Flat geometry on Riemann surfaces and Teichmüller
dynamics in moduli spaces
Anton Zorich
Obergurgl, December 15, 2008
From Billiards to Flat Surfaces
2
Billiard in a polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Closed trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Motivation for studying billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Gas of two molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Unfolding billiard trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Flat surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Rational polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Billiard in a rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Unfolding rational billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Flat pretzel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Surfaces which are more flat than the others. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Very flat surfaces
Very flat surfaces: construction from a polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of very flat surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conical singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Families of flat surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Family of flat tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
17
19
20
21
22
Holomorphic 1-forms versus very flat surfaces
From flat to complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
From complex to flat structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Concise geometro-analytic dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flat surfaces and quadratic differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Volume element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Group action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Masur—Veech Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hope for a magic wand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relation to the Teichmüller metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
24
25
26
27
28
29
30
31
32
Generic geodesics on very flat surfaces
33
Asymptotic cycle for a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1
Asymptotic cycle for general surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Asymptotic flag: empirical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Saddle connections and closed geodesics
Counting of closed geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exact quadratic asymptotics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phenomenon of multiple saddle connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Homologous saddle connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
43
44
45
46
Billiards in rectangular polygons
L-shaped billiard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rectangular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Billiards versus quadratic differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Number of generalized diagonals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Naive intuition does not help... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Billiard players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
50
51
52
53
54
55
2
From Billiards to Flat Surfaces
2 / 55
Billiard in a polygon
Moon Duchin
plays billiard
on an L-shaped
table
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 billiard
trajectory (it is called Fagnano trajectory ). Show that it realizes an inscribed triangle of minimal
perimeter.
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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. Schwartz
and 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?
5 / 55
Motivation to study billiards: gas of two molecules in a one-dimensional chamber
Consider two elastic beads (“molecules”) sliding along a rod. They are bounded from two sides by
solid walls. All collisions are ideal — without loss of energy.
x2
a
m1 m2
x1
0
x
x2
a
|
a
x1
Neglecting the sizes of the beads we can describe the configuration space of our system using
coordinates 0 < x1 ≤ x2 ≤ a of the beads, where a is the distance between the walls. This gives a
right isosceles triangle.
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4
Gas of two molecules
Rescaling the coordinates by square roots (
of masses
√
x̃1 = m1 x1
√
x̃2 = m2 x2
we get a new right triangle ∆ as a configuration space.
x̃2 =
√
m2 x2
m1 m2
x1
x2
x
0
x̃1 =
√
m1 x1
Lemma In coordinates (x̃1 , x̃2 ) trajectories of the system of two beads on a rod correspond to billiard
trajectories in the triangle ∆.
7 / 55
Unfolding billiard trajectories
Identifying the boundary of two triangles we get a flat sphere. A billiard trajectory unfolds to a geodesic
on 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 not
have singularities on the edges. After parallel transport around a conical singularity a vector comes
back pointing to a direction different from the initial one, so this flat metric has nontrivial holonomy.
9 / 55
Rational polygons
p
A polygon Π is called rational if all the angles of Π are rational multiples qi π of π . Properties of
i
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 a
straight 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 to
four copies of the billiard table; other copies can be obtained from these four by a parallel translation:
D
A
A
C
B
D
B
A
A
D
C
D
B
A
C
D
11 / 55
Billiard in a rectangle versus flow on a torus
Identifying the equivalent patterns by a parallel translation we obtain a torus; the billiard trajectory
unfolds 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 generic
trajectory in such billiard table has 16 directions (instead of 4 for a rectangle). Using 16 copies of the
triangle we unfold the billiard into a regular octagon with opposite sides identified by parallel
translations.
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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 a
torus with a single square hole. Identifying two opposite sides of the hole we get a torus with two
distinct 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 the
sides were performed by parallel translations. As before, a billiard trajectory is unfolded to a geodesic
on the surface. But now geodesics resemble geodesics on the torus: they do not have
self-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 .
~v2
~v3
~v4
~v1
~v1
~v4
~v2
~v3
and another one constructed from the same vectors taken in another order.
the two broken lines do not intersect and form a polygon.
If we are lucky enough
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9
Very flat surfaces: construction from a polygon
~v2
~v3
~v4
~v1
~v1
~v4
~v2
~v3
Identifying the corresponding pairs of sides by parallel translations we get a closed surface endowed
with 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 the
vertices of the polygon).
• The flat metric has trivial holonomy, i.e. parallel transport along any closed path brings a tangent
vector 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 a
part of the “very flat structure”.
For example, a surface obtained from a rotated polygon is considered as a different very flat
surface.
• 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 this
conical 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 the
combinatorial geometry of the resulting flat surface (its genus g , the number m and types
2π(d1 + 1), . . . , 2π(dm + 1) of the resulting conical singularities) does not change under small
deformations of the vectors ~
vj . This allows to consider a flat surface as an element of a family of flat
surfaces sharing common combinatorial geometry.
As an example of such family one can consider a family of flat tori of area one, which can be identified
with 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 a
cusp = subset of
tori having short
closed geodesic
The corresponding “modular surface” is not compact: flat tori representing points, which are close to
the cusp, are almost degenerate: they have a very short closed geodesic.
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Holomorphic 1-forms and quadratic differentials versus very flat surfaces
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 the
parallel translations which we use to identify the sides of the polygon are represented as
z ′ = z + const.
Since this correspondence is holomorphic, our flat surface S with punctured conical points inherits the
complex 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 defined
on the surface S . However, since the changes of local coordinates are defined as z ′ = z + const, we
see that dz = dz ′ . Thus, the holomorphic 1-form dz on C defines a holomorphic 1-form ω on S which
in 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:
R
z=
ω.
• In a neighborhood of a zero a holomorphic 1-form can be represented as w d dw , where d is a
degree of the zero. The form ω has a zero of degree d at a conical point with cone angle
2π(d + 1). Moreover, d1 + · · · + dm = 2g − 2.
• The moduli space Hg of pairs (complex structure, holomorphic 1-form) is a Cg -vector bundle
over the moduli space Mg of complex structures.
• The space Hg is naturally stratified by the strata H(d1 , . . . , dm ) enumerated by unordered
partitions d1 + · · · + dm = 2g − 2.
• Any holomorphic 1-form corresponding to a fixed stratum H(d1 , . . . , dm ) has exactly m zeroes
of degrees d1 , . . . , dm .
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Concise geometro-analytic dictionary
flat structure (including a choice
of the vertical direction)
complex structure and a choice
of a holomorphic 1-form ω
conical point
with a cone angle 2π(d + 1)
zero of degree d
of the holomorphic 1-form ω
(in local coordinates ω = w d dw )
side ~
vj of a polygon
family of flat surfaces sharing
the same cone angles
R Pj+1
ω=
of the 1-form ω
relative period
Pj
R
~
vj
dz
stratum H(d1 , . . . , dm ) in the
moduli space of holomorphic 1-forms
2π(d1 + 1), . . . , 2π(dm + 1)
local coordinates in the family:
vectors ~
vi
defining the polygon
local coordinates in H(d1 , . . . , dm ) :
relative periods of ω in
H 1 (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 flat
metric 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 stratum
Q(2, −1, −1).
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Volume element
1
Note that the vector space H√
(S, {P1 , . . . , Pm } ; C) contains a natural integer lattice
1
H (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 hypersurface
H1 (d1 , . . . , dm ) ⊂ H(d1 , . . . , dm ) defined by the equation area(S) = 1. The volume element dν
can 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
acts on the “unit hyperboloid”
linear transformations
H1 (d1 , . . . , dm ). The diagonal subgroup
et 0
0 e−t
⊂ SL(2, R) induces a natural flow on the
stratum, which is called the Teichmüller geodesic flow.
t
e
0
Key Theorem (H. Masur; W. A. Veech) The action of the groups SL(2, R) and
0 e−t
preserves the measure dν1 . Both actions are ergodic with respect to this measure on each connected
component 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 it
horizontally 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 the
way 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 the
GL+ (2, R)-orbit of S in H(d1 , . . . , dm ). Knowledge about the structure of K(S) gives a
comprehensive information about geometry and dynamics on the initial flat surface S . Delicate
numerical 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 nice
complex-analytic variety. (Proved in genus g = 2 by C. McMullen; under additional assumptions
proved by M. Möller.)
Theorem (M. Kontsevich) Suppose that the closure K(S) of a GL+ (2, R)-orbit of some flat surface
S is a complex-analytic subvariety. Then in cohomological coordinates it is represented by an affine
subspace.
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Relation to the Teichmüller metric
Teichmüller Theorem. For any two complex structures C1 , C2 on the same closed topological surface
one can find a unique “very flat metric” (= holomorphic quadratic differential) on C1 and a unique “very
flat metric” (= holomorphic quadratic differential) on C2 such that the corresponding
flat surfaces
et 0
S1 . The
0 e−t
corresponding map realizes the extremal quasiconformal map and the value |t|/2 represents the
S1 , S2 are obtained one from the other by contraction-expansion: S2 =
distance between the two complex structures in the Teichmüller metric.
Projection of any orbit of GL(2, R) to the Teichmüller space (called a Teichmüller disc) is a hyperbolic
plane H2 = GL(2, R)/C∗ . Any such projection is an isometric embedding with respect to the
Teichmüller metric. Thus, Teichmüller discs can be considered as complex geodesics for the
Teichmüller metric.
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16
Generic geodesics on very flat surfaces
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Asymptotic cycle for a torus
Consider an “irrational” geodesic on a torus. Choose a short transversal segment X . Each time when
the 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
= c ∈ H1 (T2 ; R)
n
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 any
direction 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 all
points of the surface.
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17
Asymptotic flag: empirical description
S
Direction of the
asymptotic cycle
x2g
cN
To study a deviation of cycles
cN from the asymptotic cycle
consider their projections
to an orthogonal hyperscreen
x5
x4
H1 (S; R) ≃ R2g
x3
x1
x2
36 / 55
Asymptotic flag: empirical description
S
Direction of the
asymptotic cycle
x2g
The projections accumulate
along a straight line
inside the hyperscreen
x5
x4
cN
H1 (S; R) ≃ R2g
x3
x2
x1
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18
Asymptotic flag: empirical description
S
Direction of the
asymptotic cycle
cN
x2g
x5
x4
Asymptotic plane L2
H1 (S; R) ≃ R2g
x3
x1
x2
38 / 55
Asymptotic flag: empirical description
S
Direction of the
asymptotic cycle
kcN kν2
cN
kcN kν3
x2g
x5
x4
Asymptotic plane L2
H1 (S; R) ≃ R2g
x3
x2
x1
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19
Asymptotic flag
For almost any surface S in any stratum H1 (d1 , . . . , dm ) there exists a flag of subspaces
L1 ⊂ L2 ⊂ · · · ⊂ Lg ⊂ H1 (S; R) such that for any j = 1, . . . , g − 1 one has
Theorem
lim sup
N →∞
log dist(cN , Lj )
= νj+1
log N
dist(cN , Lg ) ≤ const ,
where the constant depends only on S and on the choice of the Euclidean structure in the homology
and
space.
The numbers 2 > 1 + ν2 > · · · > 1 + νg are the top g Lyapunov exponents of the Teichmüller
geodesic 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 and
in full generality by A. Avila and M. Viana.
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Lyapunov exponents
We have a natural vector bundle with a fiber H 1 (S, R) over a point S of a stratum H(d1 , . . . , dm ).
This vector bundle is endowed with
connection. Let us carry a fiber along a
a flat Gauss—Manin
et 0
. When at some time tk the trajectory passes near the
0 e−t
starting point we can evaluate the eigenvalues λ1 (tk ) ≥ · · · ≥ λ2g (tk ) of the monodromy matrix, take
their logarithms and normalize by the length tk of our piece of trajectory.
trajectory of the diagonal group
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” are
called 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
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Counting of closed geodesics
A saddle connection is a geodesic segment joining a pair of conical singularities or a conical singularity
to itself without any singularities in its interior.
Similar to the torus case regular closed geodesics on a flat surface always appear in families; any such
family fills a maximal cylinder bounded on each side by a closed saddle connection or by a chain of
parallel saddle connections.
Let Nsc (S, L) be the number of saddle connections of length at most L on a flat surface S . Let
Ncg (S, L) be the number of maximal cylinders filled with closed regular geodesics of length at most L
on 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 ) the
counting 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 = [P1 P2 ] with an endpoint at P1 . Memorize its direction, say, let
it be the North-West direction. Let us launch a geodesic from the same starting point P1 in one of the
remaining k − 1 North-West directions. Let us study how big is the chance to hit P2 ones again, and
how big is the chance to hit it after passing the same distance as before.
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Phenomenon of multiple saddle connections
Theorem (A. Eskin, H. Masur, A. Zorich) For almost any flat surface S in any stratum and for any
pair P1 , P2 of conical singularities on S the function N2 (S, L) counting the number of pairs of parallel
saddle connections of the same length joining P1 to P2 also has exact quadratic asymptotics
N2 (S, L)
= c2 > 0.
L→∞
πL2
lim
For almost all flat surfaces S in any stratum one cannot find neither a single pair of parallel saddle
connections on S of different length, nor a single pair of parallel saddle connections joining different
pairs of singularities.
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Homologous saddle connections
S3
S3′
S2
S2′
S3
S2
γ3
γ2 γ1
S1′
S1
Saddle connections γ1 , γ2 , γ3 are homologous. Hence, under any small deformation of the flat metric
this 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 configurations
C 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 a
combinatorial type C of homologous saddle connections of length at most L on a flat surface S (called
the Siegel–Veech constant) can be evaluated by the following formula:
1 Vol(“ε-neighborhood of the cusp C ”)
=
ε→0 πε2
Vol H1comp (β)
c(C) = lim
= (explicit combinatorial factor) ·
Qk
′
j=1 Vol H1 (βk )
Vol H1comp (β)
Remark. Sums ν1 + · · · + νg of the Lyapunov exponents can be evaluated in terms of Siegel–Veech
constants.
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More artistic picture of a decomposition
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Applications: billiards in rectangular polygons. (With J. Athreya and
A. Eskin)
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L-shaped billiard
Moon Duchin
playing billiard
on a L-shaped table.
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Rectangular polygons
P4
P7
P3
P6
P5
P1
P2
By a rectangular polygon Π we call a topological disc endowed with a flat metric, such that the
boundary ∂Π is presented by a finite broken line of geodesic segments and such that the angle
between any two consecutive sides equals kπ/2, where k ∈ N.
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Billiards in rectangular polygons versus quadratic differentials 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 length
bounded 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 by
the boundary is endowed with a flat metric. In the case of a “rectangular polygon” the flat metric has
holonomy in Z/(2Z), or, in other words, it corresponds to a meromorphic quadratic differential with
simple poles on CP1 .
We have seen that geodesics on this flat sphere project to billiard trajectories. Thus, to count billiard
trajectories we may count geodesics on flat spheres!
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Number of generalized diagonals
Pj
Pj
Pi
Pi
We prove that for a generic rectangular polygon with angles π/2 and 3π/2 the number of trajectories
joining any two fixed corners with right angles is “approximately” the same as for a rectangle:
1 (bound for the length)2
·
2π
area of the table
and does not depend on the shape of the polygon.
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Naive intuition does not help...
P4
P5
P0
P3
P1
P2
However, say, for a typical L-shaped polygon the number of trajectories joining the corner with the
angle 3π/2 to some other corner is “approximately”
2 (bound for the length)2
·
π
area of the table
which is 4 times (and not 3) times bigger than the number of trajectories joining a fixed pair of right
corners...
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Billiard in a rectangular polygon, artistic view
Varvara Stepanova. Billiard players. Thyssen museum, Madrid
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