Lyapunov exponents of
Teichmüller flows
Marcelo Viana
IMPA - Rio de Janeiro
Lyapunov exponents ofTeichmüller flows – p. 1/6
Lecture # 1
Geodesic flows on translation surfaces
Lyapunov exponents ofTeichmüller flows – p. 2/6
on a
Abelian differential = holomorphic -form
(compact) Riemann surface.
Abelian differentials
Lyapunov exponents ofTeichmüller flows – p. 3/6
on a
Adapted local coordinates:
Abelian differential = holomorphic -form
(compact) Riemann surface.
Abelian differentials
near a zero with multiplicity
:
then
then
Lyapunov exponents ofTeichmüller flows – p. 3/6
on a
Adapted local coordinates:
Abelian differential = holomorphic -form
(compact) Riemann surface.
Abelian differentials
near a zero with multiplicity
:
then
then
Adapted coordinates form a translation atlas: coordinate
changes near any regular point have the form
Lyapunov exponents ofTeichmüller flows – p. 3/6
Translation surfaces
The translation atlas defines
a flat metric with a finite number of conical singularities:
PSfrag replacements
a parallel unit vector field (the “upward” direction) on the
complement of the singularities.
Conversely, the flat metric and the parallel unit vector field
characterize the translation structure completely.
Lyapunov exponents ofTeichmüller flows – p. 4/6
Geometric representation
Consider any planar polygon with even number of sides,
organized in pairs of parallel sides with the same length.
S
E
W
N
PSfrag replacements
Identifying sides in the same pair, by translation, yields a
translation surface. Singularities arise from the vertices.
Every translation surface can be represented in this way,
but not uniquely.
Lyapunov exponents ofTeichmüller flows – p. 5/6
Moduli spaces
of Riemann
a fiber bundle over the moduli space
surfaces of genus
a complex orbifold of dimension
moduli space of Abelian differentials on Riemann
surfaces of genus
Lyapunov exponents ofTeichmüller flows – p. 6/6
Moduli spaces
moduli space of Abelian differentials on Riemann
surfaces of genus
a complex orbifold of dimension
of Riemann
complex orbifold with
stratum of Abelian differentials having
zeroes, with multiplicities
a fiber bundle over the moduli space
surfaces of genus
Lyapunov exponents ofTeichmüller flows – p. 6/6
on the fiber
:
The Teichmüller flow is the natural action
by the diagonal subgroup of
bundle
Teichmüller geodesic flow
Geometrically:
The area of the surface
rag replacements
is constant on orbits of this flow.
Lyapunov exponents ofTeichmüller flows – p. 7/6
Ergodicity
Masur, Veech:
Each stratum of
carries a canonical volume measure,
which is finite and invariant under the Teichmüller flow.
The Teichmüller flow
is ergodic on every connected
component of every stratum, restricted to any hypersurface
of constant area.
Kontsevich, Zorich:
Classified the connected components of all strata. Each
stratum has at most connected components.
Eskin, Okounkov, Pandharipande
Computed the volumes of all the strata.
Lyapunov exponents ofTeichmüller flows – p. 8/6
Lyapunov exponents
Theorem (Avila, Viana).
Fix a connected component of a stratum. The Lyapunov
spectrum of the Teichmüller flow has the form
.
exponents
(exactly
(flow is non-uniformly hyperbolic)
Forni proved
Veech proved
Conjectured by Zorich, Kontsevich.
)
Before discussing the proof, let us describe an application.
Lyapunov exponents ofTeichmüller flows – p. 9/6
Asymptotic cycles
in a given direction,
:
Given
any long geodesic segment
PSfrag
replacements
of
“close” it to get an element
Kerckhoff, Masur, Smillie: The geodesic flow in almost
every direction is uniquely ergodic.
converges uniformly to some
Then
when the length
, where the asymptotic cycle
depends only on the surface and the direction.
Lyapunov exponents ofTeichmüller flows – p. 10/6
, such that
for all
is bounded
(
from
genus of
).
#
the deviation of
"
!
has amplitude
from
the deviation of
with
of
Corollary (Zorich). There are subspaces
Asymptotic flag in homology
PSfrag replacements
Lyapunov exponents ofTeichmüller flows – p. 11/6
Zippered rectangles
Almost every translation surface may be represented in the
form of zippered rectangles (minimal number of rectangles
):
required is
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Lyapunov exponents ofTeichmüller flows – p. 12/6
in the stratum:
describes the combinatorics of
This defines local coordinates
Coordinates in the stratum
the associated interval exchange map
are the horizontal coordinates of
the saddle-connections (= widths of the rectangles)
of the rectangles are
The heights
linear functions of
are the vertical components of the
saddle-connections
has complex dimension
Lyapunov exponents ofTeichmüller flows – p. 13/6
Poincaré return map - 1st step
We consider the return map of the Teichmüller flow to the
.
cross section
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Lyapunov exponents ofTeichmüller flows – p. 14/6
Poincaré return map - 1st step
We consider the return map of the Teichmüller flow to the
.
cross section
PSfrag replacements
In this example: the top rectangle is the widest of the two.
Lyapunov exponents ofTeichmüller flows – p. 15/6
Poincaré return map - 1st step
We consider the return map of the Teichmüller flow to the
.
cross section
PSfrag replacements
Lyapunov exponents ofTeichmüller flows – p. 16/6
, with
is the linear operator
and analogously for .
Write
where
defined in this way. Then
except that
same for
except that
(top case)
This 1st step corresponds to
Rauzy-Veech induction
Lyapunov exponents ofTeichmüller flows – p. 17/6
Poincaré return map - 2nd step
We consider the return map of the Teichmüller flow to the
.
cross section
PSfrag replacements
Lyapunov exponents ofTeichmüller flows – p. 18/6
Poincaré return map - 2nd step
We consider the return map of the Teichmüller flow to the
.
cross section
PSfrag replacements
Lyapunov exponents ofTeichmüller flows – p. 19/6
,
The Poincaré return map
(invertible) Rauzy-Veech renormalization.
is the normalizing factor.
where
and
This 2nd step corresponds to
with
Rauzy-Veech renormalization
is called
Lyapunov exponents ofTeichmüller flows – p. 20/6
Rauzy-Veech cocycle and long geodesics
defined by
Consider the linear cocycle over the map
Lyapunov exponents ofTeichmüller flows – p. 21/6
Rauzy-Veech cocycle and long geodesics
PSfrag replacements
defined by
Consider the linear cocycle over the map
Consider a geodesic segment crossing each rectangle
vertically. “Close” it by joining the endpoints to some fixed
point in the cross-section. This defines some
.
Lyapunov exponents ofTeichmüller flows – p. 21/6
Rauzy-Veech cocycle and long geodesics
PSfrag replacements
defined by
Consider the linear cocycle over the map
Lyapunov exponents ofTeichmüller flows – p. 22/6
Rauzy-Veech cocycle and long geodesics
.
In other words,
except that
PSfrag replacements
defined by
Consider the linear cocycle over the map
“builds” long geodesics.
Lyapunov exponents ofTeichmüller flows – p. 22/6
Lyapunov exponents and asymptotic flag
So, it seems likely that
the behavior of long geodesics be described by the
asymptotics of the Rauzy-Veech cocycle, the flag
being defined by the Oseledets
subspaces of
with positive Lyapunov exponents.
the Lyapunov exponents of
(for an appropriate
invariant measure) be related to the Lyapunov
exponents of the Teichmüller flow.
has positive
This suggests we should try to prove that
Lyapunov exponents and they are all distinct. There is one
technical difficulty, however:
Lyapunov exponents ofTeichmüller flows – p. 23/6
Zorich cocycle
The Rauzy-Veech renormalization does have a natural
invariant measure (on each stratum) related to the invariant
volume of the Teichmüller flow. But this measure is infinite.
Lyapunov exponents ofTeichmüller flows – p. 24/6
Zorich cocycle
The Rauzy-Veech renormalization does have a natural
invariant measure (on each stratum) related to the invariant
volume of the Teichmüller flow. But this measure is infinite.
and
Zorich introduced an accelerated renormalization and an
accelerated cocycle
where
is smallest such that the Rauzy iteration
changes from “top” to “bottom” or vice-versa.
This map has a natural invariant volume probability (on
each stratum) and this probability is ergodic.
Lyapunov exponents ofTeichmüller flows – p. 24/6
has the
The Lyapunov spectrum of the Zorich cocycle
form
Zorich cocycle
and it is related to the spectrum of the Teichmüller flow by
Lyapunov exponents ofTeichmüller flows – p. 25/6
has the
The Lyapunov spectrum of the Zorich cocycle
form
Zorich cocycle
Forni: proved
Theorem (Avila, Viana).
and it is related to the spectrum of the Teichmüller flow by
.
(cocycle is non-uniformly hyperbolic)
Lyapunov exponents ofTeichmüller flows – p. 25/6
Lecture # 2
Simplicity criterium for Lyapunov spectra
Lyapunov exponents ofTeichmüller flows – p. 26/6
Linear cocycles
is an extension
. Note
.
be an -invariant ergodic probability on
such that
. Then the Oseledets theorem applies.
Let
where
with
A linear cocycle over a map
We say has simple Lyapunov spectrum if all subspaces in
the Oseledets decomposition are -dimensional.
Lyapunov exponents ofTeichmüller flows – p. 27/6
Hypotheses
, where the
is the shift map on
alphabet is at most countable
Assume that
for all
for all
for all
the cylinder of sequences
.
Denote by
such that
Denote
Lyapunov exponents ofTeichmüller flows – p. 28/6
Hypotheses
Assume that
.
and every
for every
the ergodic probability has bounded distortion: is
such that
positive on cylinders and there exists
Lyapunov exponents ofTeichmüller flows – p. 29/6
Hypotheses
Assume that
is locally constant:
.
and every
for every
the ergodic probability has bounded distortion: is
such that
positive on cylinders and there exists
More generally: is continuous and admits stable and
unstable holonomies.
Lyapunov exponents ofTeichmüller flows – p. 29/6
be the monoid generated by the
. We call the cocycle
,
for every
.
(pinching) if for every
there exists some
such that the -eccentricity
Let
Pinching
acements
1
Lyapunov exponents ofTeichmüller flows – p. 30/6
be the monoid generated by the
. We call the cocycle
,
and any finite family
there exists some
for all
.
such that
twisting if for any
Let
Twisting
Lyapunov exponents ofTeichmüller flows – p. 31/6
be the monoid generated by the
. We call the cocycle
,
and any finite family
there exists some
for all
.
is simple if and only if
Lemma.
is both pinching and twisting.
simple if
such that
twisting if for any
Let
Twisting
is simple.
Lyapunov exponents ofTeichmüller flows – p. 31/6
Criterium for simplicity
Theorem 1 (Bonatti-V, Avila-V). If the cocycle
Lyapunov spectrum is simple.
is simple then its
Guivarc’h, Raugi obtained a simplicity condition for products
of independent random matrices, and this was improved by
Gol’dsheid, Margulis.
Later we shall apply this criterium to the Zorich cocycles.
Right now let us describe some main ideas in the proof.
Lyapunov exponents ofTeichmüller flows – p. 32/6
Comments on the strategy
If the first exponents are strictly larger, the corresponding
is an “attractor” for the action
subbundle
:
of on the Grassmannian bundle
is constant on local unstable sets
Notice that
PSfrag replacements
.
Lyapunov exponents ofTeichmüller flows – p. 33/6
is simple. Then there
such that
Proposition 1. Fix . Assume the cocycle
exists a measurable section
of the most
, we have that
3. for any
replacements
at -almost every point.
and the image
expanded -subspace converges to
2.
is constant on local unstable sets and -invariant, that is,
at -almost every point.
1.
is transverse to
PSfrag
Invariant section
Lyapunov exponents ofTeichmüller flows – p. 34/6
Proof of Theorem 1
Recall
is simple if and only if
Applying the proposition to the inverse cocycle and to
, we find another invariant section
the dimension
with similar properties for
.
is simple.
Lyapunov exponents ofTeichmüller flows – p. 35/6
Proof of Theorem 1
is simple if and only if
is simple.
Recall
Applying the proposition to the inverse cocycle and to
, we find another invariant section
the dimension
with similar properties for
.
Keeping in mind that
is constant on unstable sets
is constant on stable sets, part 3 implies that
and
at -almost every point.
Lyapunov exponents ofTeichmüller flows – p. 35/6
Proof of
.
such that
.
for all
and
. This contradicts part 3.
. Then
Define
is not transverse to on
By local product structure, there exist points
has positive -measure, where
with
Suppose the claim fails on a set
PSfrag replacements
Lyapunov exponents ofTeichmüller flows – p. 36/6
Proof of Theorem 1
From part 2 we get that the Lyapunov exponents of
are strictly larger than the ones of
.
g replacements
Using that
is close to
for large , one finds an
for the cocycle induced by
invariant cone field around
on some convenient subset (with large return times).
Hence, there are Lyapunov exponents (for either cocycle)
which are larger than the other ones. Theorem 1 follows.
Lyapunov exponents ofTeichmüller flows – p. 37/6
0 2
6 7
5
5
7
5
5
45
< ;
$
7
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# &
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-
,+*
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and
45
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4 8 ;1
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4 982 1
Sfrag replacements
#
$
.
A probability on
down to and there is
/
/0
for every
!
"
-states
is a -state if it projects
such that
.
Lyapunov exponents ofTeichmüller flows – p. 38/6
-states
is a -state iff it may be disintegrated as
-
, with
where
is equivalent to
whenever
derivative uniformly bounded by .
+ *
on
.
Equivalently,
Lyapunov exponents ofTeichmüller flows – p. 39/6
-states
is a -state iff it may be disintegrated as
, with
for any
-states always exist: for instance,
probability in the Grassmannian.
where
is equivalent to
whenever
derivative uniformly bounded by .
+ *
-
on
.
Equivalently,
.
-
+ *
-
+ *
.
any Cesaro limit of iterates of a -state is invariant
under the cocycle
.
Lyapunov exponents ofTeichmüller flows – p. 39/6
Proof of Proposition 1
be its projection to
-
+ *
.
be an invariant -state and
Proposition 2. Let
. Then
converge almost
.
0
PSfrag replacements
the push-forwards
of under
surely to a Dirac measure at some point
the support of is not contained in any hyperplane section of the
Grassmannian
Lyapunov exponents ofTeichmüller flows – p. 40/6
Proof of Proposition 1
.
.
Proposition 1 follows from Proposition 2 together with
-
.
converge to a Dirac measure
then the
-
If the push-forwards
+ *
-
+ *
.
,+*
.
Lemma. Let
be a sequence of linear isomorphisms and
be a probability measure on
which is not supported in a
hyperplane section
of the most
-
and the images
eccentricity
expanded -subspace converge to .
Lyapunov exponents ofTeichmüller flows – p. 41/6
Part 1:
Proof of Proposition 2
is not supported in any hyperplane section.
Part 2: the sequence
This follows from the twisting hypothesis.
converges to a Dirac measure.
The proof has a few steps. The most delicate is to show
that some subsequence converges to a Dirac measure:
Lyapunov exponents ofTeichmüller flows – p. 42/6
such that
For almost every there exist
converges to a Dirac measure.
Proof of Proposition 2
Lyapunov exponents ofTeichmüller flows – p. 43/6
such that
For almost every there exist
converges to a Dirac measure.
Proof of Proposition 2
'
#
/
0
/0
Sfrag replacements
#
%
#
By hypothesis, there is some strongly pinching
. The
are chosen so that
for some
and large
. This uses ergodicity.
convenient
Lyapunov exponents ofTeichmüller flows – p. 43/6
.
-
converges almost surely to some
.
+ *
The sequence
on
probability
Let
denote the projection to the Grassmannian of
the normalized restriction of to the cylinder
that contains .
Proof of Proposition 2
This follows from a simple martingale argument.
Lyapunov exponents ofTeichmüller flows – p. 44/6
-
.
converges almost surely to some
.
+ *
The sequence
on
probability
Let
denote the projection to the Grassmannian of
the normalized restriction of to the cylinder
that contains .
Proof of Proposition 2
for all , with
Moreover,
This follows from a simple martingale argument.
.
This is a consequence of the definition of -state.
Lyapunov exponents ofTeichmüller flows – p. 44/6
-
.
converges almost surely to some
.
+ *
The sequence
on
probability
Let
denote the projection to the Grassmannian of
the normalized restriction of to the cylinder
that contains .
Proof of Proposition 2
for all , with
Moreover,
This follows from a simple martingale argument.
.
Thus,
This is a consequence of the definition of -state.
exists and is a Dirac measure, as stated.
Lyapunov exponents ofTeichmüller flows – p. 44/6
Lecture # 3
Zorich cocycles are pinching and twisting
Lyapunov exponents ofTeichmüller flows – p. 45/6
we introduced a system of
, where
#
On each stratum
local coordinates
Strata and Rauzy classes
,
live in convenient subsets of
and
.
is a pair of permutations of the alphabet
.
The pair determines the stratum and even its connected
component.
An (extended) Rauzy class is the set of all corresponding
to a given connected component of a stratum (there is also
an intrinsic combinatorial definition).
Lyapunov exponents ofTeichmüller flows – p. 46/6
and consider
Fix some (irreducible) Rauzy class
Zorich cocycle
.
with
,
Previously, we introduced the Zorich renormalization and
the Zorich cocycle for translation surfaces
Lyapunov exponents ofTeichmüller flows – p. 47/6
and consider
Fix some (irreducible) Rauzy class
Zorich cocycle
.
with
,
Previously, we introduced the Zorich renormalization and
the Zorich cocycle for translation surfaces
For our purposes,
Dropping , we find the Zorich renormalization and Zorich
cocycle for interval exchange maps
is equivalent to the inverse limit of .
Lyapunov exponents ofTeichmüller flows – p. 47/6
#
#
is a Markov map: There exists a finite partition
and a countable partition
such that
for every
.
#
Checking the hypotheses
Lyapunov exponents ofTeichmüller flows – p. 48/6
#
#
is a Markov map: There exists a finite partition
and a countable partition
such that
for every
.
#
Checking the hypotheses
There exists a -invariant probability which is ergodic
and equivalent to volume . Moreover, has bounded
distortion.
Lyapunov exponents ofTeichmüller flows – p. 48/6
#
#
is a Markov map: There exists a finite partition
and a countable partition
such that
for every
.
#
Checking the hypotheses
There exists a -invariant probability which is ergodic
and equivalent to volume . Moreover, has bounded
distortion.
, and it is
The cocycle
is constant on every atom of
integrable with respect to .
Lyapunov exponents ofTeichmüller flows – p. 48/6
.
of
There is an -invariant subbundle
the vector bundle
, such that the
Zorich cocycle
and they depend only on the
fibers have dimension
permutation pair:
Lyapunov exponents of the cocycle
in the directions
vanish (in a trivial fashion)
transverse to
restricted Zorich cocycle
preserves a symplectic
on this subbundle
form
Thus, the Lyapunov spectrum of the Zorich cocycle has the
form
Lyapunov exponents ofTeichmüller flows – p. 49/6
Zorich cocycles are simple
The goal is to prove that the are all distinct and positive.
By Theorem 1 and previous observations, all we need is
Theorem 2. Every restricted Zorich cocycle is twisting and pinching.
The proof is by induction on the complexity (
number of
genus of the surface) of the stratum:
singularities,
we look for orbits of
that spend a long time close to the
boundary of each stratum (hence, close to “simpler” strata).
Lyapunov exponents ofTeichmüller flows – p. 50/6
Hierarchy of strata
Let
be any stratum. Collapsing two or
more singularities of an
together (multiplicities add)
one obtains an Abelian differential in another stratum, on
.
the boundary of
but
if
can
by collapsing singularities.
We write
be attained from
6
PSfrag replacements
.
Lyapunov exponents ofTeichmüller flows – p. 51/6
,
is easy:
.
, with
and
% and bottom case is
Top case is
There is only one permutation pair
,
The case
Starting the induction
Lyapunov exponents ofTeichmüller flows – p. 52/6
,
is easy:
.
, with
and
% % and bottom case is
Top case is
There is only one permutation pair
,
The case
Starting the induction
is hyperbolic and so its powers
have arbitrarily large eccentricity. This proves pinching.
Lyapunov exponents ofTeichmüller flows – p. 52/6
+ *
and
To prove twisting, consider
Starting the induction
.
% , that is,
Then
for all and any sufficiently large .
coincides with any of
% Fix large enough so that no
the eigenspaces of .
% PSfrag replacements
Lyapunov exponents ofTeichmüller flows – p. 53/6
Inductive step
#
#
#
the submonoid of
such
Fix any pair
and denote by
corresponding to orbit segments
.
that
on the space
is
It suffices to prove that the action of
simple. The proof is by induction on the complexity of the
stratum, with respect to the order and to the genus.
This is easier to implement at the level of interval exchange
transformations. In that setting, approaching the boundary
corresponds to making some coefficient
very small.
Then it remains small for a long time, under iteration by the
renormalization operator.
Lyapunov exponents ofTeichmüller flows – p. 54/6
Reduction and extension
6
6
6
6
6
6
6
0
6
0
6
0
6
6
0
6
We consider operations of simple reduction/extension:
or
is a simple extension of
.
such that
there exists
Lemma. Given any
. Moreover, either
extension: inserted letter can not be last in either row and can not
be first in both rows simultaneously
Lyapunov exponents ofTeichmüller flows – p. 55/6
Proof of Theorem 2
twists isotropic subspaces of
That is, the twisting property holds for the subspaces
which the symplectic form vanishes identically.
.
Proposition 3. The action of
For proving twsting and pinching, we take advantage of the
symplectic structure:
on
To compensate for this weaker twisting statement, we prove
a stronger form of pinching:
Lyapunov exponents ofTeichmüller flows – p. 56/6
on
Proposition 4. The action of
is strongly pinching.
there exist
in the monoid for
&
for all
.
and
which
That is, given any
Proof of Theorem 2
Lemma. isotropic twisting & strong pinching
For symplectic actions in
, twisting = isotropic twisting
and pinching = strong pinching. In any dimension,
twisting & pinching.
This reduces Theorem 2 to proving the two propositions.
Lyapunov exponents ofTeichmüller flows – p. 57/6
.
Therefore, by induction,
then there is a symplectic isomorphism
that conjugates the action of
on
to the
on
.
action of
If
is a simple extension of
such that
Take
Proof of Proposition 3
does twist isotropic subspaces.
Lyapunov exponents ofTeichmüller flows – p. 58/6
, there is some symplectic reduction
of
and some symplectic isomorphism
that conjugates the action of
on
to the action
on
.
induced by
If
Proof of Proposition 3
symplectic reduction:
quotient by of the symplectic
orthogonal of , where
.
induced action: the action on
of the stabilizer of , that
is, the submonoid of elements of that preserve .
Lyapunov exponents ofTeichmüller flows – p. 59/6
, there is some symplectic reduction
of
and some symplectic isomorphism
that conjugates the action of
on
to the action
on
.
induced by
If
Proof of Proposition 3
symplectic reduction:
quotient by of the symplectic
orthogonal of , where
.
induced action: the action on
of the stabilizer of , that
is, the submonoid of elements of that preserve .
The action of
on the projective space
minimal, that is, all its orbits are dense.
Lemma. If the action of on
is minimal and its induced action on
twists isotropic subspaces, then twists isotropic subspaces of .
is indeed
Lyapunov exponents ofTeichmüller flows – p. 59/6
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Lyapunov exponents ofTeichmüller flows – p. 60/6
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