Statistical Regularities of Geodesics on Negatively Curved ...galton.uchicago.edu/~lalley/Talks/winnepeg-short.pdfSuspension Flows Symbolic Coding for the Modular Surface Symbolic

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



Acknowledgment.

Thanks to SI TANG forassistance in drawingthe figures.




The Plan:

I Hyperbolic SurfacesI Symbolic DynamicsI Orbit StatisticsI Self-intersections




Compact Orientable Surfaces

Surfaces of genus 2,3, . . . admit hyperbolic metrics.Surface of genus 1 admits a flat metric.




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.




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




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).




Geodesics in H are Euclidean circles or lines that intersect theideal boundary (the x−axis) orthogonally.




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




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




Geodesics in H are Euclidean circles or lines that intersect thecircle at∞ orthogonally.




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).




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.




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




Example: Punctured torus

The images of thefundamental polygonobtained by mappingby elements of Γ give atessellation of H bycongruent polygons.




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




Example: Modular SurfaceModular Group: Γ = PSL(2,Z)Modular Surface: H/Γ.




Example: Modular Surface




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




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






Successiveapplications of the shiftmapping on cuttingsequences determinesuccessive sequencesof the geodesic on thesurface.

ABB






Successiveapplications of the shiftmapping on cuttingsequences determinesuccessive sequencesof the geodesic on thesurface.

ABB

BBA






Successiveapplications of the shiftmapping σ on cuttingsequences determinesuccessive sequencesof the geodesic on thesurface.

ABB

BBA

BAB






Periodic sequencescorrespond to closedgeodesics. For aperiodic sequence xthe sequencesx , σx , σ2x , . . . allrepresent the sameclosed geodesic.

ABB

BBA

BAB





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





Suspension Flows

The geodesic flow on ahyperbolic surface is(semi-)conjugate to asuspension flow over atwo-sided shift of finitetype.

x σx





Suspension Flows

The length of a periodicorbit corresponding toa periodic sequence xof period m is

Smh(x) =n∑

i=1

h(σix)

x σx




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.






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 .






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.





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).





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)

).





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).





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.





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.





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.





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.





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.




Law of Large NumbersIntersection KernelThat’s all!

Negative Curvature: Geodesics Typically Self-Intersect





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





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





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.






NϕL =

NL∑i=1

ϕ(xi).


NϕL

L2 −→ κϕ := κ

{∫Sϕ(x) dx

/area(S)

}







NϕL =

NL∑i=1

ϕ(xi).


NϕL

L2 −→ κϕ := κ

{∫Sϕ(x) dx

/area(S)

}






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





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.





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)).







Nm = N(γ[0,m]) =12

m/δ∑i=1

m/δ∑j=1

Hδ(γ(i), γ(j)).







Nm = N(γ[0,m]) =12

m/δ∑i=1

m/δ∑j=1

Hδ(γ(i), γ(j)).





Intersection Kernel

Hδ = 0 Hδ = 1





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).






Hδ1(u) :=









Hδ1(u) :=








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.





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.





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

.






Nm = N(γ([0,m])) =12

m/δ∑i=1

m/δ∑j=1


=12

m/δ∑i=1

m/δ∑j=1

∞∑k=1


= κm2 +mδ

∞∑k=2

λk

1√m/δ

m/δ∑i=1

ϕk (γ(iδ))

2

.






Nm = N(γ([0,m])) =12

m/δ∑i=1

m/δ∑j=1


=12

m/δ∑i=1

m/δ∑j=1

∞∑k=1


= κm2 +mδ

∞∑k=2

λk

1√m/δ

m/δ∑i=1

ϕk (γ(iδ))

2

.






Nm = N(γ([0,m])) =12

m/δ∑i=1

m/δ∑j=1


=12

m/δ∑i=1

m/δ∑j=1

∞∑k=1


= 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 .






Nm = N(γ([0,m])) =12

m/δ∑i=1

m/δ∑j=1


=12

m/δ∑i=1

m/δ∑j=1

∞∑k=1


= 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.






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Statistical Regularities of Geodesics on Negatively Curved ...galton.uchicago.edu/~lalley/Talks/winnepeg-short.pdfSuspension Flows Symbolic Coding for the Modular Surface Symbolic