statistical regularities of geodesics on negatively curved...
TRANSCRIPT
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Statistical Regularities of Geodesics onNegatively Curved Surfaces
Steve Lalley
University of Chicago
February 2016
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Acknowledgment.
Thanks to SI TANG forassistance in drawingthe figures.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
The Plan:
I Hyperbolic SurfacesI Symbolic DynamicsI Orbit StatisticsI Self-intersections
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-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 SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
A Flat Surface: The Torus
A geodesic on a flat torusis the projection of astraight line. A closedgeodesic is the projectionof a line with rationalslope.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Hyperbolic Geometry: Upper Halfplane Model
γThe hyperbolic length of aparametrized curveγ(t) = x(t) + iy(t) is∫ 1
0
√x(t)2 + y(t)2
y(t)dt
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Orientation-Preserving Isometries of H are the linear fractionaltransformations
z 7→ az + bcz + d
where a,b, c,d ∈ R and ad − bc = 1. Composition of two linearfractional transformations is gotten by matrix multiplication of
the corresponding matrices(
a bc d
). Thus,
Isom(H) = PSL(2,R).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Geodesics in H are Euclidean circles or lines that intersect theideal boundary (the x−axis) orthogonally.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Hyperbolic Geometry: The Poincaré Disk Model D
γ The hyperbolic lengthof a parametrizedcurve γ : [0,1]→ H is∫ 1
0
2|γ′(t)|1− |γ(t)|2
dt
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Hyperbolic Geometry: The Poincaré Disk Model D
γThe upper halfplanemodel and thePoincaré disk modelare isometric by themap Φ : D→ H givenby
Φ(z) = − iz + iz − 1
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Geodesics in H are Euclidean circles or lines that intersect thecircle at∞ orthogonally.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
The orientation-preserving isometries of D are the linearfractional transformations
z 7→ az + ccz + a
where |a|2 − |c|2 = 1,
and composition of linear fractional transformations is by
multiplication of the representing matrices(
a cc a
). Therefore,
Isom(D) = SU(1,1).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Hyperbolic SurfacesA hyperbolic surface is a quotient space H/Γ where Γ is adiscrete subgroup of Isom(H). Every hyperbolic surface can beobtained from a geodesic polygon by identifying boundarygeodesic segments in pairs.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Example: Punctured Torus
Identifying the two blueedges and the two rededges gives a puncturedtorus. The group Γgenerated by theisometries A and B is thefree group on twogenerators.
AB
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Example: Punctured torus
The images of thefundamental polygonobtained by mappingby elements of Γ give atessellation of H bycongruent polygons.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Geodesics on Hyperbolic Surfaces
Geodesics in thehyperbolic planeproject to geodesics ona hyperbolic surfaceH/Γ, and geodesics ona hyperbolic surfaceH/Γ lift to geodesic inthe hyperbolic plane.
ABB
BBA
BAB
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Example: Modular SurfaceModular Group: Γ = PSL(2,Z)Modular Surface: H/Γ.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Example: Modular Surface
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
The Prime Geodesic TheoremTheorem: On any compact, negatively curved surface M thereare countably many closed geodesics. Let L(t) be the numberof 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 curvatureLalley: infinite area hyperbolic surfaces
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Symbolic CodingSuspension Flows
Symbolic Coding of Geodesics
Geodesics on ahyperbolic surface H/Γare determined by theircutting sequence.Every (two-sided)cutting sequenceuniquely determines ageodesic.
a
Ab
B
a
A b
B
AaBb
B b
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Symbolic CodingSuspension Flows
Symbolic Coding of Geodesics
Successiveapplications of the shiftmapping on cuttingsequences determinesuccessive sequencesof the geodesic on thesurface.
ABB
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Symbolic CodingSuspension Flows
Symbolic Coding of Geodesics
Successiveapplications of the shiftmapping on cuttingsequences determinesuccessive sequencesof the geodesic on thesurface.
ABB
BBA
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Symbolic CodingSuspension Flows
Symbolic Coding of Geodesics
Successiveapplications of the shiftmapping σ on cuttingsequences determinesuccessive sequencesof the geodesic on thesurface.
ABB
BBA
BAB
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Symbolic CodingSuspension Flows
Symbolic Coding of Geodesics
Periodic sequencescorrespond to closedgeodesics. For aperiodic sequence xthe sequencesx , σx , σ2x , . . . allrepresent the sameclosed geodesic.
ABB
BBA
BAB
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Symbolic CodingSuspension Flows
Symbolic Coding for the Modular Surface
Symbolic coding for a geodesic on the modular surface is givenby the continued fraction expansions of the two ideal endpoints.Closed geodesics are those for which the continued fractionexpansions are periodic. Figure by C. Series
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Symbolic CodingSuspension Flows
Suspension Flows
The geodesic flow on ahyperbolic surface is(semi-)conjugate to asuspension flow over atwo-sided shift of finitetype.
x σx
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Symbolic CodingSuspension Flows
Suspension Flows
The length of a periodicorbit corresponding toa periodic sequence xof period m is
Smh(x) =n∑
i=1
h(σix)
x σx
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Example: The Bernoulli Flow
When the sequencespace is Σ = {0,1}Zand the height functionh depends only on thefirst entry of thesequence, thesuspension flow iscalled a Bernoulli flow.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Example: The Bernoulli Flow
The length of a periodicorbit corresponding toa periodic sequence xof minimal period m is
Smh(x) = mh(0) + (h(1)− h(0)n∑
i=1
xi .
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Example: The Bernoulli Flow
The number N(L) of(periodic) sequencesthat correspond toperiodic orbits of length≤ L satisfies therecursive 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 SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Example: The Bernoulli FlowTransformation to a Renewal Equation. Let β > 0 be the uniquesolution of
e−βh(0) + e−βh(1) = 1.
SetZ (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 withsuccess parameter p = e−βh(1).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Example: The Bernoulli FlowSolution of the Renewal Equation. The recursive equationholds only for L > h(1). For L ∈ [−h(1),h(1)] there is anadditive correction z(L); thus,
Z (L) = EZ (L− h(ξ)) + z(L),
Iteration =⇒
Z (L) =∞∑
n=0
Ez
(L−
n∑i=1
h(ξi)
).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Example: The Bernoulli FlowBlackwell’s Renewal Theorem implies that if h(0)/h(1) 6∈ Qthen there exists C > 0 such that
limL→∞
Z (L) = C =⇒ N(L) ∼ CeβL.
The Law of Large Numbers implies that most sequencescounted in Z (L) look like i.i.d. Bernoulli - p = e−βh(1), and somost have minimal period
≈ L/Eh(ξ).
Therefore, the number N∗(L) of periodic orbits of the Bernoulliflow with minimal period ≤ L satisfies
N∗(L) ∼ CeβL
L/Eh(ξ).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Equidistribution of Periodic OrbitsThe Law of Large Numbers implies that most sequencescounted in N(L) look like i.i.d. Bernoulli - p = e−βh(1).Therefore, most periodic orbits of length ≤ L will be nearlyequi-distributed according to the suspension νp of the Bernoulli- p measure on sequence space.
Theorem: (Bowen; Lalley) Let g : Σh → R be a continuousfunction and for any periodic orbit γ let Avg(g; γ) be the meanvalue of g along γ. Then for any ε > 0, as L→∞,
#{γ : Length(γ) ≤ L and |Avg(g; γ)−∫
g dνp| < ε}#{γ : Length(γ) ≤ L}
−→ 1.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Equidistribution of Periodic OrbitsThis extends to closed geodesics on hyperbolic surfaces.
Theorem: (Bowen; Lalley) Let S = H/Γ be a compacthyperbolic surface and g : S → R a continuous function. Forany periodic orbit γ let Avg(g; γ) be the mean value of g alongγ. Then for any ε > 0, as L→∞,
#{γ : Length(γ) ≤ L and |Avg(g; γ)−∫
g dµ| < ε}#{γ : Length(γ) ≤ L}
−→ 1.
where µ = normalized surface area.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
Cohomology and the CLTTheorem: (Ratner 1972) Let S = H/Γ be a compact hyperbolicsurface and let f : S → R be smooth. If γ(t ; x , θ) is geodesicwith randomly chosen initial point x and direction θ then ast →∞,
1√t
(∫ t
0f (γ(s; u)) ds − t
∫S
f dµ)D−→ Gaussian(0, σ2
f )
and σf > 0 if and only if f is not cohomologous to a constant.
Cohomology: A function g is a coboundary for the geodesicflow if it integrates to 0 on every closed geodesic. A function fis cohomologous to a constant α if f − α is a coboundary.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
CLT for Closed GeodesicsTheorem: (La 1986) Let f : S → R be smooth. If γL is randomlychosen from among all closed geodesics of length ≤ L then asL→∞,
√L(
Avg(f ; γL)−∫
Sf dµ
)D−→ Gaussian(0, σ2
f )
Note: The results of Bowen, Lalley, and Ratner all generalize tosurfaces of variable negative curvature; normalized surfacearea measure is replaced by the maximal entropy invariantmeasure for the geodesic flow.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Renewal TheoryEquidistribution of OrbitsCentral Limit Theory
CLT for Closed GeodesicsTheorem: (La 1986) Let f : S → R be smooth. If γL is randomlychosen from among all closed geodesics of length ≤ L then asL→∞,
√L(
Avg(f ; γL)−∫
Sf dµ
)D−→ Gaussian(0, σ2
f )
Note: The results of Bowen, Lalley, and Ratner all generalize tosurfaces of variable negative curvature; normalized surfacearea measure is replaced by the maximal entropy invariantmeasure for the geodesic flow.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Negative Curvature: Geodesics Typically Self-Intersect
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Law of Large Numbers IQuestion: How many times does a closed geodesic of length≈ L self-intersect? How many times does a random geodesicsegment of length L self-intersect?
Theorem: Let NL be the number of self-intersections of arandom geodesic segment of length L on a hyperbolic surfaceS = H/Γ. Then with probability→ 1 as L→∞,
NL
L2 −→ κS = (π|S|)−1
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Law of Large Numbers IHeuristic Explanation: The geodesic segment γ[0,L] consists ofn = L/δ segments of length δ. Because the geodesic flow ismixing, these look like independent geodesic segments. Thereare
(n2
)pairs. Therefore, NL ∼ κL2 where
κ =12
P{two independent segments meet}/δ2
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Law of Large Numbers IIDenote by xi ∈ S the location of the i th self-intersection. Forany smooth, nonnegative function ϕ : S → R+ define theϕ−weighted self-intersection count by
NϕL =
NL∑i=1
ϕ(xi).
Theorem: With probability→ 1, for each smooth functionϕ : M → R,
NϕL
L2 −→ κϕ := κ
{∫Sϕ(x) dx
/area(S)
}
Consequently, the self-intersections of a random geodesic rayare asymptotically uniformly distributed on the surface.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Law of Large Numbers IIDenote by xi ∈ S the location of the i th self-intersection. Forany smooth, nonnegative function ϕ : S → R+ define theϕ−weighted self-intersection count by
NϕL =
NL∑i=1
ϕ(xi).
Theorem: With probability→ 1, for each smooth functionϕ : M → R,
NϕL
L2 −→ κϕ := κ
{∫Sϕ(x) dx
/area(S)
}
Consequently, the self-intersections of a random geodesic rayare asymptotically uniformly distributed on the surface.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Law of Large Numbers IIDenote by xi ∈ S the location of the i th self-intersection. Forany smooth, nonnegative function ϕ : S → R+ define theϕ−weighted self-intersection count by
NϕL =
NL∑i=1
ϕ(xi).
Theorem: With probability→ 1, for each smooth functionϕ : M → R,
NϕL
L2 −→ κϕ := κ
{∫Sϕ(x) dx
/area(S)
}
Consequently, the self-intersections of a random geodesic rayare asymptotically uniformly distributed on the surface.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Self-Intersections: First-Order AsymptoticsTheorem: (La 1996) For any compact, negatively curvedsurface S = H/Γ there exists a constant κ∗ > 0 such that forany ε > 0, if t is sufficiently large then the number Kt ofself-intersections of a randomly chosen closed geodesic oflength ≤ t satisfies
P{|Kt − t2κ∗| > εt2} < ε
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Self-Intersections: First-Order AsymptoticsTheorem: (La 1996) For any compact, negatively curvedsurface S = H/Γ there exists a constant κ∗ > 0 such that forany ε > 0, if t is sufficiently large then the number Kt ofself-intersections of a randomly chosen closed geodesic oflength ≤ t satisfies
P{|Kt − t2κ∗| > εt2} < ε
Furthermore:
(A) If S has constant negative curvature then κ∗ = κS.(B) In general, the locations of self-intersections areasymptotically distributed according to the (projection to S ofthe) maximal entropy invariant probability measure for thegeodesic flow.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Intersection KernelFix δ > 0 so small that if two geodesic segments of length δintersect transversally then they intersect in only one point.
Intersection Kernel: Nonnegative, symmetric functionHδ : SM × SM → {0,1} that takes value Hδ(u, v) = 1 ifgeodesic segments of length δ based at u, v intersecttransversally, and Hδ(u, v) = 0 if not.
Nm = N(γ[0,m]) =12
m/δ∑i=1
m/δ∑j=1
Hδ(γ(i), γ(j)).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Intersection KernelFix δ > 0 so small that if two geodesic segments of length δintersect transversally then they intersect in only one point.
Intersection Kernel: Nonnegative, symmetric functionHδ : SM × SM → {0,1} that takes value Hδ(u, v) = 1 ifgeodesic segments of length δ based at u, v intersecttransversally, and Hδ(u, v) = 0 if not.
Nm = N(γ[0,m]) =12
m/δ∑i=1
m/δ∑j=1
Hδ(γ(i), γ(j)).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Intersection KernelFix δ > 0 so small that if two geodesic segments of length δintersect transversally then they intersect in only one point.
Intersection Kernel: Nonnegative, symmetric functionHδ : SM × SM → {0,1} that takes value Hδ(u, v) = 1 ifgeodesic segments of length δ based at u, v intersecttransversally, and Hδ(u, v) = 0 if not.
Nm = N(γ[0,m]) =12
m/δ∑i=1
m/δ∑j=1
Hδ(γ(i), γ(j)).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Intersection Kernel
Hδ = 0 Hδ = 1
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Properties of the Intersection KernelKey Lemma: For sufficiently small δ, the constant function 1 isan eigenvector of the integral operator on L2(SM, νL) inducedby the intersection kernel Hδ:
Hδ1(u) :=
∫Hδ(u, v) dνL(v) = δ2κM
Note 1: The kernel Hδ is symmetric and u 7→ Hδ(u, ·) iscontinuous in L2(νL), so the spectrum is a sequence of realeigenvalues converging to 0.
Note 2: Let γ(t ; u) be the geodesic ray with initial tangent vectoru ∈ SM. Then Hδ1(u) = probability that a randomly chosengeodesic segment of length δ intersects γ([0, δ]; u).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Properties of the Intersection KernelKey Lemma: For sufficiently small δ, the constant function 1 isan eigenvector of the integral operator on L2(SM, νL) inducedby the intersection kernel Hδ:
Hδ1(u) :=
∫Hδ(u, v) dνL(v) = δ2κM
Note 1: The kernel Hδ is symmetric and u 7→ Hδ(u, ·) iscontinuous in L2(νL), so the spectrum is a sequence of realeigenvalues converging to 0.
Note 2: Let γ(t ; u) be the geodesic ray with initial tangent vectoru ∈ SM. Then Hδ1(u) = probability that a randomly chosengeodesic segment of length δ intersects γ([0, δ]; u).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Properties of the Intersection KernelKey Lemma: For sufficiently small δ, the constant function 1 isan eigenvector of the integral operator on L2(SM, νL) inducedby the intersection kernel Hδ:
Hδ1(u) :=
∫Hδ(u, v) dνL(v) = δ2κM
Note 1: The kernel Hδ is symmetric and u 7→ Hδ(u, ·) iscontinuous in L2(νL), so the spectrum is a sequence of realeigenvalues converging to 0.
Note 2: Let γ(t ; u) be the geodesic ray with initial tangent vectoru ∈ SM. Then Hδ1(u) = probability that a randomly chosengeodesic segment of length δ intersects γ([0, δ]; u).
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Properties of the Intersection KernelLemma 2: The eigenvalue λ1 = δκ is simple, and all othereigenvalues λ2, λ3, . . . are smaller in absolute value. Therefore,all nonconstant eigenfunctions ψ2, ψ3, . . . are orthogonal to 1:∫
ψj(u) dνL(u) = 0.
Proof: The normalized intersection kernel (δκ)−1Hδ(u, v) is aMarkov kernel with the Doeblin property.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Properties of the Intersection KernelLemma 2: The eigenvalue λ1 = δκ is simple, and all othereigenvalues λ2, λ3, . . . are smaller in absolute value. Therefore,all nonconstant eigenfunctions ψ2, ψ3, . . . are orthogonal to 1:∫
ψj(u) dνL(u) = 0.
Proof: The normalized intersection kernel (δκ)−1Hδ(u, v) is aMarkov kernel with the Doeblin property.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Eigenfunction Expansion
Nm = N(γ([0,m])) =12
m/δ∑i=1
m/δ∑j=1
Hδ(γ(iδ), γ(jδ))
=12
m/δ∑i=1
m/δ∑j=1
∞∑k=1
λkϕk (γ(iδ))ϕk (γ(jδ))
= κm2 +mδ
∞∑k=2
λk
1√m/δ
m/δ∑i=1
ϕk (γ(iδ))
2
.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Eigenfunction Expansion
Nm = N(γ([0,m])) =12
m/δ∑i=1
m/δ∑j=1
Hδ(γ(iδ), γ(jδ))
=12
m/δ∑i=1
m/δ∑j=1
∞∑k=1
λkϕk (γ(iδ))ϕk (γ(jδ))
= κm2 +mδ
∞∑k=2
λk
1√m/δ
m/δ∑i=1
ϕk (γ(iδ))
2
.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Eigenfunction Expansion
Nm = N(γ([0,m])) =12
m/δ∑i=1
m/δ∑j=1
Hδ(γ(iδ), γ(jδ))
=12
m/δ∑i=1
m/δ∑j=1
∞∑k=1
λkϕk (γ(iδ))ϕk (γ(jδ))
= κm2 +mδ
∞∑k=2
λk
1√m/δ
m/δ∑i=1
ϕk (γ(iδ))
2
.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Eigenfunction Expansion
Nm = N(γ([0,m])) =12
m/δ∑i=1
m/δ∑j=1
Hδ(γ(iδ), γ(jδ))
=12
m/δ∑i=1
m/δ∑j=1
∞∑k=1
λkϕk (γ(iδ))ϕk (γ(jδ))
= κm2 +mδ
∞∑k=2
λk
1√m/δ
m/δ∑i=1
ϕk (γ(iδ))
2
.
Central Limit Theorem for geodesic flow implies that eachinterior sum has a limiting Gaussian distribution. These may becorrelated for different k .
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Eigenfunction Expansion
Nm = N(γ([0,m])) =12
m/δ∑i=1
m/δ∑j=1
Hδ(γ(iδ), γ(jδ))
=12
m/δ∑i=1
m/δ∑j=1
∞∑k=1
λkϕk (γ(iδ))ϕk (γ(jδ))
= κm2 +mδ
∞∑k=2
λk
1√m/δ
m/δ∑i=1
ϕk (γ(iδ))
2
.
Unfortunately, there is no justification for the convergence of theeigenfunction expansion.
Steve Lalley Statistics of Geodesics
Hyperbolic SurfacesSymbolic Dynamics
Orbit StatisticsSelf-intersections
Law of Large NumbersIntersection KernelThat’s all!
Steve Lalley Statistics of Geodesics