LECTURES ON THE MULTIPLICATIVE ERGODIC
THEOREM
SIMION FILIP
Abstract. The Oseledets Multiplicative Ergodic theorem is a
basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as
well as subsequent generalizations. They are based on lectures at
summer schools in Brazil and France.
Contents
Introduction
1. Lecture 1: Oseledets Multiplicative Ergodic Theorem
1.1. Basic ergodic theory
1.2. The Oseledets theorem
1.3. Proof of the Oseledets theorem
1.4. Proof of Lemma 1.7
1.5. Proof of Lemma 1.9
2. Lecture 2: The Geometric version of the Oseledets theorem
2.1. Structure theory of Lie groups
2.2. Symmetric spaces
2.3. Symmetric spaces and the Oseledets theorem
2.4. Geometric form of the Oseledets theorem
3. Lecture 3: The noncommutative ergodic theorem
3.1. Horofunctions
3.2. Buseman functions
3.3. Noncommutative ergodic theorem
3.4. Proof of the Noncommutative Ergodic Theorem
3.5. An example.
References
Revised July 7, 2015 .
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Introduction
The Oseledets multiplicative ergodic theorem is a basic result with
applications throughout dynamical systems. It was first proved by
Oseledets [Ose68], with previous work on random multiplication of
matrices by Furstenberg and Furstenberg-Kesten [FK60]. In its basic
form, it describes the asymptotic behavior of a product of matrices,
sampled from a dynamical system.
The theorem was later interpreted in terms of symmetric spaces by
Kaimanovich [Kaı̆87], and then extended to CAT(0)-spaces by KarlssonMargulis [KM99]. It was subsequently generalized by Karlsson-Ledrappier [KL06] to include all proper metric spaces.
A classical treatment with many applications is in the lecture notes
of Ledrappier [Led84]. Some monographs which deal with this circle
of questions are the ones by Katok-Hasselblatt [KH95], Mañé [Mañ87],
Viana [Via14], and Zimmer [Zim84]. These lectures also borrow from
the notes of Karlsson [Kar].
Outline of Lectures. Lecture 1 contains a statement and proof of the
classical form of the Oseledets multiplicative ergodic theorem. Lecture
2 contains an introduction to the structure of Lie groups, symmetric
spaces, and Kaimanovich’s interpretation of the Oseledets in terms of
symmetric spaces. Finally, Lecture 3 discusses the generalization by
Karlsson-Ledrappier to general metric spaces.
1. Lecture 1: Oseledets Multiplicative Ergodic
Theorem
1.1. Basic ergodic theory
For a basic introduction to the notions below, one can consult the
monographs of Katok-Hasselblatt [KH95], Einsiedler-Ward [EW11], or
Walters [Wal75].
Notation. Let Ω be a separable, second-countable metric space, B its
Borel σ-algebra and µ a probability measure on Ω. Let T : Ω → Ω be a
measurable transformation, i.e. if A is measurable then T −1 (A) is also
measurable. Define the push-forward measure by
T∗ µ(A) := µ T −1 (A) where A ∈ B
The measure µ is T -invariant if T∗ µ = µ and in this case (Ω, B, µ, T ) is
called a probability measure-preserving system.
Ergodicity. By definition (Ω, B, µ, T ) is ergodic if the only measurable
T -invariant sets have either full or null measure. In other words, if
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T −1 (A) = A then µ(A) = 0 or 1. Equivalently, if Ω = A B is
a T -invariant decomposition into measurable sets, then µ(A) = 1 or
µ(B) = 1. Using the ergodic decomposition ([EW11, Ch. 4.2]) most
questions can be reduced to ergodic systems.
Birkhoff ergodic theorem. A starting point for many results is the
Birkhoff, or Pointwise Ergodic Theorem.
`
Theorem 1.1 (Birkhoff pointwise Ergodic Theorem) Assume (Ω, B, µ, T )
is an ergodic probability measure-preserving system and let f ∈ L1 (Ω, µ)
be an integrable function. Then for a.e. ω ∈ Ω we have
Z
−1
1 NX
i
lim
f (T ω) → f dµ
N →∞ N
Ω
i=0
The quantity on the right is the “space average” of f , while the
quantity on the left is the “time average”.
Remark 1.2 One can slightly relax the L1 -integrability assumption
above. Let f = f + + f − where f + such that everywhere on Ω we have
f + ≥ 0 and f − ≤ 0. If f + ∈ L1 (Ω, µ), then the same conclusion in
Theorem 1.1 holds, except that −∞ is allowed as a limit.
Variant. Instead of additive averages, one can also consider multiplicative ones. With the notation as in the theorem, set g := exp(f )
(equivalently, f = log g). Then for a.e. ω ∈ Ω we have
lim
N →∞
g(ω)g(T ω) · · · g(T
N −1
1/N
ω)
→ exp
Z
log gdµ
Ω
1.2. The Oseledets theorem
Example. Suppose that M is a smooth manifold and F : M → M is a
smooth diffeomorphism preserving a measure µ (for instance, a volume
form). The diffeomorphism induces a map on the tangent bundle of M ;
if p ∈ M is a point then
(1.1)
Dp F : Tp M → TF (p) M is a linear map.
In analogy with the Birkhoff ergodic theorem, one can inquire about
the asymptotic behavior of the N -fold composition
(1.2)
DF N −1 (p) F ◦ · · · ◦ Dp F : Tp M → TF N (p) M
An answer is given by the Oseledets Multiplicative Ergodic theorem.
Vector bundles. For (Ω, B, µ) consider vector bundles V → Ω. By
definition, this is a collection of sets Uα which cover Ω, as well as gluing
maps φα,β : Uα ∩ Uβ → GLn R satisfying the compatibility condition
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φγ,α ◦ φβ,γ ◦ φα,β = 1. The vector bundle V is then defined by gluing
the pieces Uα × Rn using the identifications:
!
(1.3)
V :=
a
Uα × R
α
n
/(x, v) ∼ (x, φα,β v) for x ∈ Uα ∩ Uβ
The sets Uα can be measurable or open. Similarly, the gluing maps can
be measurable, continuous, smooth, depending on the qualities of Ω.
Any vector bundle can be measurably trivialized, i.e. it is measurably
isomorphic to Ω × Rn → Ω. However, intrinsic notation will be more
useful in the sequel. For ω ∈ Ω using it as subscript, e.g. Vω , will
denote the fiber of V over ω.
Cocycles. Let T : Ω → Ω be a probability measure preserving transformation, and V → Ω a vector bundle. Then V is a cocycle over T
if the action of T lifts to V by linear transformations. In other words,
there are linear maps
Tω : Vω → VT ω
and these maps vary measurably, continuously, or smoothly with ω.
For simplicity, assume that Tω are always invertible.
Oseledets Multiplicative Ergodic Theorem. To motivate the next
result, consider a single matrix A acting on Rn , say with eigenvalues
λ1 > λ2 > · · · > λk (perhaps
with
multiplicities). For a vector v ∈ Rn ,
N consider the behavior of A v as N gets large. For typical v, this will
grow at rate λN
1 . But for v contained in the
span
of eigenvectors with
N eigenvalues λ2 or less, the growth rate of A v will be different. Repeating the analysis gives a filtration of Rn by subspaces with different
order of growth under iteration of A.
Remark 1.3 Throughout these notes, a metric on a vector space or
vector bundle will mean a symmetric, possitive-definite bilinear form.
In other words, it is a positive-definite inner product.
The next result if a generalization of the Birkhoff theorem to cocycles
over general vector bundles.
Theorem 1.4 (Oseledets Multiplicative Ergodic Theorem) Suppose
V → (Ω, B, µ, T ) is a cocycle over an ergodic probability measurepreserving system. Assume that V is equipped with a metric k−k on
each fiber such that
Z
(1.4)
Ω
+
log+ kTω kop dµ(ω) < ∞
Here log (x) := max(0, log x) and k−kop denotes the operator norm of
a linear map between normed vector spaces.
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Then there exist real numbers λ1 > λ2 > · · · > λk and T -invariant
subbundles of V defined for a.e. ω ∈ Ω:
0 ( V ≤λk ( · · · ( V ≤λ1 = V
such that for vectors v ∈ Vω≤λi \ Vω≤λi+1 we have
1
log T N v → λi
N →∞ N
lim
Remark 1.5
(i) The invariance of the subbundles V ≤λi means that Tω : Vω →
≤λ•
i
VT ω takes Vω≤λi → VT≤λ
will be called
ω . The filtration V
the forward Oseledets filtration. The numbers {λi } are called
Lyapunov exponents.
(ii) The multiplicity of an exponent λi is defined to be dim V ≤λi −
dim V ≤λi+1 . Later it will be convenient to list exponents repeating them with their appropriate multiplicity.
Variant. Suppose now that T : Ω → Ω is an invertible map, and the
cocycle on V for the map T −1 satisfies the same assumptions as in the
Oseledets Theorem 1.4. Applying the result to the inverse operator
gives Lyapunov exponents ηj and the backwards Oseledets filtration
V ≤ηj . By construction if v ∈ V ≤ηj \ V ≤ηj+1 then
1
log T −N v → ηj
N →∞ N
The only way for this to be compatible with the forward behavior of the
vectors is if ηj = −λk+1−j . Moreover, defining V λj := V ≤λj ∪ V ηk+1−j
gives a T -invariant direct sum decomposition
(1.5)
(1.6)
lim
V = V λ1 ⊕ · · · ⊕ V λk
The defining dynamical property of this decomposition is that
1
(1.7)
0 6= v ∈ V λi ⇔ lim
log T N v = λi
N →±∞ N
Note that when N goes to −∞, the sign in the
1
N
factor changes.
Remark 1.6 Suppose that the vector bundle V is 1-dimensional. The
Oseledets theorem is then equivalent to the Birkhoff theorem. To see
this, define
f (ω) := log
kTω vk
which is independent of the choice of 0 6= v ∈ Vω
kvk
The integrability condition on the cocycle from Eq. (1.4) is equivalent
to the integrability of f + := max(0, f ) ∈ L1 (Ω, µ). Then the Birkhoff
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theorem gives
(1.8)
N Z
T v 1 1
N −1
f (ω) + · · · + f (T
ω) =
log
→ f dµ
N
N
kvk
Ω
It follows that
(1.9)
Z
1
N log T v → f dµ = λ1
N
Ω
Conversely, given f ∈ L1 (Ω, µ), define T : Ω × R → Ω × R by
(1.10)
T (ω, v) = (T ω, exp(f (ω)) · v)
and use the standard norm on R to deduce the Birkhoff theorem.
1.3. Proof of the Oseledets theorem
The proof of Theorem 1.4 will involve two preliminary results. These
will be proved in separate sections below. The setup and notation is
from Theorem 1.4.
Lemma 1.7 At least one, but perhaps both, of the following possibilities
occurs.
(i) There exists λ ∈ R such that for a.e. ω ∈ Ω and for all v ∈ Vω
1
log T N v → λ
N →∞ N
(ii) There exists a nontrivial proper T -invariant subbundle E ( V
which is defined at a.e. ω.
In other words, either there is a nontrivial subbundle with 0 < dim E <
dim V , or vectors in V exhibit growth with just one Lyapunov exponent.
(1.11)
lim
Remark 1.8 Suppose V is a cocycle over Ω such that Eq. (1.4) holds.
Suppose further that E ⊂ V is an a.e. defined T -invariant subbundle.
Then the same boundedness condition (1.4) holds for the bundles E and
V /E, equipped with the natural norms.
Lemma 1.9 Consider a short exact sequence of cocycles over Ω
p
0→E→V →
− F →0
Assume there exist λE , λF ∈ R such that for a.e. ω ∈ Ω
1
log T N e → λE
N
1
∀f ∈ Fω we have
log T N f → λF
N
∀e ∈ Eω we have
(1.12)
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If λE > λF then there exists a splitting (i.e. a linear map)
(1.13)
σ : F → V such that V = E ⊕ σ(F ) and p ◦ σ|F = 1F
and this decomposition is T -invariant (recall that p : V → F is the
quotient map).
Remark 1.10
(i) The splitting constructed in Lemma 1.9 will be tempered, i.e.
σ(T N v)
1
lim
log
→0
N →∞ N
kT N vk
The Lyapunov exponent of F and σ(F ) will thus be the same.
(ii) In general, one expects that the maximal Lyapunov exponent on
a subbundle should be less than the one on the entire bundle.
What Lemma 1.9 says is that if this is not the case, then the
cocycle must be a direct sum.
Proof of Theorem 1.4. Using Lemma 1.7 iteratively gives a filtration
on V into subbundles
0 ( Vk ( · · · ( V1 = V
such that in each Vi /Vi+1 , the growth rate of every vector is given by
a single exponent λi . In principle, it could happen that λi < λi+1 for
some i. Applying Lemma 1.9 repeatedly gives another filtration, where
the exponents are sorted as required by the Oseledets theorem.
1.4. Proof of Lemma 1.7
Consider the bundle V → Ω and the associated projective space bundle
π
P(V ) →
− Ω. Since the transformation T acts on V by linear maps, its
action extends to P(V ) by projective-linear transformations.
Define the space of probability measures on P(V ) which project to
the measure µ on Ω:
(1.14)
M1 (P(V ), µ) := {λ prob. measure on P(V ) with π∗ λ = µ}
Since µ is T -invariant, the action of T on P(V ) naturally extends to an
action on M1 (P(V ), µ).
Exercise 1.11 Krylov-Bogoliubov. Let S : X → X be a homeomorphism of a compact metric space. Prove that the space of probability
measures on X is weak-* compact, and that it has an S-invariant measure.
Krylov-Bogoliubov in families. Prove that the space M1 (P(V ), µ)
is weak-* compact, and that it has at least one T -invariant measure.
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Define now the function f : P(V ) → R by
kT vk
f ([v]) := log
kvk
(1.15)
!
Integrating against f (−) gives a continuous function on the space of
measures:
Z
f : M1 (P(V ), µ) → R
(1.16)
λ 7→
Z
f dλ
P(V )
The set of T -invariant measures, denoted M1 (P(V ), µ)T is also compact, so this function achieves a minimum on it. Moreover, the set
of measures η on which the minimum is achieved is a closed convex
subset. Therefore, an extremal point of this convex set exists. This is
a measure η ∈ M1 (P(V ), µ) which is also ergodic for the T -action on
P(V ).
Applying the Birkhoff ergodic theorem to the function f and the
measure η, it follows that for η-a.e. [v] ∈ P(V )
(1.17)


N Z
T v 1 1
N −1
→
f ([v]) + · · · + f (T
[v]) =
log 
f dη =: λ
N
N
kvk
P(V )
In each fiber P(Vω ) ⊂ P(V ) consider the set M of vectors for which
the above limit holds. This is a T -invariant set, and for µ-a.e. ω this
set is non-empty. Moreover, the fiberwise span of the [v] ∈ M gives a
T -invariant subbundle E ⊂ V .
If this is a proper subbundle, then we are in case (ii) of Lemma 1.7.
Suppose therefore that E = V .
In this case, any vector in V can be written as a linear combination
of vectors with asymptotic norm growth rate λ. Thus, their asymptotic
growth rate is at most λ. In fact, by construction their asymptotic
growth rate has to be exactly λ. For this, consider the set of vectors
which don’t satisfy this property:
1
(1.18)
M := {[v] ∈ P(V )| lim sup log T N v ≤ λ − }
N
This is a T -invariant set, and the set of ω ∈ Ω for which Mω is not
empty has either full, or null measure. If it has null measure, the proof
is complete.
Otherwise, using the same technique as in Lemma 3.9 below, one
constructs
a T -invariant measure η 0 on P(V ), projecting to µ, for which
R
f dη 0 ≤ λ − . This is a contradiction.
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The proof of Lemma 1.7 shows that the filtration constructed will,
in fact, give the correct sequence of Lyapunov exponents. However,
Lemma 1.9 might be of intrinsic interest.
1.5. Proof of Lemma 1.9
The setup is a short exact sequence of vector bundles
p
0→E→V →
− F →0
(1.19)
Pick any lift σ0 : F → V such that V = E ⊕ σ0 (F ). For instance, since
∼
V has a metric, there is a natural identification F −
→ E⊥ ⊂ V .
The cocycle map T now takes the form
"
T
U
T = E
0 TF
(1.20)
#
with the following linear maps
TE,ω : Eω → ET ω
TF,ω : Fω → FT ω
(1.21)
Uω : Fω → ET ω
Any other possible lift σ : F → V differs from σ0 by a map τ : F →
E. Indeed, the difference τ = σ − σ0 is a map F → V which after
composition back with the projection p : V → F is the zero map.
Therefore τ has image in E, and can be regarded as a map τ : F → E.
The condition that σ = σ0 + τ is a splitting that diagonalizes T is
explicit.
Work in the decomposition V = E ⊕ σ0 (F ). Then a vector
" #
e
is in σ(F ) if and only if e = τ (f ). But applying T gives
f
"
(1.22)
#
"
#
τ (f )
T ◦ τ (f ) + Uω (f )
= E,ω ω
∈ VT ω
T
f
TF,ω (f )
The condition that this vector is in σT ω (F ) is that
(1.23)
TE,ω ◦ τω (f ) + Uω (f ) = τT ω ◦ TF,ω (f )
Note that this is an equality of maps from Fω to ET ω . The equation
can be equivalently rewritten
(1.24)
−1
−1
τω = TE,ω
◦ τT ω ◦ TF,ω − TE,ω
◦ Uω
A formal solution of this equation is given by
(1.25)
τω =
∞
X
n=0
−(n+1)
TE,ω
n
◦ UT n ω ◦ TF,ω
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which uses the linear operators
(1.26)
n
TF,ω
: Fω → FT n ω
UT n ω : FT n ω → ET n+1 ω
−(n+1)
TE,ω
: ET n+1 ω → Eω
However, recall the assumption λE > λF , or equivalently λF − λE < 0.
Each vector in E and F respectively grows exponentially at rate λE
and λF respectively.
Therefore, the formal sum defined above converges uniformly and
gives the desired linear map. Indeed, the operator norm of Uω is an
L1 function, and by (1.27) below it does not affect the discussion.
−(n+1)
n
The operator norm of TF,ω
is of size enλF while that of TE,ω
is of
−nλE
size e
. Therefore, the formal solution in (1.25) is bounded by a
convergent geometric series.
Remark 1.12
(i) Given f ∈ L1 (Ω, µ), from the Birkhoff theorem it follows that
for µ-a.e. ω
1
(1.27)
f (T N ω) → 0
N
(ii) The initial splitting σ0 was bounded, while the norm of τ is in
L1 (Ω, µ). Therefore, the splitting provided by σ does not change
Lyapunov exponents.
2. Lecture 2: The Geometric version of the
Oseledets theorem
The point of view on the Oseledets theorem developed in this section
goes back at least to Kaimanovich [Kaı̆87]. The work of Karlsson and
Margulis [KM99] provides a link between the point of view in this and
the next sections.
Standard operations of linear algebra. Starting with a collection
of vector bundles, standard linear algebra operations produce new ones.
This applies in particular to cocycles over dynamical systems.
Suppose that L and N are two cocycles over the map T : (Ω, µ) →
(Ω, µ), with Lyapunov exponents λ1 ≥ · · · λl and η1 ≥ · · · ≥ ηn . The
Lyapunov exponents are listed with multiplicities, and this will be the
convention from now on.
Here is a list of constructions and corresponding Lyapunov exponents:
L ⊗ N:
Exponents are {λi + ηj } with i = 1..l, j = 1..n.
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L∨ (dual): Exponents are {−λi } with i = 1..l.
Hom(L, N ): Exponents are {ηj − λi } with i = 1..l, j = 1..n.
Λk (L):
Exponents are {λi1 + · · · + λik } with i1 < · · · < ik .
The Oseledets filtrations of the new cocycles can also be explicitly
described in terms of those for the cocycles L and N .
Volume-preserving diffeomorphisms. Suppose that F : M → M
is a diffeomorphism of a manifold which preserves a measure given by
a volume form µ. Thus µ is a section of the top exterior power of the
tangent bundle of M , denoted Λn (T M ). Because µ is preserved by T
the Lyapunov exponent of Λn (T M ) is zero. On the other hand, by the
preceding discussion the exponent of Λn (T M ) is also the sum of the
Lyapunov exponents on the tangent bundle T M .
Symplectic cocycles. Suppose the rank 2g vector bundle V → Ω
carries a symplectic pairing denoted h−, −i which is preserved by the
linear maps Tω . Then the Lyapunov exponents of V have the symmetry
(2.1)
λ1 ≥ · · · ≥ λg ≥ −λg ≥ · · · ≥ −λ1
In fact, if V ≤λi denotes the forward Oseledets filtration, then
(2.2)
The symplecic orthogonal of V ≤λi+1 is V ≤−λi
For the symmetry of the Lyapunov spectrum, it suffices to note that
the symplectic form gives an isomorphism of the cocycle V and its dual
cocycle V ∨ . The exponents of V ∨ are the negatives of that for V , so
the claim follows.
The same construction gives the claim about filtrations. The isomorphism given by the symplectic form respects the Oseledets filtrations
on V and V ∨ . On the other hand, the Oseledets filtration on V ∨ can
be described in the general case as the dual, via annihilators, of the
Oseledets filtration on V .
Remark 2.1
(i) Consider a matrix in the symplectic group A ∈ Sp2 g(R). Suppose that eλ is an eigenvalue of A. Then e−λ is also an eigenvalue of A, and this can be seen by considering the symmetry of
the characteristic polynomial of A. This is a different explanation for the symmetry of the Lyapunov spectrum of symplectic
cocycles.
(ii) There are some other symmetries the cocycle might have. If
it preserves a symmetric bilinear form of signature (p, q) with
p ≥ q, then the spectrum has the form
(2.3)
λ1 ≥ · · · ≥ λq ≥ 0 · · · ≥ 0 ≥ −λq ≥ · · · ≥ −λ1
In particular, there are at least p − q zero exponents.
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(i) The Oseledets theorem holds for both real and complex vector
bundles (with hermitian metrics). The Lyapunov exponents are
real numbers in both cases.
2.1. Structure theory of Lie groups
Setup. Throughout this section G will be a real semisimple Lie group
with Lie algebra g. Fix a maximal compact subgroup K ⊂ G with Lie
algebra k. Then G can be equipped with a Cartan involution σ : G → G
whose action on g gives the decomposition into eigenspaces
(2.4)
g = k ⊕ p where σ|k = +1 and σ|p = −1
Recall also the Killing form on g given by hx, yi = tr(adx ◦ ady ). It is
negative-definite on k and positive-definite on p.
Example of SLn R. In this case
(2.5)
g = sln R = {a ∈ Matn×n R| tr a = 0}
The maximal compact subgroup is K = SOn R with Lie algebra son R.
The involution σ acts on SLn R by g 7→ (g −1 )t and on g by x 7→ −xt .
The subspace p on which σ acts by −1 is the space of symmetric
matrices:
(2.6)
p = {x ∈ sln R|x = xt }
Remark 2.2 In fact, any real semisimple Lie algebra g admits an
embedding g ,→ sln R such that the Cartan involution on g is that of
sln R restricted to g. In this case k = g ∩ son R and similarly for p.
Split maximal Cartan algebra and Polar Decomposition. Pick
a maximal abelian subalgebra a ⊂ g. By definition, any two elements
of a commute, and a is maximal with this property. This is called
a split Cartan subalgebra. It carries an action of a reflection group
whose description is omitted. The reflection hyperplanes divide a into
chambers; fix one such a+ ⊂ a, called a Weyl chamber. Using the
exponential map, a gives the Lie subgroup A ⊂ G; the chamber a+
determines a semigroup A+ ⊂ A.
The polar (or Iwasawa, or KAK) decomposition of an element g ∈ G
describes it as a product
(2.7)
g = k1 ak2 where ki ∈ K and a ∈ A+
In other words G = KAK. The decomposition of an element is typically,
though not always, unique.
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The example of SLn R. Recall first the spectral theorem. If M is a
symmetric n × n matrix, then it has a basis of orthogonal eigenvectors. Thus, M is conjugate to a diagonal matrix via an orthogonal
transformation:
(2.8)
M = kdk t where k ∈ SOn R and d is diagonal.
Because permutation matrices are orthogonal, one can assume the
entries of d are in increasing order.
Let now g ∈ SLn R and consider the elements gg t and g t g. Both of
them are symmetric, so the spectral theorem gives
gg t = k1 d1 k1t
g t g = k2 d2 k2t
√
One can check that d1 = d2 = d and then that g = k1 dk2 . One can
take the entries of d to be positive, at the expense of adjusting the signs
of k1 and k2 .
A maximal split Cartan subalgebra for sln R is given by diagonal
matrices
(2.9)
a = {a = diag(λ1 , . . . , λn )|λ1 + . . . + λn = 0}
A Weyl chamber is given by
(2.10)
a+ = {a = diag(λ1 , . . . , λn ) ∈ a|λ1 ≥ · · · ≥ λn }
2.2. Symmetric spaces
Setup. Consider the quotient X := G/K, equipped with a left Gaction. At the distinguished basepoint e = 1K, the tangent space is
canonically identified with p. Here are some properties of the quotient:
(i) It admits a canonical G-invariant metric, coming from the restriction of the Killing form to p.
(ii) The space X is diffeomorphic to p. The exponential map exp :
p → G → G/K exhibits the diffeomorphism.
(iii) The canonical metric has non-positive sectional curvature. At
the basepoint, in the direction of normalized vectors x, y ∈ p, it
involves the commutator of x and y and is given by
1
(2.11)
K(x, y) = − k[x, y]k2
2
In particular, if x and y commute, the curvature is 0. The
abelian subalgebra a determines, via the exponential map, a
totally geodesic embedding of a (with Euclidean metric) into
X.
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SIMION FILIP
The example of SL2 R. Recall that the hyperbolic unit disc can be
described as a quotient D = SL2 R/ SO2 R. The canonical symmetric
space metric will in this case be the hyperbolic metric.
Remark 2.3 The action of G on X is transitive, i.e. for any x, y ∈ X
there is a g ∈ G such that gx = y. This action also preserves distances.
In the hyperbolic plane D, for any x1 , x2 and y1 , y2 such that d(x1 , x2 ) =
d(y1 , y2 ), there is a transformation g ∈ G such that gxi = yi .
However, for a general X the action does not act transitively on pairs
of points at the same distance. The number of parameters that needs to
be “matched” is equal to the dimension of a. In the case of SL2 R, this
dimension is 1 and the parameter corresponds to distance.
Regularity. A sequence of points {xn } in X is regular if there exists
a geodesic γ : [0, ∞) → X and θ ≥ 0 such that
(2.12)
dist(xn , γ(θ · n)) = o(n)
In other words the quantity
1
n
dist(xn , γ(θ · n)) tends to zero.
Remark 2.4
(i) If the parameter θ is zero, the sequence satisfies d(xn , e) = o(n).
Recall that e ∈ X is the distinguished basepoint.
(ii) A sequence gn ∈ G is regular if the sequence of points {gn e} is
regular.
Geodesics in X. Given a point x ∈ X = G/K, using the decomposition G = KAK it follows that x = kx ax e, where kx ∈ K is to be
thought of as a “direction” and ax as a “distance”. All geodesics in X
starting at the basepoint e can be parametrized by
(2.13)
γ(t) = k exp(t · α)e where k ∈ K, α ∈ a+
The geodesic is unit speed if kαk2 = 1. Note that if α is on a wall
of the Weyl chamber (for sln R, this means some eigenvalues coincide)
then different k ∈ K can give the same geodesic. This is also the way
in which the KAK decomposition can fail to be unique.
Cartan projection. For x ∈ X let r(x) ∈ a+ denote the unique
element such that k exp(r(x))e = x for some k ∈ K. Although k might
not be unique, the element r(x) is. It can be viewed as a “generalized
radius” (in the literature, also called a Cartan projection). Its norm in
a is equal to the Riemannian distance from e to x.
Kaimanovich’s characterization of regularity. The regularity of
a sequence in the symmetric space X can be characterized by rather
simple conditions. The non-positive curvature assumption is crucial for
this description to hold.
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Theorem 2.5 (Kaimanovich [Kaı̆87]) A sequence of points {xn } in
X = G/K is regular if and only if the following two conditions are
satisfied:
Small steps:
dist(xn , xn+1 ) = o(n)
Distances converge: α := lim r(xnn ) exists in a+
Remark 2.6
(i) In R2 , this characterization of regular sequences fails. There
exists a sequence of points {xn } ⊂ R2 with dist(xn , xn+1 ) =
O(n), lim |xnn | exists, but the angles of xn to the origin don’t
converge. One can take the points to be on a logarithmic spiral.
(ii) It is instructive to verify Theorem 2.5 in the case when X is a
tree, and r(x) is simply the distance to a fixed basepoint.
Proof of Theorem 2.5. For simplicity, consider the case of the hyperbolic plane D = SL2 R/ SO2 R, equipped with the metric of constant
negative curvature. This case contains the main idea and is easier
notationally.
First, recall the Law of Sines in constant negative curvature. Consider
a triangle with angles of sizes α, β, γ and opposite sides of lengths a, b, c.
These quantities are related by
(2.14)
sin β
sin γ
sin α
=
=
sinh a
sinh b
sinh c
Therefore sin α = sin β ·
(2.15)
sinh a
.
sinh b
Recall that we have the estimates
dist(xn , xn+1 ) = o(n)
dist(e, xn ) = θn + o(x)
Assume that θ > 0, otherwise the claim follows directly.
Define the angle, when viewed from the origin, between successive
points: φn := ](xn exn+1 ). Define also the angle βn+1 := ](xn xn+1 e).
The law of sines then gives
(2.16)
sin φn = sin βn+1 ·
sinh(dist(xn , xn+1 ))
sinh(dist(e, xn+1 ))
1
Applying the bounds 10
|x| ≤ | sin x| (for |x| ≤ π/10) and | sin x| ≤ 1
to φn and βn+1 respectively, combined with the assumed bounds on
distance, yields
(2.17)
|φn | ≤ 10 ·
sinh(dist(xn , xn+1 ))
eo(n)
≤ θn+o(n)
sinh(dist(e, xn+1 ))
e
≤ e−θn+o(n)
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SIMION FILIP
P
Therefore, the total angle φ := n≥0 φn converges uniformly. The
geodesic launched at angle φ and speed θ will be o(n)-close to xn
(again, by an application of the law of sines).
Remark 2.7
(i) To generalize the above proof to all symmetric spaces X = G/K,
one needs two further ingredients. The first is a comparison
theorem for triangles in manifolds with all sectional curvatures
≤ κ ≤ 0. The second is a more detailed use of the structure
theory of Lie groups.
(ii) A general symmetric space X = G/K will contain geodesically
embedded copies of Euclidean spaces Rdim a , called flats. These
arise from taking the exponential map of maximal abelian subalgebras inside p. If the sequence of points {xn } is contained
in such a flat, the condition on “convergence of distance” from
Theorem 2.5 is both necessary and sufficient for the existence of
the geodesic.
2.3. Symmetric spaces and the Oseledets theorem
An introduction to the formalism used below is available in the book
of Zimmer [Zim84]. For this section we consider more general real
reductive Lie groups G. This class includes GLn R, not just SLn R, so
it allows factors such as R× .
For most considerations, including symmetric spaces and the Oseledets
theorem, everything can be reduced to semisimple groups. Indeed, for
a cocycle in GLn R, the maps can be rescaled to assume the cocycle
takes values in SLn R.
Setup. Recall that T : (Ω, µ) → (Ω, µ) is an ergodic probability measure preserving transformation and E → Ω is a vector bundle equipped
with a metric k−k. The extension of T to a cocycle on E is a collection
of linear maps between the fibers of E:
(2.18)
Tω : Eω → ET ω
If G is a reductive Lie group, what does it mean to say that the maps
Tω belong to this group? After all, the maps are between distinct vector
spaces.
Principal bundles. Suppose that E → Ω is a rank n vector bundle.
For each fiber Eω consider the set of isomorphisms to a fixed vector
space
(2.19)
Pω := Isom(Rn , Eω )
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Note that Pω carries a right action of GLn R by precomposing the
isomorphism. This action makes it isomorphic to GLn R, except that
it does not have a distinguished basepoint. The spaces Pω glue to give
a fiber bundle P → Ω.
More generally, if E has some extra structure, e.g. a symplectic form,
then Px can be the set of isomorphisms respecting the extra structure
on the source and target. It will be isomorphic to a subgroup of GLn R,
e.g. the symplectic group if E carries a symplectic form.
By definition, a principal G-bundle over Ω is a space P with a map
π
P →
− Ω and a right action of G on P such that π(pg) = π(p). Moreover,
each fiber Pω must be isomorphic to G with the right G-action.
Induced bundles. Let P → Ω be a principal G-bundle, and suppose
that G acts on another space F . The associated bundle over Ω with
fiber F is the quotient
(2.20)
P ×G F := {(p, f ) ∈ P × F } /(p, f ) ∼ (pg, g −1 f )
Cocycles on principal bundles. Suppose that Ω carries a T -action
π
and a principal G-bundle P →
− Ω. Then a cocycle T : P → P is a lift
of the T -action from Ω to P which commutes with the G-action of P
on the right.
If G acts on a space F , then the cocycle T on P extends to a natural
action of T on P ×G F . For instance, if G has a representation on
F = Rn , then P ×G Rn is a vector bundle with a linear cocycle.
Example 2.8 Suppose that G = GLn R and K = SOn R. Then X =
G/K is the space of metrics on Rn , with distinguished basepoint the
euclidean metric (corresponding to the coset eK). Call this the standard
metric k−kstd . Then for x = gK ∈ G/K the metric is given by kvkx =
kgvkstd .
Now, a vector bundle E → Ω gives rise to a principal G = GLn Rbundle P → Ω. Let K = SOn R and consider the bundle X := P ×G
K → Ω. A fiber Xω is the space of metrics on Eω . Thus, a metric on
π
E → Ω is the same as a choice of point in each fiber of X →
− Ω, i.e. a
map σ : Ω → X such that π ◦ σ(ω) = ω.
If E is a cocycle over T : Ω → Ω, the action of T extends to P and
X . Typically it will not preserve the metric on E but rather give an
action σ 7→ T σ on the space of metrics; for v ∈ Eω the new metric is
defined by the action of T :
(2.21)
kvkT σ(ω) := kT vkσ(T ω)
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SIMION FILIP
Proposition 2.9 Consider the sequence of vector spaces and linear
maps
(2.22)
T
T
Vω −
→ VT ω −
→ VT 2 ω → · · ·
Recall that each space VT • ω carries a metric k−k. Then the following
statements are equivalent:
(i) The Oseledets theorem holds for Eω , i.e. there exist numbers
λ1 > · · · λk and a filtration V ≤λi such that each v ∈ V ≤λi \V ≤λi+1
has the asymptotic growth
1
log T N v = λi
N →∞ N
(2.23)
lim
(ii) The sequence of metrics k−kN on Vω defined by kvkN := T N v is a regular sequence (see (2.12)) in the symmetric space GL(Vω )/ SO(Vω ).
Here SO(Vω ) is the group of orthogonal transformations preserving the initial metric on Vω .
(iii) There exists a liner map Λ : Vω → Vω which is symmetric and
self-adjoint (for the fixed metric on Vω ) and such that ∀v ∈ Vω
satisfies
(2.24)
D
E
| Λ−2n (T n )† T n v, v | = o(n)
Above (T n )† denotes the adjoint for the initial metrics on Vω
and VT n ω .
Proof. For the equivalence of (i) and (iii), assume first (i). Given the
≤λ
λ
Oseledets filtration Vω j , define Vω j as the orthogonal complement of
λ
≤λ
Vω j+1 inside Vω j . These spaces give a direct sum decomposition of Vω .
λ
Declare Λ to act as the scalar eλj on Vω j . Then Λ satifies (iii).
λ
≤λ
Conversely, given Λ, define Vω j using the eigenspaces of Λ and Vω j
λ
using the partial sums of Vω j . The asymptotic behavior guaranteed by
≤λ
(iii) shows that Vω j satisfies the properties of the Oseledets filtration.
Finally, the equivalence of (ii) and (iii) follows from the definition
of a regular sequence in (2.12) and the description of a geodesic in
(2.13).
Remark 2.10 Proposition 2.9 shows the Oseledets theorem for the
cocycle V → Ω is equivalent to a statement on the associated bundle of
symmetric space X := P ×G X where X = G/K and G = GLn R, K =
SOn R.
Namely, we have the linear maps T N : Vω → VT N ω and the initial
norms k−k on the corresponding spaces. The dynamics defines new
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norms kvkN := T N v on Vω and it suffices to show that for a.e. ω ∈ Ω,
this sequence of norms is regular in the symmetric space Xω .
However, Kaimanovich’s Theorem 2.5 gives a simple criterion to
check regularity of a sequence (see Theorem 2.11 below).
Singular values. Suppose that T : V → W is a linear map, and each
of V and W is equipped with a metric. The superscript (−)† denotes
the adjoint of an operator.
Consider the symmetric self-adjoint linear operator T † T : V → V .
By the spectral theorem, it can be diagonalized with eigenvalues
σ1 (T ) ≥ · · · ≥ σn (T )
(2.25)
These are called the singular values of T . The top singular value also
computes the operator norm of T :
(2.26)
σ1 (T ) = kT kop := sup
06=v∈V
kT vk
kvk
Moreover, one has the induced operators on exterior powers Λk T :
Λk V → Λk W . Then the singular values of Λk T and T can be related,
in particular the largest one is given by
(2.27)
σ1 (Λk T ) = σ1 (T ) · · · σk (T )
In particular, the operator norm of Λk T on Λk V gives the product of
the first k singular values.
Cartan projections. Suppose that T : Rn → Rn is a matrix in
G = GLn R, with maximal compact K = SOn R and split Cartan
subalgebra the diagonal matrices exp(a) = A ⊂ GLn R. Consider the
point t ∈ G/K obtained by applying T to the basepoint e ∈ G/K.
Then the Cartan projection r(t) ∈ a+ is the diagonal matrix with ith
entry log(σi (T )). Thus, convergence of the Cartan projection r(gn e)
is equivalent to the convergence of all singular values log(σi (gn )) for
matrices gn ∈ GLn R.
2.4. Geometric form of the Oseledets theorem
Theorem 2.11 Let E → Ω be a cocycle over an ergodic measurepreserving transformation T : (Ω, µ) → (Ω, µ). Suppose that E carries
a metric k−k such that (see (1.4))
(2.28)
Z
Ω
log+ kTω kop dµ(ω) < ∞
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Consider the associated symmetric space bundle X whose fiber over
ω ∈ Ω is the space of metrics
on Eω . Then the sequence of metrics
N defined by kvkN := T v is a regular sequence in Xω , for µ-a.e. ω ∈ Ω.
From Proposition 2.9, the above theorem is equivalent to the usual
form of the Oseledets Theorem 1.4. The proof of Theorem 2.11 will
make use of the Subadditive Ergodic Theorem, recalled below. A proof
is available in [Led84, Mañ87, Via14].
Theorem 2.12 (Kingman Subadditive Ergodic Theorem) Let T :
(Ω, µ) → (Ω, µ) be an ergodic probability measure-preserving transformation. Suppose that {fi } is a sequence of functions on Ω with
f1 ∈ L1 (Ω, µ) and satisfying the subadditivity condition
(2.29)
fi (ω) + fj (T i ω) ≥ fi+j (ω) ∀ω ∈ Ω,
Then for µ-a.e. ω the limit
1
f (ω)
N N
∀i, j ≥ 1
exists and can be computed as
1
1 Z
(2.30)
lim
fN (ω) = inf
fN (ω)dµ(ω)
N →∞ N
N N Ω
Note that integrating the subadditivity condition (2.29) it follows
that in the statement of the Subadditive Ergodic Theorem, each fi is
in L1 (Ω, µ).
Proof of Theorem 2.11. Define the sequence of functions
(2.31)
fn (ω) := log TωN op
The operator norms are computed for the initial norm k−k on Eω and
ET N ω .
Since for two linear maps kA ◦ Bkop ≤ kAkop · kBkop , the sequence
fi satisfies the subadditivity condition (2.29) in the Kingman Theorem.
Since f1 ∈ L1 (Ω, µ), Theorem 2.12 applies and so there exists λ1 such
that
1
(2.32)
lim log TωN = λ1
op
N
Finally, recall that the operator norm is the same as the first singular
value of Tω (see (2.26)). Thus the quantity N1 log(σ1 (TωN )) converges
µ-a.e.
Apply now the same construction to the exterior power bundles Λk V
to find that in fact all (normalized) singular values converge. Finally,
recall that the Cartan projection r(xN ) was given by considering the
singular values of the corresponding operator, and so N1 r(xN ) converge
(where xN denote the pull-back metrics after N steps).
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To check the small steps condition in Theorem 2.5, recall that if
f ∈ L1 (Ω, µ)by the Birkhoff Theorem 1.1 it follows that
1
f (T N ω) → 0
N
Using for f (ω) the quantity log+ kTω kop gives the desired bound.
(2.33)
for µ-a.e. ω ∈ Ω
3. Lecture 3: The noncommutative ergodic
theorem
This lecture follows closely the notes of Karlsson [Kar]. The main result
was proved by Karlsson-Ledrappier [KL06] and recently extended by
Gouëzel-Karlsson to semicontractions. An introduction to the geometry
of non-positively curved spaces is available in the monograph of BridsonHaefliger [BH99].
3.1. Horofunctions
Setup. Let (X, d) be a metric space which is proper, i.e. balls of
bounded radius are compact. Let C 0 (X) be the space of continuous
functions on X, with the sup-norm. Fix a basepoint x0 ∈ X. This
defines an embedding
(3.1)
Φ :X → C 0 (X)
x 7→ Φ(x) = d(x, −) − d(x, x0 )
To simplify notation, Φ(x) will also be denoted by hx .
Remark 3.1
(i) The map Φ is 1-Lipschitz, since
|Φ(x)(y) − Φ(x)(z)| = |d(x, y) − d(x, z)| ≤ d(y, z)
(ii) The map is normalized to have Φ(x0 ) ∼
= 0 and it is moreover
injective. Indeed, if d(x, x0 ) ≥ d(y, x0 ) then Φ(y)(x)−Φ(x)(x) ≥
d(x, y) > 0.
(3.2)
Define the metric bordification of X by taking its closure inside
C (X):
0
(3.3)
X := Φ(X) = X ∩ ∂X
If X is proper, then X is (sequentially) compact by Arzela-Ascoli.
Functions in ∂X are called horofunctions on X.
Example 3.2
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SIMION FILIP
(i) If X = R2 with Euclidean distance, then ∂R2 equals the linear
functions of norm 1.
(ii) If D2 ∼
= H2 is the hyperbolic plane, then the boundary is isomorphic to RP1 . In the upper halfplane model,
√ corresponding to
∞ ∈ ∂H is the function − log y (with x0 = −1 ∈ H).
Proposition 3.3 The action of the isometry group of X extends continuously to an action on X. Given an isometry g and a function
h ∈ X, the action is by the formula
g · h(z) := h(g −1 z) − h(g −1 x0 )
(3.4)
Proof. The action described in (3.4) is continuous on the space of functions, so it suffices to check its compatibility with the action on X.
Suppose that x ∈ X has corresponding function hx . We have the chain
of equalities
hgx (z) = d(gx, z) − d(gx, x0 )
= d(x, g −1 z) − d(x, x0 ) − (d(x, g −1 x0 ) − d(x, x0 ))
= hx (g −1 z) − hz (g −1 x0 )
This shows the compatibility of actions.
Remark 3.4 Up to homeomorphism, the space X is independent of
the choice of basepoint x0 ∈ X. Consider the quotient C 0 (X) →
C 0 (X)/{const.} of the space of continuous functions by the constant
ones. Then the image of Φ(X) under the quotient is independent of the
basepoint x0 , and is still an embedding.
Linear drift. Let {an } be a subadditive sequence, i.e. assume that
an+m ≤ an + am
(3.5)
Then the sequence
1
a
N N
∀n, m ≥ 1
has a limit, given by
1
1
an = inf an
N N
N N
Now, let f be a semicontraction of X, i.e. d(f x, f y) ≤ d(x, y) for all
x, y ∈ X. Consider the quantity d(x0 , f n x0 ) and apply the triangle
inequality with the semicontraction property to find
(3.6)
lim
d(x0 , f n+m x0 ) ≤ d(x0 , f n x0 ) + d(f n x0 , f n+m x0 )
≤ d(x0 , f n x0 ) + d(x0 , f m x0 )
Thus the limit N1 d(x0 , f N x0 ) exists, and is called the linear drift of the
semicontraction f .
In fact, the drift can be detected by a single horofunction.
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Proposition 3.5 (Karlsson) Let (X, d) be a proper metric space and
f a semicontraction. Then there exists h ∈ Φ(X) such that
h(f k x0 ) ≤ −l · k
−1
(3.8)
h(f N x) = l
∀x ∈ X lim
N N
Exercise 3.6 Suppose an+m ≤ an + am and lim N1 aN = l. Then there
exists a subsequence {mj } of 1, 2, 3, . . . such that for any further subsequence {ni } of {mj } we have
(3.7)
∀k ≥ 0
(3.9)
lim inf (ani −k − ani ) ≤ −l · k
ni
Proof of Proposition 3.5. Set an := d(x0 , f n x0 ) and pick the subsequence {mj } provided by Exercise 3.6. Since X is compact, pick a
further subsequence {ni } such that f ni x0 converge to an element h ∈ X.
Next, observe that −ak ≤ h(f k (x0 )) since from the definition of h
we have
(3.10) h(f k (x0 )) = lim d(f ni x0 , f k x0 ) − d(f ni x0 , x0 ) ≥ −d(x0 , f k x0 )
ni
Using the definition of h again, we have
h(f k (x0 )) = lim
d(f ni x0 , f k x0 ) − d(f ni x0 , x0 )
n
i
≤ lim inf (ani −k − ani )
ni
It follows that
(3.11)
−ak ≤ h(f k (x0 )) ≤ −l · k
and this gives the first part. Moreover, since k1 h(f k x0 ) → l and
d(f k (x0 ), f k (x)) stays bounded, it follows the limit (3.8) holds for all
x ∈ X.
3.2. Buseman functions
Let γ : [0, ∞) → X be a geodesic ray starting at x0 . Define
(3.12)
hγ (x) := lim d(γ(t), x) − t
t→∞
Equivalently, one can take the limit of Φ(γ(t)) in the compactification
X defined in Eq. (3.3). The function hγ obtained this way is called a
Buseman function.
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SIMION FILIP
CAT(0) spaces. Let (X, d) be a geodesic metric space. It is called a
CAT(0) space if the following holds for all triples of points x, y, z ∈ X.
Connect the points x, y, z by geodesics, and pick two points p, q on
distinct geodesics. In R2 , there is a unique up to isometry triangle with
vertices x0 , y 0 , z 0 which has the same side lengths as the triangle formed
by x, y, z. There is a unique map preserving lengths between the sides
of x, y, z and x0 , y 0 , z 0 and let p0 , q 0 be the images of p, q. Then we must
have
(3.13)
dX (p, q) ≤ dR2 (p0 , q 0 )
For CAT(0)-spaces, any point boundary of the metric bordification
∂X ⊂ X comes from a Buseman function hγ for some geodesic ray γ.
Gromov-hyperbolic spaces. For a geodesic metric space (X, d) denote by [a, b] the geodesic between any two points a, b ∈ X. Then X is
δ-hyperbolic if its triangles are δ-thin, i.e. for any x, y, z ∈ X we have
(3.14)
[x, y] ⊂ Nδ ([x, z] ∪ [z, y])
where Nδ (−) is the δ-neighborhood of a set.
For example, trees are δ-hyperbolic. In fact, they are 0-hyperbolic
(where the 0-neighborhood of a set is the set itself).
The unit disc D with the constant negative curvature metric is also
δ-hyperbolic, for some δ < ∞. Moreover, it is a CAT(0)-space.
However, unlike the CAT(0)-property which holds at all scales, and
in particular locally, δ-hyperbolicity is coarse and does not restrict the
local geometry of a space. For example, a connected graph whose loops
have uniformly bounded length is also δ-hyperbolic.
The metric (or horocycle) bordification ∂hor X of a δ-hyperbolic space
is in general too large. A more natural boundary is formed by quasiisometry classes of rays ∂ray X and there is a natural map ∂hor X →
∂ray X. Moreover, if two different h, h0 ∈ ∂hor X have the same image in
the ray compactification, then h−h0 is bounded by a constant uniformly
on X.
Buseman functions for SLn R/ SOn R. Consider the space p := {M ∈
Matn×n (R)|M = M t , tr M = 0}. Recall we also have the symmetric
space
(3.15)
X = SLn R/ SOn R ∼
= Pn (R) := {M ∈ Matn×n (R)|M = M t , det M = 1}
The exponential map takes p diffeomorphically to Pn (R). Note that
SLn R acts on Pn (R) by g · M = gM g t .
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Pick now α ∈ p; after a conjugation by an orthogonal matrix, we can
assume that α is diagonal, with eigenvalues occurring with multiplicity
(n1 , · · · , nk ).
Using the partition (n1 , · · · , nk ) of n define
(3.16)

Pn1 (R)
0

Pn2 (R)
 0
F (α) = 
..
..


.
.
0
···
(3.17)

1n1 (R)
∗

1n2 (R)
 0
Nα = 
..
..


.
.
0
···

···
0
···
0 

.. 
...
 a product of symmetric spaces
. 
0 Pnk (R)
∗
∗
...
0
∗
∗
∗
1nk (R)






a unipotent subgroup
This gives a set of coordinates via the isomorphism
(3.18)
Pn (R) ∼
= Nα · F (α)
The Buseman function corresponding to α is then
(3.19)
hexp(α·t) (n · f ) := − tr(α · log f )
Note that this function is constant on Nα -orbits.
3.3. Noncommutative ergodic theorem
The next theorem was proved by Karlsson-Ledrappier [KL06].
Theorem 3.7 Assume that T : (Ω, µ) → (Ω, µ) is a probability measurepreserving ergodic system. Let (X, d) be a metric space with chosen
basepoint x0 ∈ X and isometry group Isom(X, d).
Let Ω → Isom(X, d) be a measurable map, and denote by gω the
isometry corresponding to ω ∈ Ω. Assume that we have the L1 -bound
(3.20)
Z
Ω
d(gω x0 , x0 )dµ(ω) < ∞
Then we have a linear drift defined by
1 Z
(3.21)
l := inf
d(gω · · · gT N −1 ω x0 , x0 )dµ(ω)
N N Ω
In addition, for µ-a.e. ω we have
1
(3.22)
lim d(gω · · · gT N −1 ω x0 , x0 ) = l
N N
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SIMION FILIP
Moreover, there exists a measurable map
Ω→X
ω 7→ hω
(3.23)
such that for µ-a.e. ω we have
−1
hω (gω · · · gT N −1 ω x0 , x0 ) = l
N →∞ N
Remark 3.8 In general the map ω 7→ hω need not be equivariant, i.e.
hT ω need not equal gω · hω . However, in many circumstances, e.g. when
X is CAT(0) or δ-hyperbolic, this equivariance property can be arranged
in the appropriate boundary.
(3.24)
lim
3.4. Proof of the Noncommutative Ergodic Theorem
The following lemma is quite general and explains the mechanism in
the proof of Theorem 3.7.
Lemma 3.9
(i) Suppose X is a compact metric space equipped with a continuous
map T : X → X. Let f : X → R be a continuous function and
define
(3.25)
fn (x) := f (x) + · · · + f (T n−1 x) and the quantity an := sup fn (x)
x∈X
Then an+m ≤ an + am and so limN N1 aN = l exists.R Moreover,
there exists a T -invariant measure µ on X such that X f dµ ≥ l.
(ii) Suppose X is a compact metric space and T : (Ω, µ) → (Ω, µ)
is an ergodic probability measure preserving system. Let S :
Ω × X → Ω × X be of the form S(ω, x) := (T ω, gω x) where gω
is a homeomorphism of X for µ-a.e. ω. Let also f : Ω × X → R
be a function such that f (ω, −) : X → R is continuous for µ-a.e.
ω.
Define the Birkhoff averages
(3.26)
fn (ω, x) := f (ω, x) + f (S(ω, x)) + · · · + f (S n−1 (ω, x))
and their fiberwise supremum Fn (ω) := supx∈X fn (ω, x).
Then Fn+m (ω) ≤ Fn (ω) + Fm (T n ω), so the sequence satisfies the assumptions of the Kingman Theorem 2.12. Therefore
1
F (ω) tends to a limit l for µ-a.e. ω.
N N
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27
Then there exists an S-invariant probability measure η on
Ω × X, with projection to η equal to µ, and such that
Z
(3.27)
F dη ≥ l
Ω×X
Proof. The proof of (ii) is similar to that of (i), so we focus on the
latter.
Since X is compact, the continuous function fn achieves the supremum at some point xn . Define a probability measure using Dirac
delta-functions on the trajectory of xn by
(3.28)
µn :=
1
(δx + · · · + δT n−1 xn )
n n
Then by construction f dµn = n1 supx fn (x) = n1 an ≥ l.
Let µ be a weak-* limit of the µn . Then Rµ is T -invariant (by a KrylovBogoliubov type argument) and satisfies f dµ ≥ l by construction.
For the proof of (ii), one needs to select maximizers of fn in each
fiber of Ω × X → Ω (see also the end of proof of Theorem 3.7). Namely,
one constructs a map σn : Ω → X such that Fn (ω) = fn (σn (ω)). Then
the measure ηn is defined as the average along a length n S-orbit of the
measure (σn )∗ µ and the proof proceeds as before.
R
Translation distance and horocycles. For an isometry g of X, the
distance by which it moves x0 can be measured by
(3.29)
d(x0 , gx0 ) = max −h(g −1 x0 )
h∈X
Indeed, assume that h comes from a point z ∈ X. Then
(3.30)
−hz (g −1 x0 ) = d(z, x0 ) − d(z, g −1 x0 )
≤ d(x0 , g −1 x0 ) = d(x0 , gx0 )
This inequality persists in the closure X of X. Taking z = g −1 x0
achieves equality.
A cocycle function. Define therefore the function
(3.31)
F : Isom(X) × X → R
F (g, h) := −h(g −1 x0 )
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SIMION FILIP
If the function h comes from a point z ∈ X, i.e. h = hz , then the
following cocycle property holds:
(3.32)
F (g1 g2 , hz ) = − d(z, g2−1 g1−1 x0 ) − d(z, x0 )
= − d(g2 z, g1−1 x0 ) − d(g2 z, x0 ) + d(g2 z, x0 ) − d(z, x0 )
= F (g1 , g2 hz ) + F (g2 , hz )
By continuity of the action of isometries, this property extends to all
h ∈ X:
(3.33)
F (g1 g2 , h) = F (g1 , g2 h) + F (g2 , h)
The total space of the cocycle. Consider the space Ω × X → Ω.
Extend the action of T on Ω to a map S on Ω × X by S(ω, h) :=
(T ω, gω−1 h). Define now the function
f1 : Ω × X → R
(3.34)
f1 (ω, h) := F (gω−1 , h)
The Birkhoff sums for f1 and the transformation S, using the cocycle
property of F , can be expressed as:
(3.35)
fn (ω, h) :=
n−1
X
f1 (S i (ω, h))
i=0
= F (gω−1 , h) + F (gT−1ω , gω−1 h) + · · · + F (gT−1n−1 ω , gT−1n−2 ω · · · gω−1 h)
= F (gT−1n−1 ω · · · gω−1 , h)
By the definition of F , this gives
(3.36)
fn (ω, h) = −h(gω · · · gT n−1 ω x0 )
Existence of the drift. Define the functions
(3.37)
Fn (ω) := d(gT ω · · · gT n ω x0 , x0 )
Then by the triangle inequality Fn+m (ω) ≤ Fn (ω)+Fm (T n ω). Therefore
the drift l defined in (3.21) exists by subadditivity of the integrals of
Fn . The limit (3.22) exists µ-a.e. and equals l by the Kingman ergodic
theorem.
LECTURES ON MET
29
Existence of a maximizing measure. Note that by Eq. (3.29) the
functions Fn also satisfy
(3.38)
Fn (ω) = max fn (ω, x)
x∈X
Thus, by Lemma 3.9 there exists a measure η on Ω × X such that
(3.39)
Z
Ω×X
f1 dη ≥ l
Note however than since Fn (ω) ≥ supx fn (ω, x), it follows that for any
T -invariant measure the reverse inequality also holds:
(3.40)
Z
Ω×X
f1 dη ≤ l
Existence of hω . Let now M be the set of all (ω, x) for which the
Birkhoff theorem holds for the measure η and the function f1 . Then M
is of η-full measure, so over µ-a.e. ω the set M is not empty. Using a
measurable selection theorem (see e.g. [Kar, Sec. 3.8]) there is a map
ω 7→ hω ∈ M and by construction it satisfies the required properties.
3.5. An example.
This discussion is from the lecture notes [Kar]. See also the book of
Aaronson [Aar97, Prop. 2.3.1].
Consider a function D : R≥0 → R≥0 which is
• Increasing, and D(0) = 0
• D(t) → ∞ as t → ∞
• D(t)
→ 0 monotonically as t → ∞
t
This implies that
(3.41)
1
1
D(t + s) ≤ D(t)
t+s
t
Assuming that t ≤ s this gives
(3.42)
s
D(t)/t
D(s) ≤ D(t) + D(s)
D(t + s) ≤ D(t) + D(t) = D(t) +
t
D(s)/s
Therefore (R, D(−)) is a proper metric space. One can check that the
only point in the metric bordification is in this case the zero function, i.e.
X = X ∪ {h ≡ 0}. The isometry group is still R, acting by translations.
So a function f : Ω → R can be viewed as a map to the isometry group.
30
SIMION FILIP
Corollary 3.10 Assume that T : (Ω, µ) → (Ω, µ) is a probability
measure-preserving transformation, and f : Ω → R is D-integrable,
i.e.
Z
(3.43)
D(|f |)dµ < ∞
Ω
Then we have
1 D |f (ω) + · · · + f (T N −1 ω)| = 0
N
Example 3.11 (Marcinkiewic-Zygmund) Take D(t) := tp with 0 <
p < 1. Assume that f ∈ Lp (Ω, µ) (note that typically f ∈
/ L1 (Ω, µ)).
Then it follows that
(3.44)
(3.45)
lim
p
1 f (ω) + · · · + f (T N −1 ω) → 0
N
or equivalently
(3.46)
1 N −1
f
(ω)
+
·
·
·
+
f
(T
ω)
→0
N 1/p
Acknowledgements. I am grateful to the organizers of the summer
schools in Ilhabela, Brazil (January 2015, “Holomorphic Dynamics
School”) and Luminy, France (July 2015, “Translation surfaces School”).
These notes were strongly influenced by the point of view in the works
of Kaimanovich, Karlsson, Ledrappier, Margulis, Zimmer. I am also
grateful for remarks from Alex Eskin, Pascal Hubert, Misha Kapovich,
Zemer Kosloff, Erwan Lanneau, Carlos Matheus, Martin Möller, Nessim
Sibony, Misha Verbitsky, Anton Zorich.
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Department of Mathematics
University of Chicago
Chicago IL, 60615
E-mail address: [email protected]