Statistical Regularities of Geodesics on Negatively Curved Surfaces

Hyperbolic Surfaces Symbolic Dynamics Orbit Statistics Self-intersections

Steve Lalley

University of Chicago

February 2016

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Symbolic Dynamics Orbit Statistics Self-intersections Acknowledgment.

Thanks to SI 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 γ parametrized curve γ(t) = x(t) + iy(t) is

Z 1 px(t)2 + y(t)2 dt 0 y(t)

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 gotten by 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 1 2|γ0(t)| 2 dt 0 1 − |γ(t)|

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Symbolic Dynamics Orbit Statistics Self-intersections

Hyperbolic Geometry: The Poincaré Disk Model D

The upper halfplane γ model and the Poincaré disk model are isometric by the map Φ: D → H given by

iz + i Φ(z) = − z − 1

Steve Lalley Statistics of Geodesics 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 az + c¯ z 7→ where |a|2 − |c|2 = 1, cz + a¯ and composition of linear fractional transformations is by a c¯ multiplication of the representing matrices . Therefore, c a¯

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 A generated by the B isometries A and B is the free group on two generators.

Steve Lalley 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 ABB a hyperbolic surface H/Γ, and geodesics on BAB a hyperbolic surface

H/Γ lift to geodesic in BBA the hyperbolic plane.

Steve Lalley 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 → ∞,

eht L(t) ∼ 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 Symbolic Coding Orbit Statistics Suspension Flows Self-intersections Symbolic Coding of Geodesics

Geodesics on a hyperbolic surface H/Γ are determined by their B a cutting sequence. b A Every (two-sided) cutting sequence a B A b uniquely determines a B b

geodesic. a A B b

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Symbolic Dynamics Symbolic Coding Orbit Statistics Suspension Flows Self-intersections Symbolic Coding of Geodesics

Successive applications of the shift mapping on cutting ABB sequences determine successive sequences of the geodesic on the surface.

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Symbolic Dynamics Symbolic Coding Orbit Statistics Suspension Flows Self-intersections Symbolic Coding of Geodesics

Successive applications of the shift mapping on cutting ABB sequences determine successive sequences of the geodesic on the

surface. BBA

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Symbolic Dynamics Symbolic Coding Orbit Statistics Suspension Flows Self-intersections Symbolic Coding of Geodesics

Successive applications of the shift mapping σ on cutting ABB sequences determine

successive sequences BAB of the geodesic on the

surface. BBA

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Symbolic Dynamics Symbolic Coding Orbit Statistics Suspension Flows Self-intersections Symbolic Coding of Geodesics

Periodic sequences correspond to closed geodesics. For a ABB periodic sequence x

the sequences BAB x, σx, σ2x,... all

represent the same BBA closed geodesic.

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Symbolic Dynamics Symbolic Coding Orbit Statistics Suspension Flows Self-intersections 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 Symbolic Coding Orbit Statistics Suspension Flows Self-intersections 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 σx

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Symbolic Dynamics Symbolic Coding Orbit Statistics Suspension Flows Self-intersections Suspension Flows

The length of a periodic orbit corresponding to a periodic sequence x of period m is

n X i Smh(x) = h(σ x) i=1 x σx

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.

The length of a periodic orbit corresponding to a periodic sequence x of minimal period m is

n X Smh(x) = mh(0) + (h(1) − h(0) xi . i=1

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 Renewal Theory Symbolic Dynamics Equidistribution of Orbits Orbit Statistics Central Limit Theory Self-intersections 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−βLN(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 Renewal Theory Symbolic Dynamics Equidistribution of Orbits Orbit Statistics Central Limit Theory Self-intersections 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 =⇒

∞ n ! X X Z (L) = Ez L − h(ξi ) . n=0 i=1

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Renewal Theory Symbolic Dynamics Equidistribution of Orbits Orbit Statistics Central Limit Theory Self-intersections 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 =⇒ N(L) ∼ CeβL. L→∞

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). Therefore, the number N∗(L) of periodic orbits of the Bernoulli ﬂow with minimal period ≤ L satisﬁes CeβL N∗(L) ∼ . L/Eh(ξ)

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Renewal Theory Symbolic Dynamics Equidistribution of Orbits Orbit Statistics Central Limit Theory Self-intersections 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 → ∞,

#{γ : Length(γ) ≤ L and |Avg(g; γ) − R g dν | < ε} p −→ 1. #{γ : Length(γ) ≤ L}

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 → ∞,

#{γ : Length(γ) ≤ L and |Avg(g; γ) − R g dµ| < ε} −→ 1. #{γ : Length(γ) ≤ L}

where µ = normalized surface area.

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Renewal Theory Symbolic Dynamics Equidistribution of Orbits Orbit Statistics Central Limit Theory Self-intersections 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 2 √ f (γ(s; u)) ds − t f dµ −→ Gaussian(0, σf ) 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 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.

Hyperbolic Surfaces Renewal Theory Symbolic Dynamics Equidistribution of Orbits Orbit Statistics Central Limit Theory Self-intersections 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 2 L Avg(f ; γL) − f dµ −→ Gaussian(0, σf ) S

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Renewal Theory Symbolic Dynamics Equidistribution of Orbits Orbit Statistics Central Limit Theory Self-intersections 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 2 L Avg(f ; γL) − f dµ −→ Gaussian(0, σf ) 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 Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Negative Curvature: Geodesics Typically Self-Intersect

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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 → ∞, N L −→ κ = (π|S|)−1 L2 S

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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, these look like independent geodesic segments. There n 2 are 2 pairs. Therefore, NL ∼ κL where 1 κ = P{two independent segments meet}/δ2 2

Steve Lalley Statistics of Geodesics Theorem: With probability → 1, for each smooth function ϕ : M → R, Nϕ Z L −→ κϕ¯ := κ ϕ( ) ( ) 2 x dx area S L S

Consequently, the self-intersections of a random geodesic ray are asymptotically uniformly distributed on the surface.

Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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

NL ϕ X NL = ϕ(xi ). i=1

Steve Lalley Statistics of Geodesics Consequently, the self-intersections of a random geodesic ray are asymptotically uniformly distributed on the surface.

Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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

NL ϕ X NL = ϕ(xi ). i=1 Theorem: With probability → 1, for each smooth function ϕ : M → R, Nϕ Z L −→ κϕ¯ := κ ϕ( ) ( ) 2 x dx area S L S

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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

NL ϕ X NL = ϕ(xi ). i=1 Theorem: With probability → 1, for each smooth function ϕ : M → R, Nϕ Z L −→ κϕ¯ := κ ϕ( ) ( ) 2 x dx area S L S

Consequently, the self-intersections of a random geodesic ray are asymptotically uniformly distributed on the surface.

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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

2 ∗ 2 P{|Kt − t κ | > εt } < 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 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 X X N = N(γ[0, m]) = H (γ(i), γ(j)). m 2 δ i=1 j=1

Steve Lalley Statistics of Geodesics m/δ m/δ 1 X X N = N(γ[0, m]) = H (γ(i), γ(j)). m 2 δ i=1 j=1

Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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 Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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 X X N = N(γ[0, m]) = H (γ(i), γ(j)). m 2 δ i=1 j=1

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Intersection Kernel

Hδ = 0 Hδ = 1

Steve Lalley Statistics of Geodesics Note 1: The kernel Hδ is symmetric and u 7→ Hδ(u, ·) is 2 continuous in L (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 Hδ1(u) = probability that a randomly chosen geodesic segment of length δ intersects γ([0, δ]; u).

Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Properties of the Intersection Kernel Key Lemma: For sufﬁciently small δ, the constant function 1 is 2 an eigenvector of the integral operator on L (SM, νL) induced by the intersection kernel Hδ: Z 2 Hδ1(u) := Hδ(u, v) dνL(v) = δ κM

Steve Lalley Statistics of Geodesics Note 2: Let γ(t; u) be the geodesic ray with initial tangent vector u ∈ SM. Then Hδ1(u) = probability that a randomly chosen geodesic segment of length δ intersects γ([0, δ]; u).

Note 1: The kernel Hδ is symmetric and u 7→ Hδ(u, ·) is 2 continuous in L (νL), so the spectrum is a sequence of real eigenvalues converging to 0.

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Properties of the Intersection Kernel Key Lemma: For sufﬁciently small δ, the constant function 1 is 2 an eigenvector of the integral operator on L (SM, νL) induced by the intersection kernel Hδ: Z 2 Hδ1(u) := Hδ(u, v) dνL(v) = δ κM

Note 1: The kernel Hδ is symmetric and u 7→ Hδ(u, ·) is 2 continuous in L (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 Hδ1(u) = probability that a randomly chosen geodesic segment of length δ intersects γ([0, δ]; u).

Steve Lalley Statistics of Geodesics −1 Proof: The normalized intersection kernel (δκ) Hδ(u, v) is a Markov kernel with the Doeblin property.

Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections 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.

−1 Proof: The normalized intersection kernel (δκ) Hδ(u, v) is a Markov kernel with the Doeblin property.

Steve Lalley Statistics of Geodesics m/δ m/δ ∞ 1 X X X = λ ϕ (γ(iδ))ϕ (γ(jδ)) 2 k k k i=1 j=1 k=1 2 ∞  m/δ  2 m X 1 X = κm + λk  ϕk (γ(iδ)) . δ p /δ k=2 m i=1

Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Eigenfunction Expansion

m/δ m/δ 1 X X N = N(γ([0, m])) = H (γ(iδ), γ(jδ)) m 2 δ i=1 j=1

Steve Lalley Statistics of Geodesics 2 ∞  m/δ  2 m X 1 X = κm + λk  ϕk (γ(iδ)) . δ p /δ k=2 m i=1

Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Eigenfunction Expansion

m/δ m/δ 1 X X N = N(γ([0, m])) = H (γ(iδ), γ(jδ)) m 2 δ i=1 j=1 m/δ m/δ ∞ 1 X X X = λ ϕ (γ(iδ))ϕ (γ(jδ)) 2 k k k i=1 j=1 k=1

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Eigenfunction Expansion

m/δ m/δ 1 X X N = N(γ([0, m])) = H (γ(iδ), γ(jδ)) m 2 δ i=1 j=1 m/δ m/δ ∞ 1 X X X = λ ϕ (γ(iδ))ϕ (γ(jδ)) 2 k k k i=1 j=1 k=1 2 ∞  m/δ  2 m X 1 X = κm + λk  ϕk (γ(iδ)) . δ p /δ k=2 m i=1

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Eigenfunction Expansion

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 Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections Eigenfunction Expansion

Unfortunately, there is no justiﬁcation for the convergence of the eigenfunction expansion.

Steve Lalley Statistics of Geodesics Hyperbolic Surfaces Law of Large Numbers Symbolic Dynamics Intersection Kernel Orbit Statistics That’s all! Self-intersections

Steve Lalley Statistics of Geodesics