Flat geometry on Riemann surfaces and Teichmullerdynamics in moduli spaces

Anton Zorich

Obergurgl, December 15, 2008

From Billiards to Flat Surfaces 2Billiard in a polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Closed trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Motivation for studying billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Gas of two molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Unfolding billiard trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Flat surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Rational polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Billiard in a rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Unfolding rational billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Flat pretzel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Surfaces which are more flat than the others. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Very flat surfaces 16Very flat surfaces: construction from a polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Properties of very flat surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Conical singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Families of flat surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Family of flat tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Holomorphic 1-forms versus very flat surfaces 23From flat to complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24From complex to flat structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Concise geometro-analytic dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Flat surfaces and quadratic differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Volume element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Group action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Masur—Veech Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Hope for a magic wand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Relation to the Teichmuller metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Generic geodesics on very flat surfaces 33Asymptotic cycle for a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1

Asymptotic cycle for general surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Asymptotic flag: empirical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Saddle connections and closed geodesics 42Counting of closed geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Exact quadratic asymptotics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Phenomenon of multiple saddle connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Homologous saddle connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Billiards in rectangular polygons 49L-shaped billiard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Rectangular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Billiards versus quadratic differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Number of generalized diagonals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Naive intuition does not help... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Billiard players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2

From Billiards to Flat Surfaces 2 / 55

Billiard in a polygon

Moon Duchinplays billiardon an L-shapedtable

3 / 55

Closed trajectories

It is easy to find a closed billiard trajectory in an acute triangle.

Exercise. Prove that the broken line joining the bases of hights of an acute triangle is a billiardtrajectory (it is called Fagnano trajectory ). Show that it realizes an inscribed triangle of minimalperimeter.

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3

Challenge

It is difficult to believe, but a similar problem for an obtuse triangle is open.

Open problem. Is there at least one closed trajectory for (almost) any obtuse triangle?

It seems like the answer is affirmative (see an extensive computer search performed by R. Schwartzand P. Hooper: www.math.brown.edu/∼res/Billiards/index.html)

Then, one can ask further questions:

• Estimate the number N(L) of periodic trajectories of length bounded by L ≫ 1 when L → +∞.

• Is the billiard flow ergodic for almost any triangle?

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Motivation to study billiards: gas of two molecules in a one- dimensional cham-ber

Consider two elastic beads (“molecules”) sliding along a rod. They are bounded from two sides bysolid walls. All collisions are ideal — without loss of energy.

m1 m2

x1 x2x

0 a

x1

|a

x2

a

Neglecting the sizes of the beads we can describe the configuration space of our system usingcoordinates 0 < x1 ≤ x2 ≤ a of the beads, where a is the distance between the walls. This gives aright isosceles triangle.

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4

Gas of two molecules

Rescaling the coordinates by square roots of masses{

x1 =√

m1x1

x2 =√

m2x2

we get a new right triangle ∆ as a configuration space.

m1 m2

x1 x2x

0x1 =

√m1 x1

x2 =√

m2 x2

Lemma In coordinates (x1, x2) trajectories of the system of two beads on a rod correspond to billiardtrajectories in the triangle ∆.

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Unfolding billiard trajectories

Identifying the boundary of two triangles we get a flat sphere. A billiard trajectory unfolds to a geodesicon this flat sphere.

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5

Flat surfaces

The surface of the cube represents a flat sphere with eight conical singularities. The metric does nothave singularities on the edges. After parallel transport around a conical singularity a vector comesback pointing to a direction different from the initial one, so this flat metric has nontrivial holonomy.

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Rational polygons

A polygon Π is called rational if all the angles of Π are rational multiples pi

qiπ of π. Properties of

billiards in rational polygons are known much better. Consider a model case of a rectangular billiard.

As before instead of reflecting the trajectory we can reflect the billiard table. The trajectory unfolds to astraight line. Folding back the copies of the billiard table we project this line to the actual trajectory.

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6

Billiard in a rectangle

Fix a (generic) trajectory. At any moment the ball moves in one of four directions. They correspond tofour copies of the billiard table; other copies can be obtained from these four by a parallel translation:

C

D

DA

A

A

A

A

C

D

D

D

B

B BC

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Billiard in a rectangle versus flow on a torus

Identifying the equivalent patterns by a parallel translation we obtain a torus; the billiard trajectoryunfolds to a “straight line” on the corresponding torus.

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7

Unfolding rational billiards

We can apply an analogous procedure to any rational billiard.

Consider, for example, a triangle with angles π/8, 3π/8, π/2. It is easy to check that a generictrajectory in such billiard table has 16 directions (instead of 4 for a rectangle). Using 16 copies of thetriangle we unfold the billiard into a regular octagon with opposite sides identified by paralleltranslations.

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Flat surface of genus two

Identifying the pair of horizontal sides and then the pair of vertical sides of a regular octagon we get atorus with a single square hole. Identifying two opposite sides of the hole we get a torus with twodistinct holes. Identifying the holes we get a surface of genus two.

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8

Surfaces which are more flat than the others

Note that the flat metric on the resulting surface has trivial holonomy, since the identifications of thesides were performed by parallel translations. As before, a billiard trajectory is unfolded to a geodesicon the surface. But now geodesics resemble geodesics on the torus: they do not haveself-intersections!

We abandon rational billiards for a while and pass to a more systematic study of “very flat” surfaces.

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Very flat surfaces 16 / 55

Very flat surfaces: construction from a polygon

Consider a broken line constructed from vectors ~v1, . . . , ~vk.

~v1

~v2

~v3

~v4

~v4

~v3

~v2

~v1

and another one constructed from the same vectors taken in another order. If we are lucky enoughthe two broken lines do not intersect and form a polygon.

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9

Very flat surfaces: construction from a polygon

~v1

~v2

~v3

~v4

~v4

~v3

~v2

~v1

Identifying the corresponding pairs of sides by parallel translations we get a closed surface endowedwith a flat metric.

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Properties of very flat surfaces

• The flat metric is nonsingular outside of a finite number of conical singularities (inherited from thevertices of the polygon).

• The flat metric has trivial holonomy, i.e. parallel transport along any closed path brings a tangentvector to itself.

• In particular, all cone angles are integer multiples of 2π.

• By convention, the choice of the vertical direction (“direction to the North”) will be considered as apart of the “very flat structure”.

For example, a surface obtained from a rotated polygon is considered as a different very flatsurface.

• A conical singularity with the cone angle 2π · N has N outgoing directions to the North.

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10

Example: conical singularity with cone angle 6π

Locally a neighborhood of a conical point looks like a “monkey saddle”.

A neighborhood of a conical point with a cone angle 6π can be glued from six metric half discs. At thisconical point we have 3 distinct directions to the North.

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Families of flat surfaces

The polygon in our construction depends continuously on the vectors ~vj . This means that thecombinatorial geometry of the resulting flat surface (its genus g, the number m and types2π(d1 + 1), . . . , 2π(dm + 1) of the resulting conical singularities) does not change under smalldeformations of the vectors ~vj . This allows to consider a flat surface as an element of a family of flatsurfaces sharing common combinatorial geometry.

As an example of such family one can consider a family of flat tori of area one, which can be identifiedwith the space of lattices of area one:

\ SL(2, R) /SO(2, R) SL(2, Z) = H2/

SL(2, Z)

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11

Family of flat tori

neighborhood of acusp = subset oftori having shortclosed geodesic

The corresponding “modular surface” is not compact: flat tori representing points, which are close tothe cusp, are almost degenerate: they have a very short closed geodesic.

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Holomorphic 1-forms and quadratic differentials versus ve ry flat sur-faces 23 / 55

Holomorphic 1-form associated to a flat structure

Consider the natural coordinate z in the complex plane, where lives the polygon. In this coordinate theparallel translations which we use to identify the sides of the polygon are represented asz′ = z + const.

Since this correspondence is holomorphic, our flat surface S with punctured conical points inherits thecomplex structure. This complex structure extends to the punctured points.

Consider now a holomorphic 1-form dz in the complex plane. The coordinate z is not globally definedon the surface S. However, since the changes of local coordinates are defined as z′ = z + const, wesee that dz = dz′. Thus, the holomorphic 1-form dz on C defines a holomorphic 1-form ω on S whichin local coordinates has the form ω = dz.

The form ω has zeroes exactly at those points of S where the flat structure has conical singularities.

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12

Flat structure defined by a holomorphic 1-form

• Reciprocally a pair (Riemann surface, holomorphic 1-form) uniquely defines a flat structure:z =

∫

ω.

• In a neighborhood of a zero a holomorphic 1-form can be represented as wd dw, where d is adegree of the zero. The form ω has a zero of degree d at a conical point with cone angle2π(d + 1). Moreover, d1 + · · ·+ dm = 2g − 2.

• The moduli space Hg of pairs (complex structure, holomorphic 1-form) is a Cg-vector bundleover the moduli space Mg of complex structures.

• The space Hg is naturally stratified by the strata H(d1, . . . , dm) enumerated by unorderedpartitions d1 + · · ·+ dm = 2g − 2.

• Any holomorphic 1-form corresponding to a fixed stratum H(d1, . . . , dm) has exactly m zeroesof degrees d1, . . . , dm.

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Concise geometro-analytic dictionary

flat structure (including a choice complex structure and a choiceof the vertical direction) of a holomorphic 1-form ω

conical point zero of degree dwith a cone angle 2π(d + 1) of the holomorphic 1-form ω

(in local coordinates ω = wd dw)

side ~vj of a polygon relative period∫ Pj+1

Pjω =

∫

~vjdz

of the 1-form ω

family of flat surfaces sharing stratum H(d1, . . . , dm) in thethe same cone angles moduli space of holomorphic 1-forms

2π(d1 + 1), . . . , 2π(dm + 1)

local coordinates in the family: local coordinates in H(d1, . . . , dm) :vectors ~vi relative periods of ω in

defining the polygon H1(S, {P1, . . . , Pm}; C)

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13

Flat surfaces and quadratic differentials

Identifying pairs of sides of this polygon by isometries we obtain a surface of genus g = 1. Now the flatmetric has holonomy group Z/2Z. The cone angles are multiples of π.Flat surfaces of this type correspond to quadratic differentials.For example, the quadratic differential representing the surface from the picture belongs to the stratumQ(2,−1,−1).

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Volume element

Note that the vector space H1(S, {P1, . . . , Pm} ; C) contains a natural integer latticeH1(S, {P1, . . . , Pm} ; Z ⊕

√−1 Z). Consider a linear volume element dν normalized in such a way

that the volume of the fundamental domain in this lattice equals one. Consider now a real hypersurfaceH1(d1, . . . , dm) ⊂ H(d1, . . . , dm) defined by the equation area(S) = 1. The volume element dνcan be naturally restricted to the hypersurface defining the volume element dν1 on H1(d1, . . . , dm).

Theorem (H. Masur; W. A. Veech) The total volume Vol(H1(d1, . . . , dm)) of every stratum is finite.

The values of these volumes were computed by A. Eskin and A. Okounkov.

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14

Group action

The subgroup SL(2, R) of area preserving linear transformations acts on the “unit hyperboloid”

H1(d1, . . . , dm). The diagonal subgroup

(

et 00 e−t

)

⊂ SL(2, R) induces a natural flow on the

stratum, which is called the Teichmuller geodesic flow.

Key Theorem (H. Masur; W. A. Veech) The action of the groups SL(2, R) and

(

et 00 e−t

)

preserves the measure dν1. Both actions are ergodic with respect to this measure on each connectedcomponent of every stratum H1(d1, . . . , dm).

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Masur—Veech Theorem

Theorem of Masur and Veech claims that taking an arbitrary octagon as below we can contract ithorizontally and expand vertically by the same factor et to get arbitrary close to, say, regular octagon.

There is no paradox since we are allowed to cut-and-paste!

−→ =

The first modification of the polygon changes the flat structure while the second one just changes theway in which we unwrap the flat surface

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15

Hope for a magic wand

Suppose that we need some information about geometry or dynamics of an individual flat surface S.Consider S as a point in the space H(d1, . . . , dm) of flat surfaces. Let K(S) be the closure of theGL+(2, R)-orbit of S in H(d1, . . . , dm). Knowledge about the structure of K(S) gives acomprehensive information about geometry and dynamics on the initial flat surface S. Delicatenumerical characteristics of S can be expressed as averages of simpler characteristics of K(S).

Optimistic hope: The closure K(S) of the GL+(2, R)-orbit of any flat surface S is a nicecomplex-analytic variety. (Proved in genus g = 2 by C. McMullen; under additional assumptionsproved by M. Moller.)

Theorem (M. Kontsevich) Suppose that the closure K(S) of a GL+(2, R)-orbit of some flat surfaceS is a complex-analytic subvariety. Then in cohomological coordinates it is represented by an affinesubspace.

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Relation to the Teichmuller metric

Teichmuller Theorem. For any two complex structures C1, C2 on the same closed topological surfaceone can find a unique “very flat metric” (= holomorphic quadratic differential) on C1 and a unique “veryflat metric” (= holomorphic quadratic differential) on C2 such that the corresponding flat surfaces

S1, S2 are obtained one from the other by contraction-expansion: S2 =

(

et 00 e−t

)

S1. The

corresponding map realizes the extremal quasiconformal map and the value |t|/2 represents thedistance between the two complex structures in the Teichmuller metric.

Projection of any orbit of GL(2, R) to the Teichmuller space (called a Teichmuller disc) is a hyperbolicplane H2 = GL(2, R)/C∗. Any such projection is an isometric embedding with respect to theTeichmuller metric. Thus, Teichmuller discs can be considered as complex geodesics for theTeichmuller metric.

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16

Generic geodesics on very flat surfaces 33 / 55

Asymptotic cycle for a torus

Consider an “irrational” geodesic on a torus. Choose a short transversal segment X . Each time whenthe geodesic crosses X we join the crossing point with the point x0 along X obtaining a closed loop.Consecutive return points x1, x2, . . . define a sequence of cycles c1, c2, . . . .

The asymptotic cycle is defined as limn→∞

cn

n= c ∈ H1(T

2; R)

The asymptotic cycle is the same for all points of T2.

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Asymptotic cycle for general surfaces

Theorem (S. Kerckhoff, H. Masur, J. Smillie) For any flat surface directional flow in almost anydirection is ergodic with respect to the natural measure.

This implies that for almost any direction the asymptotic cycle exists and is the same for almost allpoints of the surface.

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17

Asymptotic flag: empirical description

cN

H1(S; R) ≃ R2g

x1

x2

x3

x4

x5

x2gTo study a deviation of cyclescN from the asymptotic cycleconsider their projectionsto an orthogonal hyperscreen

Direction of theasymptotic cycle

S

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Asymptotic flag: empirical description

cN

H1(S; R) ≃ R2g

x1

x2

x3

x4

x5

x2gThe projections accumulatealong a straight lineinside the hyperscreen

Direction of theasymptotic cycle

S

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18

Asymptotic flag: empirical description

cN

H1(S; R) ≃ R2g

x1

x2

x3

x4

x5

x2g

Asymptotic plane L2

Direction of theasymptotic cycle

S

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Asymptotic flag: empirical description

cN

‖cN‖ν2

‖cN‖ν3

H1(S; R) ≃ R2g

x1

x2

x3

x4

x5

x2g

Asymptotic plane L2

Direction of theasymptotic cycle

S

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19

Asymptotic flag

Theorem For almost any surface S in any stratum H1(d1, . . . , dm) there exists a flag of subspacesL1 ⊂ L2 ⊂ · · · ⊂ Lg ⊂ H1(S; R) such that for any j = 1, . . . , g − 1 one has

lim supN→∞

log dist(cN , Lj)

log N= νj+1

and dist(cN , Lg) ≤ const ,

where the constant depends only on S and on the choice of the Euclidean structure in the homologyspace.

The numbers 2 > 1 + ν2 > · · · > 1 + νg are the top g Lyapunov exponents of the Teichmullergeodesic flow on the corresponding connected component of the stratum H(d1, . . . , dm).

Remark The fact that the inequalities are strict is not trivial at all! It was proved partly by G. Forni andin full generality by A. Avila and M. Viana.

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Lyapunov exponents

We have a natural vector bundle with a fiber H1(S, R) over a point S of a stratum H(d1, . . . , dm).This vector bundle is endowed with a flat Gauss—Manin connection. Let us carry a fiber along a

trajectory of the diagonal group

(

et 00 e−t

)

. When at some time tk the trajectory passes near the

starting point we can evaluate the eigenvalues λ1(tk) ≥ · · · ≥ λ2g(tk) of the monodromy matrix, taketheir logarithms and normalize by the length tk of our piece of trajectory.

By Oseledets theorem the limits νj = limk→+∞

log |λj(tk)|tk

exist and are shared for almost all

starting points. These logarithms of eigenvalues of a “mean monodromy matrix along the flow” arecalled Lyapunov exponents. In our case they are symmetric since the monodromy matrix is symplectic:νj = −ν2g−j+1 for j = 1, . . . , g.

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20

Saddle connections and closed geodesics 42 / 55

Counting of closed geodesics

A saddle connection is a geodesic segment joining a pair of conical singularities or a conical singularityto itself without any singularities in its interior.

Similar to the torus case regular closed geodesics on a flat surface always appear in families; any suchfamily fills a maximal cylinder bounded on each side by a closed saddle connection or by a chain ofparallel saddle connections.

Let Nsc(S, L) be the number of saddle connections of length at most L on a flat surface S. LetNcg(S, L) be the number of maximal cylinders filled with closed regular geodesics of length at most Lon S.

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Exact quadratic asymptotics

Theorem (A. Eskin and H. Masur) For almost all flat surfaces S in any stratum H(d1, . . . , dm) thecounting functions Nsc(S, L) and Ncg(S, L) have exact quadratic asymptotics

Nsc(S, L) ∼ csc(S) · πL2 Ncg(S, L) ∼ ccg(S) · πL2

where the constants csc(S) and ccg(S) depend only on the connected component of the stratum.

Consider some saddle connection γ1 = [P1P2] with an endpoint at P1. Memorize its direction, say, letit be the North-West direction. Let us launch a geodesic from the same starting point P1 in one of theremaining k − 1 North-West directions. Let us study how big is the chance to hit P2 ones again, andhow big is the chance to hit it after passing the same distance as before.

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21

Phenomenon of multiple saddle connections

Theorem (A. Eskin, H. Masur, A. Zorich) For almost any flat surface S in any stratum and for anypair P1, P2 of conical singularities on S the function N2(S, L) counting the number of pairs of parallelsaddle connections of the same length joining P1 to P2 also has exact quadratic asymptotics

limL→∞

N2(S, L)

πL2= c2 > 0.

For almost all flat surfaces S in any stratum one cannot find neither a single pair of parallel saddleconnections on S of different length, nor a single pair of parallel saddle connections joining differentpairs of singularities.

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Homologous saddle connections

S ′3 S ′

2

S ′1

S3

S2

S1

S3S2

γ1γ2

γ3

Saddle connections γ1, γ2, γ3 are homologous . Hence, under any small deformation of the flat metricthis configuration persists and deformed saddle connections share the same length and direction.Hence such configuration is present on almost any flat surface in the embodying stratum.

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Siegel–Veech constants

Typical degenerations of flat surfaces (analogs of “typical stable curves”) correspond to configurationsC of homologous saddle connections.

Theorem (A. Eskin, H. Masur, A. Z.)A constant c in the quadratic asymptotics N(S, L) ∼ c · πL2 for the number of configurations of acombinatorial type C of homologous saddle connections of length at most L on a flat surface S (calledthe Siegel–Veech constant) can be evaluated by the following formula:

c(C) = limε→0

1

πε2

Vol(“ε-neighborhood of the cusp C ”)

VolHcomp1 (β)

=

= (explicit combinatorial factor) ·∏k

j=1VolH1(β

′k)

VolHcomp1 (β)

Remark. Sums ν1 + · · ·+ νg of the Lyapunov exponents can be evaluated in terms of Siegel–Veechconstants.

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More artistic picture of a decomposition

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Applications: billiards in rectangular polygons. (With J. Athreya andA. Eskin) 49 / 55

L-shaped billiard

Moon Duchinplaying billiardon a L-shaped table.

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Rectangular polygons

PP

PP

PP

P

7 6

5

4 3

21

By a rectangular polygon Π we call a topological disc endowed with a flat metric, such that theboundary ∂Π is presented by a finite broken line of geodesic segments and such that the anglebetween any two consecutive sides equals kπ/2, where k ∈ N.

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Billiards in rectangular polygons versus quadratic differ entials on CP1

We want to count trajectories emitted from a corner of such billiard and getting to some other corner.We also want to count closed billiard trajectories. In both cases we count trajectories of lengthbounded by L ≫ 1 and study the asymptotics as L tends to infinity.

We have already seen that a topological sphere obtained by gluing two copies of the billiard table bythe boundary is endowed with a flat metric. In the case of a “rectangular polygon” the flat metric hasholonomy in Z/(2Z), or, in other words, it corresponds to a meromorphic quadratic differential withsimple poles on CP1.

We have seen that geodesics on this flat sphere project to billiard trajectories. Thus, to count billiardtrajectories we may count geodesics on flat spheres!

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Number of generalized diagonals

Pi

Pj

Pi

Pj

We prove that for a generic rectangular polygon with angles π/2 and 3π/2 the number of trajectoriesjoining any two fixed corners with right angles is “approximately” the same as for a rectangle:

1

2π·(bound for the length)2

area of the table

and does not depend on the shape of the polygon.

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Naive intuition does not help...

P2

1P

P5

P3

P

4

0

P

However, say, for a typical L-shaped polygon the number of trajectories joining the corner with theangle 3π/2 to some other corner is “approximately”

2

π·(bound for the length)2

area of the table

which is 4 times (and not 3) times bigger than the number of trajectories joining a fixed pair of rightcorners...

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Billiard in a rectangular polygon, artistic view

Varvara Stepanova. Billiard players. Thyssen museum, Madrid

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