA
16802, USA.
E-mail address: [email protected]