Download Statistical Regularities of Geodesics on Negatively Curved Surfaces

Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Statistical Regularities of Geodesics on
Negatively Curved Surfaces
Steve Lalley
University of Chicago
February 2016
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Acknowledgment.
Thanks to S I TANG for
assistance in drawing
the figures.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
The Plan:
I
Hyperbolic Surfaces
I
Symbolic Dynamics
I
Orbit Statistics
I
Self-intersections
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Compact Orientable Surfaces
Surfaces of genus 2, 3, . . . admit hyperbolic metrics.
Surface of genus 1 admits a flat metric.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
A Flat Surface: The Torus
A geodesic on a flat torus
is the projection of a
straight line. A closed
geodesic is the projection
of a line with rational
slope.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Hyperbolic Geometry: Upper Halfplane Model
The hyperbolic length of a
parametrized curve
γ(t) = x(t) + iy (t) is
Z 1p
x(t)2 + y (t)2
dt
y (t)
0
γ
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Orientation-Preserving Isometries of H are the linear fractional
transformations
az + b
z 7→
cz + d
where a, b, c, d ∈ R and ad − bc = 1. Composition of two linear
fractional transformations is gottenby matrix multiplication of
a b
the corresponding matrices
. Thus,
c d
Isom(H) = PSL(2, R).
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Geodesics in H are Euclidean circles or lines that intersect the
ideal boundary (the x−axis) orthogonally.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Hyperbolic Geometry: The Poincaré Disk Model D
γ
The hyperbolic length
of a parametrized
curve γ : [0, 1] → H is
Z
0
Steve Lalley
1
2|γ 0 (t)|
dt
1 − |γ(t)|2
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Hyperbolic Geometry: The Poincaré Disk Model D
The upper halfplane
model and the
Poincaré disk model
are isometric by the
map Φ : D → H given
by
γ
Φ(z) = −
Steve Lalley
Statistics of Geodesics
iz + i
z −1
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Geodesics in H are Euclidean circles or lines that intersect the
circle at ∞ orthogonally.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
The orientation-preserving isometries of D are the linear
fractional transformations
z 7→
az + c̄
cz + ā
where |a|2 − |c|2 = 1,
and composition of linear fractional transformations
is by
a c̄
multiplication of the representing matrices
. Therefore,
c ā
Isom(D) = SU(1, 1).
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Hyperbolic Surfaces
A hyperbolic surface is a quotient space H/Γ where Γ is a
discrete subgroup of Isom(H). Every hyperbolic surface can be
obtained from a geodesic polygon by identifying boundary
geodesic segments in pairs.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Example: Punctured Torus
Identifying the two blue
edges and the two red
edges gives a punctured
torus. The group Γ
generated by the
isometries A and B is the
free group on two
generators.
Steve Lalley
A
B
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Example: Punctured torus
The images of the
fundamental polygon
obtained by mapping
by elements of Γ give a
tessellation of H by
congruent polygons.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Geodesics on Hyperbolic Surfaces
Geodesics in the
hyperbolic plane
project to geodesics on
a hyperbolic surface
H/Γ, and geodesics on
a hyperbolic surface
H/Γ lift to geodesic in
the hyperbolic plane.
Steve Lalley
ABB
BAB
BBA
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Example: Modular Surface
Modular Group: Γ = PSL(2, Z)
Modular Surface: H/Γ.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Example: Modular Surface
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
The Prime Geodesic Theorem
Theorem: On any compact, negatively curved surface M there
are countably many closed geodesics. Let L(t) be the number
of closed geodesics of length ≤ t. Then as t → ∞,
L(t) ∼
eht
ht
where h is the topological entropy of the geodesic flow on SM.
For constant curvature −1,
h = 1.
Delsarte-Huber-Selberg: constant curvature (hyperbolic))
Margulis: variable negative curvature
Lalley: infinite area hyperbolic surfaces
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Symbolic Coding
Suspension Flows
Symbolic Coding of Geodesics
Geodesics on a
hyperbolic surface H/Γ
are determined by their
cutting sequence.
Every (two-sided)
cutting sequence
uniquely determines a
geodesic.
Steve Lalley
B
Bb
Aa
b
A
a
B
A
B
b
Statistics of Geodesics
a
b
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Symbolic Coding
Suspension Flows
Symbolic Coding of Geodesics
Successive
applications of the shift
mapping on cutting
sequences determine
successive sequences
of the geodesic on the
surface.
Steve Lalley
ABB
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Symbolic Coding
Suspension Flows
Symbolic Coding of Geodesics
Successive
applications of the shift
mapping on cutting
sequences determine
successive sequences
of the geodesic on the
surface.
Steve Lalley
ABB
BBA
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Symbolic Coding
Suspension Flows
Symbolic Coding of Geodesics
Successive
applications of the shift
mapping σ on cutting
sequences determine
successive sequences
of the geodesic on the
surface.
Steve Lalley
ABB
BAB
BBA
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Symbolic Coding
Suspension Flows
Symbolic Coding of Geodesics
Periodic sequences
correspond to closed
geodesics. For a
periodic sequence x
the sequences
x, σx, σ 2 x, . . . all
represent the same
closed geodesic.
ABB
BAB
BBA
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Symbolic Coding
Suspension Flows
Symbolic Coding for the Modular Surface
Symbolic coding for a geodesic on the modular surface is given
by the continued fraction expansions of the two ideal endpoints.
Closed geodesics are those for which the continued fraction
expansions are periodic. Figure by C. Series
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Symbolic Coding
Suspension Flows
Suspension Flows
The geodesic flow on a
hyperbolic surface is
(semi-)conjugate to a
suspension flow over a
two-sided shift of finite
type.
x
Steve Lalley
Statistics of Geodesics
σx
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Symbolic Coding
Suspension Flows
Suspension Flows
The length of a periodic
orbit corresponding to
a periodic sequence x
of period m is
Sm h(x) =
n
X
h(σ i x)
i=1
x
Steve Lalley
Statistics of Geodesics
σx
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Example: The Bernoulli Flow
When the sequence
space is Σ = {0, 1}Z
and the height function
h depends only on the
first entry of the
sequence, the
suspension flow is
called a Bernoulli flow.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Example: The Bernoulli Flow
The length of a periodic
orbit corresponding to
a periodic sequence x
of minimal period m is
Sm h(x) = mh(0) + (h(1) − h(0)
n
X
i=1
Steve Lalley
Statistics of Geodesics
xi .
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Example: The Bernoulli Flow
The number N(L) of
(periodic) sequences
that correspond to
periodic orbits of length
≤ L satisfies the
recursive relation
N(L) = N(L−h(0))+N(L−h(1)) for L > h(1) > h(0).
This is a renewal equation in disguise.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Example: The Bernoulli Flow
Transformation to a Renewal Equation. Let β > 0 be the unique
solution of
e−βh(0) + e−βh(1) = 1.
Set
Z (L) = e−βL N(L).
Then Z (L) = e−βh(0) Z (L − h(0)) + e−βh(1) Z (L − h(1)), i.e.,
Z (L) = EZ (L − h(ξ))
for L > h(1), where ξ is a Bernoulli random variable with
success parameter p = e−βh(1) .
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Example: The Bernoulli Flow
Solution of the Renewal Equation. The recursive equation
holds only for L > h(1). For L ∈ [−h(1), h(1)] there is an
additive correction z(L); thus,
Z (L) = EZ (L − h(ξ)) + z(L),
Iteration =⇒
Z (L) =
∞
X
Ez
n=0
Steve Lalley
L−
n
X
!
h(ξi ) .
i=1
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Example: The Bernoulli Flow
Blackwell’s Renewal Theorem implies that if h(0)/h(1) 6∈ Q
then there exists C > 0 such that
lim Z (L) = C
L→∞
=⇒
N(L) ∼ CeβL .
The Law of Large Numbers implies that most sequences
counted in Z (L) look like i.i.d. Bernoulli - p = e−βh(1) , and so
most have minimal period
≈ L/Eh(ξ).
N ∗ (L)
Therefore, the number
of periodic orbits of the Bernoulli
flow with minimal period ≤ L satisfies
N ∗ (L) ∼
Steve Lalley
CeβL
.
L/Eh(ξ)
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Equidistribution of Periodic Orbits
The Law of Large Numbers implies that most sequences
counted in N(L) look like i.i.d. Bernoulli - p = e−βh(1) .
Therefore, most periodic orbits of length ≤ L will be nearly
equi-distributed according to the suspension νp of the Bernoulli
- p measure on sequence space.
Theorem: (Bowen; Lalley) Let g : Σh → R be a continuous
function and for any periodic orbit γ let Avg(g; γ) be the mean
value of g along γ. Then for any ε > 0, as L → ∞,
R
#{γ : Length(γ) ≤ L and |Avg(g; γ) − g dνp | < ε}
−→ 1.
#{γ : Length(γ) ≤ L}
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Equidistribution of Periodic Orbits
This extends to closed geodesics on hyperbolic surfaces.
Theorem: (Bowen; Lalley) Let S = H/Γ be a compact
hyperbolic surface and g : S → R a continuous function. For
any periodic orbit γ let Avg(g; γ) be the mean value of g along
γ. Then for any ε > 0, as L → ∞,
R
#{γ : Length(γ) ≤ L and |Avg(g; γ) − g dµ| < ε}
−→ 1.
#{γ : Length(γ) ≤ L}
where µ = normalized surface area.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
Cohomology and the CLT
Theorem: (Ratner 1972) Let S = H/Γ be a compact hyperbolic
surface and let f : S → R be smooth. If γ(t; x, θ) is geodesic
with randomly chosen initial point x and direction θ then as
t → ∞,
Z t
Z
1
D
√
f (γ(s; u)) ds − t
f dµ −→ Gaussian(0, σf2 )
t
0
S
and σf > 0 if and only if f is not cohomologous to a constant.
Cohomology: A function g is a coboundary for the geodesic
flow if it integrates to 0 on every closed geodesic. A function f
is cohomologous to a constant α if f − α is a coboundary.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
CLT for Closed Geodesics
Theorem: (La 1986) Let f : S → R be smooth. If γL is randomly
chosen from among all closed geodesics of length ≤ L then as
L → ∞,
Z
√ D
L Avg(f ; γL ) −
f dµ −→ Gaussian(0, σf2 )
S
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Renewal Theory
Equidistribution of Orbits
Central Limit Theory
CLT for Closed Geodesics
Theorem: (La 1986) Let f : S → R be smooth. If γL is randomly
chosen from among all closed geodesics of length ≤ L then as
L → ∞,
Z
√ D
L Avg(f ; γL ) −
f dµ −→ Gaussian(0, σf2 )
S
Note: The results of Bowen, Lalley, and Ratner all generalize to
surfaces of variable negative curvature; normalized surface
area measure is replaced by the maximal entropy invariant
measure for the geodesic flow.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Negative Curvature: Geodesics Typically Self-Intersect
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Law of Large Numbers I
Question: How many times does a closed geodesic of length
≈ L self-intersect? How many times does a random geodesic
segment of length L self-intersect?
Theorem: Let NL be the number of self-intersections of a
random geodesic segment of length L on a hyperbolic surface
S = H/Γ. Then with probability → 1 as L → ∞,
NL
−→ κS = (π|S|)−1
L2
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Law of Large Numbers I
Heuristic Explanation: The geodesic segment γ[0, L] consists of
n = L/δ segments of length δ. Because the geodesic flow is
mixing,
geodesic segments. There
these look like independent
n
2
are 2 pairs. Therefore, NL ∼ κL where
κ=
1
P{two independent segments meet}/δ 2
2
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Law of Large Numbers II
Denote by xi ∈ S the location of the ith self-intersection. For
any smooth, nonnegative function ϕ : S → R+ define the
ϕ−weighted self-intersection count by
NLϕ
=
NL
X
ϕ(xi ).
i=1
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Law of Large Numbers II
Denote by xi ∈ S the location of the ith self-intersection. For
any smooth, nonnegative function ϕ : S → R+ define the
ϕ−weighted self-intersection count by
NLϕ
=
NL
X
ϕ(xi ).
i=1
Theorem: With probability → 1, for each smooth function
ϕ : M → R,
Z
NLϕ
−→ κϕ̄ := κ
ϕ(x) dx area(S)
L2
S
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Law of Large Numbers II
Denote by xi ∈ S the location of the ith self-intersection. For
any smooth, nonnegative function ϕ : S → R+ define the
ϕ−weighted self-intersection count by
NLϕ
=
NL
X
ϕ(xi ).
i=1
Theorem: With probability → 1, for each smooth function
ϕ : M → R,
Z
NLϕ
−→ κϕ̄ := κ
ϕ(x) dx area(S)
L2
S
Consequently, the self-intersections of a random geodesic ray
are asymptotically uniformly distributed on the surface.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Self-Intersections: First-Order Asymptotics
Theorem: (La 1996) For any compact, negatively curved
surface S = H/Γ there exists a constant κ∗ > 0 such that for
any ε > 0, if t is sufficiently large then the number Kt of
self-intersections of a randomly chosen closed geodesic of
length ≤ t satisfies
P{|Kt − t 2 κ∗ | > εt 2 } < ε
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Self-Intersections: First-Order Asymptotics
Theorem: (La 1996) For any compact, negatively curved
surface S = H/Γ there exists a constant κ∗ > 0 such that for
any ε > 0, if t is sufficiently large then the number Kt of
self-intersections of a randomly chosen closed geodesic of
length ≤ t satisfies
P{|Kt − t 2 κ∗ | > εt 2 } < ε
Furthermore:
(A) If S has constant negative curvature then κ∗ = κS .
(B) In general, the locations of self-intersections are
asymptotically distributed according to the (projection to S of
the) maximal entropy invariant probability measure for the
geodesic flow.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Intersection Kernel
Fix δ > 0 so small that if two geodesic segments of length δ
intersect transversally then they intersect in only one point.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Intersection Kernel
Fix δ > 0 so small that if two geodesic segments of length δ
intersect transversally then they intersect in only one point.
Intersection Kernel: Nonnegative, symmetric function
Hδ : SM × SM → {0, 1} that takes value Hδ (u, v ) = 1 if
geodesic segments of length δ based at u, v intersect
transversally, and Hδ (u, v ) = 0 if not.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Intersection Kernel
Fix δ > 0 so small that if two geodesic segments of length δ
intersect transversally then they intersect in only one point.
Intersection Kernel: Nonnegative, symmetric function
Hδ : SM × SM → {0, 1} that takes value Hδ (u, v ) = 1 if
geodesic segments of length δ based at u, v intersect
transversally, and Hδ (u, v ) = 0 if not.
m/δ m/δ
1 XX
Nm = N(γ[0, m]) =
Hδ (γ(i), γ(j)).
2
i=1 j=1
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Intersection Kernel
Hδ = 0
Hδ = 1
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Properties of the Intersection Kernel
Key Lemma: For sufficiently small δ, the constant function 1 is
an eigenvector of the integral operator on L2 (SM, νL ) induced
by the intersection kernel Hδ :
Z
Hδ 1(u) := Hδ (u, v ) dνL (v ) = δ 2 κM
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Properties of the Intersection Kernel
Key Lemma: For sufficiently small δ, the constant function 1 is
an eigenvector of the integral operator on L2 (SM, νL ) induced
by the intersection kernel Hδ :
Z
Hδ 1(u) := Hδ (u, v ) dνL (v ) = δ 2 κM
Note 1: The kernel Hδ is symmetric and u 7→ Hδ (u, ·) is
continuous in L2 (νL ), so the spectrum is a sequence of real
eigenvalues converging to 0.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Properties of the Intersection Kernel
Key Lemma: For sufficiently small δ, the constant function 1 is
an eigenvector of the integral operator on L2 (SM, νL ) induced
by the intersection kernel Hδ :
Z
Hδ 1(u) := Hδ (u, v ) dνL (v ) = δ 2 κM
Note 1: The kernel Hδ is symmetric and u 7→ Hδ (u, ·) is
continuous in L2 (νL ), so the spectrum is a sequence of real
eigenvalues converging to 0.
Note 2: Let γ(t; u) be the geodesic ray with initial tangent vector
u ∈ SM. Then Hδ 1(u) = probability that a randomly chosen
geodesic segment of length δ intersects γ([0, δ]; u).
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Properties of the Intersection Kernel
Lemma 2: The eigenvalue λ1 = δκ is simple, and all other
eigenvalues λ2 , λ3 , . . . are smaller in absolute value. Therefore,
all nonconstant eigenfunctions ψ2 , ψ3 , . . . are orthogonal to 1:
Z
ψj (u) dνL (u) = 0.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Properties of the Intersection Kernel
Lemma 2: The eigenvalue λ1 = δκ is simple, and all other
eigenvalues λ2 , λ3 , . . . are smaller in absolute value. Therefore,
all nonconstant eigenfunctions ψ2 , ψ3 , . . . are orthogonal to 1:
Z
ψj (u) dνL (u) = 0.
Proof: The normalized intersection kernel (δκ)−1 Hδ (u, v ) is a
Markov kernel with the Doeblin property.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Eigenfunction Expansion
m/δ m/δ
1 XX
Hδ (γ(iδ), γ(jδ))
Nm = N(γ([0, m])) =
2
i=1 j=1
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Eigenfunction Expansion
m/δ m/δ
1 XX
Hδ (γ(iδ), γ(jδ))
Nm = N(γ([0, m])) =
2
i=1 j=1
m/δ m/δ ∞
=
1 XXX
λk ϕk (γ(iδ))ϕk (γ(jδ))
2
i=1 j=1 k =1
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Eigenfunction Expansion
m/δ m/δ
1 XX
Hδ (γ(iδ), γ(jδ))
Nm = N(γ([0, m])) =
2
i=1 j=1
m/δ m/δ ∞
1 XXX
λk ϕk (γ(iδ))ϕk (γ(jδ))
2
i=1 j=1 k =1
2

m/δ
∞
X
X
m
1
= κm2 +
λk  p
ϕk (γ(iδ)) .
δ
m/δ
=
k =2
Steve Lalley
Statistics of Geodesics
i=1
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Eigenfunction Expansion
m/δ m/δ
Nm = N(γ([0, m])) =
1 XX
Hδ (γ(iδ), γ(jδ))
2
i=1 j=1
=
1
2
m/δ m/δ ∞
X
XX
λk ϕk (γ(iδ))ϕk (γ(jδ))
i=1 j=1 k =1

2
m/δ
∞
X
X
m
1
= κm2 +
λk  p
ϕk (γ(iδ)) .
δ
m/δ
k =2
i=1
Central Limit Theorem for geodesic flow implies that each
interior sum has a limiting Gaussian distribution. These may be
correlated for different k .
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Eigenfunction Expansion
m/δ m/δ
1 XX
Hδ (γ(iδ), γ(jδ))
Nm = N(γ([0, m])) =
2
i=1 j=1
m/δ m/δ ∞
1 XXX
λk ϕk (γ(iδ))ϕk (γ(jδ))
2
i=1 j=1 k =1
2

m/δ
∞
X
X
m
1
= κm2 +
λk  p
ϕk (γ(iδ)) .
δ
m/δ
=
k =2
i=1
Unfortunately, there is no justification for the convergence of the
eigenfunction expansion.
Steve Lalley
Statistics of Geodesics
Hyperbolic Surfaces
Symbolic Dynamics
Orbit Statistics
Self-intersections
Law of Large Numbers
Intersection Kernel
That’s all!
Steve Lalley
Statistics of Geodesics