**Motivation, Goals, and Sources**

Kontsevich-Zorich, `Lyapunov Exponents and Hodge Theory', 1997:

We started from computer experiments with simple one-dimensional ergodic dynamical systems, and quite unexpectedly ended with topological string theory

.

**Motivating Quote**

**INTERVAL EXCHANGE MAPS, TRANSLATION SURFACES, AND LYAPUNOV EXPONENTS**

Sources:

Understand the connection between

deviation of ergodic averages for IETs

Lyapunov exponents for Teichmuller flow

Develop a strategy for

understanding asymptotics for counting special trajectories on translation surfaces

Computing Siegel-Veech constants

Goals

One of the rare examples where Lyapunov exponents can be calculated

The numbers are geometrically meaningful

Numerical experiments are crucial in discovering new phenomenon

Main Ideas:

G. Forni

,

On the Lyapunov exponents of the Kontsevich-Zorich cocycle

, Handbook of Dynamical Systems v. 1B Elsevier, 2006.

M. Kontsevich and A. Zorich

,

Lyapunov Exponents and Hodge Theory

, `The mathematical beauty of physics', Saclay, 1996.

H. Masur

,

Ergodic Theory of Translation Surfaces

, Handbook of Dynamical Systems v. 1B Elsevier, 2006.

A. Zorich

,

Deviation for interval exchange transformations

, Ergodic Theory & Dynam. Systems, v. 17, 1477--1499, 1997.

A. Eskin

,

Counting problems in moduli space

, Handbook of Dynamical Systems v. 1B Elsevier, 2006.

A. Eskin and H. Masur

,

Asymptotic Formulas on Flat Surfaces

, Ergodic Theory & Dynam. Systems, v.21, 443-478, 2001.

Counting Problems

Lyapunov Exponents and Teichmuller Flow:

**INTERVAL EXCHANGE TRANSFORMATIONS**

A orientation preserving, piecewise isometery of an interval.

What is an IET?

What parameters do we need to specify such a map?

where

is a permutation on n letters and

is a vector in

Connection to (moduli spaces of)

geometric structures on surfaces

.

Why do we care about IETs?

Rotations of the circle

: the map

Examples of IETs

mod 1

on the unit interval [0, 1).

WHAT ELSE? WHY SHOULD WE CARE?

**Given a topological surface S of genus, a**

translation structure

on S is an atlas of charts from S

(away from a finite set of points)

to the complex plane whose transition maps are translations.

translation structure

on S is an atlas of charts from S

(away from a finite set of points)

to the complex plane whose transition maps are translations.

**TRANSLATION SURFACES**

Flat tori: given a lattice

Examples of Translation Surfaces

the quotient

is a flat torus.

What happens in higher genus?

Regular octagon:

Genus 2 Translation Surfaces

The regular octagon

genus 2,

one singular point

(angle = 1080 degrees)

Exercise: More generally, regular 4g-gon leads to genus g surface with one singular point, angle (4g-2)x180 degrees.

L-shaped table:

Again, get a genus 2 surface

with one singular point, angle 1080

degrees

L-shaped table

Decagon:

Identifying parallel sides by translation

get genus 2 surface with two singular

points, each angle 720 degrees

Decagon

PARAMETERIZING TRANSLATION SURFACES

Complex Analysis

A translation surface is given by a collection of euclidean polygons (not necessarily convex) P_1, P_2, ..., P_K, sitting in the complex plane, so that the collection of sides can be divided into sets of parallel pairs of the same length, which are then identified by translation.

Euclidean geometry

Given a topological surface S of genus, a

translation structure

on S is an atlas of charts from S

(away from a finite set of points)

to the complex plane whose transition maps are translations.

GEOMETRIC STRUCTURES:

A genus g translation surface is a pair (X, w), where X is a compact genus g Riemann surface, and w is a holomorphic one-form, w= f(z)dz in local coordinates.

The zeros of w correspond to singular points for the flat metric, and a zero of order k yields a cone angle of order (k+1)360 degrees

Integrate w to get charts

Pull back dz

"developing map"

Pull back dz

cut up surface

push forward Euclidean structure

Each translation surface yields a (family of) dynamical system(s),

the directional (or straight line) flows.

Dynamics on Translation Surfaces

The connection with IETs is given by taking the

first return map

of these flows to a transverse arc.

If you play billiards in a polygon with angles rational multiples of

Billiards

You can

unfold

, and obtain a translation surface. Rationality condition yields that there will only be finitely many copies of your original polygon. The billiard flow is given by the flow on the translation surface.

For example, for the square torus, the return map for directional flow is a rotation.

**Let (X, m) be a measure space, T: X -> X a measure preserving map.**

Ergodic Theory

is the study of long-term behavior of orbits of T, and more generally of the study of orbits of measure-preserving actions of general groups

Ergodic Theory

is the study of long-term behavior of orbits of T, and more generally of the study of orbits of measure-preserving actions of general groups

**ERGODIC THEORY**

We will work with

probability spaces

: spaces (X, m) where m(X) = 1.

We say a measure-preserving transformation T: X -> X is ergodic if the space cannot be decomposed non-trivially (in the measure category) into T-invariant pieces. Formally, if A is a T-invariant subset of X, we want either

m(A) = 0 or m(A) = 1.

The most important theorem in Ergodic Theory is the

Birkhoff Ergodic Theorem

.

Ergodicity

Let T: (X, m) -> (X, m) be an ergodic transformation of a probability space X. Then for any integrable function f, we have that for a randomly chosen point, the

time averages converge to the space average

. That is, for m-almost every x,

The Birkhoff Ergodic Theorem

Theorem:

In particular, if f is the indicator function of a set A, this says that the

amount of time the orbit spends in A is proportional to the measure of A

.

This theorem is fantastic, but has one niggling problem: it can't be improved, in general, to a statement about

every point.

Unique Ergodicity

So what can we do?

Definition:

Let (X, d) be a metric space. We say a Borel-measurable map T: X -> X is unqiuely ergodic if there is a unique T-invariant Borel probability measure m on X.

Exercise: Prove that for a continuous function f on a compact metric space X, the conclusion of Birkhoff's theorem holds for every point x in X. Use weak-* compactness of the set of Borel probability measures P(X).

Question:

IETs and Unique Ergodicity

When are IETs uniquely ergodic? That is, when do they preserve

only

Lebesgue measure?

Exercise: A rotation is uniquely ergodic if and only if it is irrational.

Exercise: More generally, regular (4g+2)-gon leads to genus g surface with two singular points, angle 2g x 180 degrees.

(Non)-minimality:

Obvious Obstructions

What are some things to rule out?

(Ir)reducibility:

If the permutation fixes some leading subset {1, 2, ..., k} of {1, 2, ..., n}, we'll be able to split the interval into invariant pieces, each supporting invariant Lebesgue measure. We call a permutation that does not fix any such subset

irreducible

.

We'll only work with irreducible IETs from now on.

If there are points without dense orbits, then there can be invariant measures constructed on the orbit closures.

We say that an IET T satisfies the

i.d.o.c.

if the backward orbit of every discontinuity point of T is infinite and distinct.

Keane's I.D.O.C.

Infinite Distinct Orbit Condition:

Theorem (Keane)

If T satisfies the i.d.o.c., every well-defined forward orbit is dense. We say in this case that T is

minimal

.

Exercise: prove this theorem, and prove that if the lengths of intervals are linearly independent over the rationals, then T satisfies the i.d.o.c.

Observation:

The Boshernitzan Criterion

If T is an irreducible IET of n intervals,

T^k is an irreducible IET of roughly kn intervals.

Theorem (Boshernitzan):

Let m_k denote the shortest interval for T^k. Then if T is not uniquely ergodic,

km_k -> 0

as k tends to infinity.

This implies the following, a solution to a conjecture of Keane (resolved originally by Masur and Veech, independently):

Generic IETs

Corollary:

Given an irreducible permutation on n letters, for Lebesgue almost-every choice of parameters, the associated IET is uniquely ergodic.

There is a crucial `renormalization' relationship between

dynamics on a translation surface

, and the

orbit of the translation surface in an appropriate dynamical system on the space of translation surfaces

.

**DYNAMICS OF TRANSLATION SURFACES**

Let

k = (k_1, k_2, ..., k_m)

denote a positive integer partition of 2g-2. The stratum

H(k)

consists of all translation surfaces of genus g with singular points with angles (k_i +1)x360 degrees.

Equivalently, it consists of the space of pairs (X, w), where X is a genus g Riemann surface and w a holomorphic 1-form with zeros of order (k_1, k_2, ..., k_m). We say two pairs (X_1, w_1) and (X_2, w_2) are equivalent if there is a biholomorphism f: X_1 -> X_2 with f*w_2 = w_1.

Konstevich-Zorich showed that each stratum contains at most 3 connected components, and that `most' are connected.

Strata of Translation Surfaces

Real and Imaginary Foliations

In complex analytic language, the orbits of the straight line flows are:

vertical: leaves of Re(w) =0

horizontal: leaves of Im(w) = 0

Direction t: vertical on exp(it)w

There is a natural

SL(2, R) action

on the space of translation surfaces, preserving connected components of strata, given simply by linear post-composition with charts.

Equivalently given a matrix G and a collection of polygons (P_1, P_2, ..., P_k) representing a translation surface, we build a new translation surface out of (GP_1, GP_2, ..., GP_k).

Why SL(2, R)? We work with

unit-area

translation surfaces.

SL(2, R)-action

Slight lie: up to a homothety.

Examples of the action:

Below, the two surfaces are the same: the matrix acts by an affine diffeomorphsim. In fact, for the torus,

any

matrix in SL(2, Z) works.

A saddle connection

Saddle Connections

Definition:

on a translation surface (X, w) is a geodesic in the flat metric connecting two singular points (zeros) with none in its interior.

Example:

The holonomy vector of a saddle connection

is the integral

Given a translation surface (X, w), define

V_w

to be the set of holonomy vectors of saddle connections on (X, w)

Equivariance

This assignment is SL(2, R)-equivariant. Given G in SL(2, R):

V_Gw = G V_w

Exercise: the set V_w is a discrete subset of the plane.

Compactness:

Compactness

Given a stratum H, let l: H -> R^+ be the function

l(X, w) = length of the shortest vector in V_w.

For any c>0, the set

{(X, w): l(X, w)>c}

is pre-compact.

Let

Teichmuller Geodesic Flow

.

The action of g_t on a stratum is called

Teichmuller geodesic flow

, since the projection of an orbit to the moduli space of Riemann surfaces is a geodesic in the Teichmuller metric.

g_t is a renormalization flow for the flow on a translation surface- it `speeds up' the vertical flow.

An important example of a stratum is

H() = SL(2, R)/SL(2, Z).

Masur's Criterion (weak version):

Masur's Criterion

Suppose the vertical flow on (X, w) is not uniquely ergodic. Then l(X_t, w_t) -> 0 as t tends to infinity.

Write (X_t, w_t) for g_t(X,w)

That is, non-divergence

of

the orbit of the translation surface to unique ergodicity

on

the translation surface.

In the case of the torus, H(), Masur's criterion checks if the vertical flow has closed leaves- the only obstruction to having unique ergodicity.

In general, however, you can have minimal, non-uniquely ergodic behavior. For example, for the billiard table with a barrier, for appropriate choices of the height, you can have a big set of minimal, non-uniquely ergodic directions (Veech, Masur, Cheung, Hubert-Cheung-Masur)

Suppose T is a uniquely ergodic IET. Then for any subinterval J, we know that for any point x with a well-defined forward orbit,

|{0<=i<=n: T^i x in J}| = n|J| + o(N)

DEVIATION OF ERGODIC AVERAGES

Zorich was motivated by the following:

Leaves of 1-forms

Zorich asked:

What is the behavior of the error term?

Question:

How do the leaves of a closed 1-form wind around a surface?

In our setting, take w to be a holomorphic 1-form, and consider the vertical foliation. Start at a point x, follow the leaf through x until time T. Close the trajectory up with some bounded length segment , to get a closed curve and a homology class.

How does this vector in homology grow with T?

Zorich observed, using computer experiments, that the error term seemed to satisfy a

power law:

There was a number

Zorich's Observations:

so that

In fact, Zorich noticed `turtles all the way down'- if you kept subtracting previous error terms, you got more power laws until you finally got a bounded error.

Motivating question:

What are these numbers? What is their geometric meaning?

**Gave a Translation Surface version of**

Boshernitzan's criterion

(

Masur's criterion

)

Explained what

Rauzy Induction

is

Explained what

Lyapunov Exponents

are.

Hinted at connection between

deviations of ergodic averages for IETs

and

Lyapunov exponents for KZ cocycle

.

Boshernitzan's criterion

(

Masur's criterion

)

Explained what

Rauzy Induction

is

Explained what

Lyapunov Exponents

are.

Hinted at connection between

deviations of ergodic averages for IETs

and

Lyapunov exponents for KZ cocycle

.

**Renormalization Review:**

**Luminy Summer School 2015**

Jayadev S. Athreya

Mathematics Department, University of Illinois

Jayadev S. Athreya

Mathematics Department, University of Illinois

Counting Agenda

Defined

IETs,

Translation Surfaces

.

Discussed

moduli space of translation surfaces

.

Discussed

Ergodicity and Unique Ergodicity

.

Stated

Boshernitzan's Criterion

.

Two Invariant Measures

If not minimal, have vertical saddle connection.

If minimal, not uniquely ergodic, we have (at least)

two invariant measures.

We have two orbits (or leaves) that

behave differently

.

The fact that

lim inf l(g_t w)>0

means there is a limit surface

Limit surfaces and Rectangles

The existence of two measures means there is a rectangle Q so that

m_1(Q)

m_2(Q)

There are leaves through points points x, y that visit Q with frequencies m_1(Q), m_2(Q) respectively

Renormalizing

Applying g_t `speeds up' the vertical flow.

On limit surface, can find points corresponding to x,y which bound a rectangle with no singularities.

So leaves have to hit Q in the same way!

We got (qualitative) information about flows on translation surfaces from Teichmuller flow, which `renormalized' them.

Is there a similar strategy that can gives us

quantitative

information for IETs?

RAUZY INDUCTION

Can build a surface out of an IET, by putting rectangles on top of intervals, zipping up, and zipping down.

Building surfaces

To speed up IET, cut off an interval at the end, look at first return to what's left

We get a new IET by inducing, what happens to the data ?

Inducing

Induction Type 1

Induction Type 2

Depends on which interval we cut off. Our procedure is to cut off the shortest interval possible at the right, that is, either of length

Change in Data

changes by a linear transformation, ones on the diagonal and one off-diagonal -1

can be easily computed as well. The set of permutations you can get is called the

Rauzy class

.

To each IET, by iteration, get path in Rauzy graph, and sequence of 0-1 matrices A_1, A_2, ... A_n so that the original lengths are related to the lengths at stage N by the product

A_1 A_2... A_N

Paths, Matrices and the Kontsevich-Zorich cocycle

Observation:

The `limiting' eigenvalues of this matrix are deviation exponents for IET.

Exercise:

compute Rauzy induction for rotations.

compute the Rauzy graph above

What are these limiting eigenvalues? What does this even mean?

LYAPUNOV EXPONENTS

Let M_1, M_2,... M_k denote your favorite collection of matrices in SL(m, R). Let p_1, p_2, ...,p_k be probabilities,

p_1 + p_2 + ... + p_k = 1

Let X_1, X_2, ... X_n, ... be an i.i.d. sequence with P(X_1 = M_i) = p_i. Let || . || denote some matrix norm.

Furstenberg-Kesten showed that, with probability 1, as N goes to infinity,

1/N log ||X_n... X_1||

has (the same) limit.

Random Matrix Products

Vastly, and beautifully, generalized by Oseledets:

Oseledet's Theorem

Actually, any matrix-valued cocycle A satisfying the finite first moment condition works:

The crucial lemma in both of these is:

Kingman's Subadditive Ergodic Theorem

These numbers are called

Lyapunov Exponents

.

Rauzy Induction is a map

Invariant measures and speed-ups

from the simplex (cross the Rauzy class) to itself. Veech constructed an

(infinite) invariant measure,

and showed the map was

ergodic.

Zorich `sped up' the map, by doing all type I or type II transformations in one giant step. He showed this has a

finite (ergodic) invariant measure

, and that the cocycle has

finite first moment,

which yielded existence of Lyapunov exponents!

Exercise: work out these invariant measures for case of rotations.

Key Relationship:

Rauzy Induction and Teichmuller Flow

Rauzy induction is a `discrete version' of Teichmuller flow. Precisely, the Teichmuller flow is a

suspension flow

over the

natural extension

of (fast) Rauzy induction.

For a concrete example, the geodesic flow on

SL(2, R)/SL(2,Z) = H()

is a suspension flow over the natural extension of the

Farey Map

F: [0, 1) -> [0, 1),

Gauss Map

G: [0,1) -> [0, 1)

G(x) = {1/x} = 1/x - [1/x]

Definitions Review

Renormalization Agenda

Give a Translation Surface version of

Boshernitzan's criterion

(

Masur's criterion

)

Exlplain what

Rauzy Induction

is

Explain what

Lyapunov Exponents

are.

Hinted at connection between

deviations of ergodic averages for IETs

and

Lyapunov exponents for KZ cocycle

.

Discuss

counting problems

on translation surfaces.

Discuss Eskin-Masur/Veech strategy to link to

equidistribution on

and

volumes of

strata.

State miraculous connection to

Lyapunov exponents.

**Goal:**

**COUNTING PROBLEMS**

Given translation surface (X, w), compute asymptotics of

N(w, R) = |{v in V_w: |v| < R}|

**Example:**

For the square torus T, putting a fake singularity at 0, saddle connections are

primitive integer vectors

, so

N(T, R) = vol(B(o,R))/zeta(2) + lower order terms

This can be proved by tiling arguments + a little elementary number theory.

For more general surfaces, we need a different strategy.

The Siegel-Veech Machine

Let Q be the

quadrilateral

with vertices at (1, 1), (-1, 1), (1/2, 1/2), (-1/2, -1/2).

The Trapezoid Trick

(True) Mathematical Folklore:

This strategy was developed by

Veech

after his mind wandered when he was attending a lecture of

Margulis

, while

extremely jetlagged

.

The Siegel-Veech transform:

So N(w, R) is the S-V transform of the indicator function of the ball of radius R.

Then we have, for t>>0

So if we put R^2 = e^t, then this integral gives a weight of R^-2 to each vector v with R/2 < |v| < R

So N(R) - N(R/2) is roughly the sum of the above integral over all holonomy vectors, multiplied by R^2.

If we could show that the measures supported on the pieces of orbit

`spread out evenly'

, that is,

equidistribute

according to some measure on the stratum H, we could solve our counting problem... almost.

Integrals on Strata

That is, we have:

Illustrating the Trapezoid Trick

Coordinates

We first need a measure on strata!

We can put coordinates on the stratum H(k_1, ..., k_m) by

fixing a basis a_1, ... a_g, b_1, ..., b_g, r_1, ... r_m for homology

relative

to the singular points.

Taking periods

That is, we view w as an element of

relative cohomology

Masur-Veech Measure

In these coordinates,

area is a quadratic form.

Theorem (Masur, Veech):

The Lebesgue measure on this area 1 `hypersurface' is a finite ergodic SL(2,R)-invariant measure.

Let SL(2, R) act ergodically on a (topological) probability space (X, m). Then for almost every x in X, and every continuous compactly supported function f on X,

Nevo's Ergodic Theorem

How do we use this?

So using Nevo's Theorem, we get, for m-almost every w in a stratum H,

Applying Nevo's Theorem

where m is Masur-Veech measure.

Actually, this works for any ergodic

SL(2, R)- invariant probability measure on the stratum.

But how do we evaluate the integral?

Let m be an SL(2, R) invariant measure on a stratum H. Suppose that for

The Siegel-Veech formula

Theorem (Veech):

Then there is a constant c = c_m (

the Siegel-Veech Constant

) so that

Corollary (Eskin-Masur):

Let m be Masur-Veech measure. Then for m-almost every w in H,

N(w, R)/R^2 --> c_H

as R --> 00. Here c_H denotes the S-V constant for the Masur-Veech measure m.

Proof: N(R) = (N(R)- N(R/2)) + (N(R/2)- n(R/4)) + ...

Applying our machine, the terms are each behaving like

2^-k x c_H x area(Q) = 2^-k x 3/4 c_H.

Summing the series in k gives a 4/3, so we get c_H, as desired.

To prove the Siegel-Veech formula, consider the map

Proving Siegel-Veech

This is a

linear, SL(2,R)-invariant functional

on C_c(R^2).

So it comes from integrating f against an

SL(2, R)-invariant measure on R^2

. Only two (up to scaling)-

delta measure at 0

, and

Lebesgue

. Can rule out delta measure at 0, so get scalar multiple of Lebesgue.

Question:

Can you compute the constant c_H?

In many cases, yes.

For some examples (like the torus), we have a big stabilizer in SL(2, R). Following McMullen, we use SL(X,w) to denote this group.

Lattice Surfaces

Definition

If SL(X,w) is a lattice, we say (X, w) is a

lattice surface.

Exercise: SL(X, w) is a discrete subgroup of SL(2, R)

Smillie showed that the SL(2, R) orbit of (X,w) is closed if and only if (X,w) is a lattice surface.

In this case, we can take our measure to be Haar measure on SL(2, R)/SL(X, w) and our S-V constant is the natural volume. Applied to the torus, we get our zeta(2) factor.

What about the constant c_H for Masur-Veech measure? Using a slightly different constant c_area (associated to counting cylinders weighted by area), Eskin-Kontsevich-Zorich proved the amazing:

SV Constants and Lyapunov Exponents

Thus, counting problems are also related to Lyapunov Exponents!!!!!!!

**Thanks to**

Alex Eskin

,

Samuel Lelievre

,

Thierry Monteil,

and

Anton Zorich

, for use of images.

Alex Eskin

,

Samuel Lelievre

,

Thierry Monteil,

and

Anton Zorich

, for use of images.

**ACKNOWLEDGMENTS**

**Thanks to Pascal, Erwan, and Anton for an incredible job organizing the meeting.**