DISTRIBUTIONAL LIMITS FOR HYPERBOLIC,

INFINITE VOLUME GEODESIC FLOWS

Jon Aaronson and Manfred Denker

1. Geodesic flows on surfaces of constant negative curvature.

Let H denote the two-dimensional hyperbolic space and φ^{t} (t ∈ IR) the geodesic flow on H × TT, where TT is the natural identification of directions. Throughout this note we work with the model of the Poincaré disk instead of the Poincaré upper half-plane (sometimes also called the Lobachevsky plane). So we consider H = {z ∈ C I : |z| < 1}.

Let Γ be a discrete group of isometries of H, and let H/Γ denote the surface defined by Γ equipped with the metric induced by the hyperbolic metric ρ (see [Be]). The space of line elements of H/Γ is X_{Γ} := (H/Γ) × TT = (H × TT)/Γ (also equipped with the induced metric) and the geodesic flow transformations on X_{Γ} are defined by

φ_{Γ}^{t}Γ(ω) = Γφ^{t}(ω).

We consider the measure m =hyperbolic area × normalised Lebesgue measure on H × TT and the corresponding induced measure m_{Γ} on (H/Γ) × TT.

For a compact surface the dynamical system (H/Γ, (φ_{Γ}^{t})_{t∈IR}) is an Anosov system ([An1], [An2]), the measure m_{Γ} is finite and φ_{Γ} is a Bernoulli flow. This is proven by using the existence of expanding and contracting flow invariant foliations (also used by Anosov and Sinai to show that φ_{Γ} is a K-flow (cf. [An2])) and applying the Ornstein isomorphism theory (Ornstein, Weiss [O-W]). Here we are mainly interested in the non-compact case and our result is for conservative geodesic flows φ_{Γ}. The following characterisation of these dynamical systems is given in the work of Hopf and Tsuji (cf. [Ho1], [Ho2], [Ts1] and [Ts2]), which uses methods from potential theory.

Theorem HT.

The geodesic flow φ_{Γ} is either totally dissipative, or conservative and ergodic.

The geodesic flow is conservative iff

∑_{γ∈Γ} e^{−ρ(x,γy)} = ∞.

Typeset by AMS-TEX

This research was supported by a grant from G.I.F., the German-Israel Foundation for Scientific Research and Development. .

A new formulation of these conditions in terms of Brownian motion has been given by Sullivan (see [Su]).

The series ∑_{γ∈Γ} e^{−ρ(x,γy)} is called the Poincaré series. The asymptotic Poincaré series is defined by

a_{Γ}(x, y; t) := ∑_{γ∈Γ;ρ(x,γy)≤t} e^{−ρ(x,γy)} = ∫_{0}^{t} a_{Γ}(x, y; ds),

where a_{Γ}(x, y; ·) denotes the distribution function of the measure on IR putting mass |{γ ∈ Γ : ρ(x, γy) = s}| exp[−s] on the point s ∈ IR. The asymptotic Poincaré series is up to asymptotic equivalence independent of x and y and denoted by a_{Γ}(t). Here we use a_{Γ}(ds) = a_{Γ}(0, 0, ds). For surfaces H/Γ of finite volume, the Poincaré series always diverges, and indeed, a_{Γ}(t) ∝ t (as can be deduced from the ergodic theorem). There are also surfaces of infinite volume with divergent Poincaré series, and the following theorem is shown in [A-S]:

Theorem AS. Any conservative geodesic flow φ_{Γ} is rationally ergodic with return sequence proportional to a_{Γ}(t).

In fact, the proportionality factor turns out to be 8π, when the specific measure dA×dθ (as introduced above) is used. This can be deduced from our proofs below. The proof of the theorem relies on the estimate

∀ x, y ∈ H/Γ, ϵ > 0, ∃ M ∈ IR_{+} ∋

∫_{Δ(y,ϵ)} S_{t}(1_{Δ(x,ϵ)})^{2}dm_{Γ} ≤ M(∫_{Δ(y,ϵ)} S_{t}(1_{Δ(x,ϵ)})dm_{Γ})^{2} ∀ t > 0,

where Δ(z, ϵ) is defined as below. In this note, we prove an extension of this estimate for p-th moments, and this provides the distributional limit theorem:

Theorem AD. Let φ_{Γ} be a conservative geodesic flow, whose return sequence a(t) is regularly varying with index α ∈ [0, 1]. Then for any f ∈ L_{+}^{1}(m_{Γ}) the sequence S_{t}(f)/a(t) converges in distribution to a random variable Y_{α} ∫_{X_{Γ}} f dm_{Γ} where

S_{t}(f) := ∫_{0}^{t} f ∘ φ_{Γ}^{s}ds,

and Y_{α} has the Mittag-Leffler distribution of order α given by

E(exp(zY_{α})) = ∑_{n=0}^{∞} \frac{Γ(1 + α)^{n}z^{n}}{Γ(1 + αn)}

for z ∈ C I when 0 < α ≤ 1 and for |z| < 1 when α = 0.

Here, convergence in distribution means that for any bounded continuous function g : [0, ∞] → IR and any probability measure q ≪ m, we have

∫_{X_{Γ}} g(\frac{S_{t}(f)}{a(t)}) dq → E(g(Y_{α}m_{Γ}(f)))

as t → ∞ where m_{Γ}(f) = ∫_{X} f dm_{Γ}.

The theorem is applicable to Abelian covers of compact surfaces. It follows from estimations of numbers of closed geodesics that a_{Γ}(t) ∝ \sqrt{t} when H/Γ is a ZZ-cover of a compact surface, and that a_{Γ}(t) ∝ log t when H/Γ is a ZZ^{2}-cover of a compact surface (see [Ad-S], [Ph-S], [La], [Po-S]).

The proof of the theorem is by the method of Darling-Kac [D-K], (see also [Aa]). To our knowledge, this is the first attempt to apply this method in the absence of some convenient Frobenius-Perron operator.

2. Preliminaries from hyperbolic geometry and geodesic flows.

Consider the hyperbolic space H := {z ∈ C I : |z| < 1} equipped with the arclength

ds(u, v) := 2\frac{\sqrt{du^{2} + dv^{2}}}{1 − u^{2} − v^{2}} ,

and the area

dA(u, v) := 4\frac{dudv}{(1 − u^{2} − v^{2})^{2}} .

The hyperbolic distance (cf. [Be]) between x, y ∈ H is denoted by

ρ(x, y) = inf \{∫_{γ} ds : γ is an arc joining x and y\} = 2 tanh^{−1} \frac{|x − y|}{|1 − \overline{x}y|}.

Note that with this metric H/Γ has curvature −1, while the metric used in [A-S] gives curvature −4.

For x ∈ H, and ϵ > 0, set

N_{ρ}(x, ϵ) = {y ∈ H : ρ(x, y) < ϵ}, Δ(x, ϵ) := N_{ρ}(x, ϵ) × TT.

Consider the angle set subtended by N_{ρ}(y, ϵ) at 0 ∉ N_{ρ}(y, ϵ),

Λ(y, ϵ) := {θ ∈ [0, 2π] : ∃ r ∈ (0, 1) ∋ ρ(y, re^{iθ}) < ϵ}.

We note that

(1)  Λ(y, ϵ) = \{θ ∈ [0, 2π] : ∥θ − arg y∥ < sin^{−1}(\frac{(1 − |y|^{2}) tanh \frac{ϵ}{2}}{|y|(1 − tanh^{2} \frac{ϵ}{2})} )\},

4 JON AARONSON AND MANFRED DENKER

where ∥θ∥ := θ ∧ (2π − θ) θ ∈ [0, 2π). In order to see this, let δ = tanh \frac{ϵ}{2}. Then

N_{ρ}(y, ϵ) = B(\frac{(1 − δ^{2})y}{1 − δ^{2}|y|^{2}} , \frac{δ(1 − |y|^{2})}{1 − δ^{2}|y|^{2}})

where B(x, r) is the Euclidean ball of radius r,

B(x, r) = {y ∈ C I : |x − y| < r}.

We’ll write

∥Λ(y, ϵ)∥ = 2 sin^{−1}(\frac{(1 − |y|^{2}) tanh \frac{ϵ}{2}}{|y|(1 − tanh^{2} \frac{ϵ}{2})} )

∼ \frac{2(1 − |y|^{2}) tanh \frac{ϵ}{2}}{1 − tanh^{2} \frac{ϵ}{2}} as |y| → 1.

We also need the following fact: Suppose that x, y ∈ H, |y| > |x|, and ∥ arg y − arg x∥ = θ, then

(2)  ρ(0, y) ≥ ρ(0, x) + ρ(x, y) − \frac{2θ}{1 − |x|^{2}} .

This can easily be seen as follows: Let x′ ∈ H, |x′| = |x|, and arg x′ = arg y, then

ρ(0, y) = ρ(0, x′) + ρ(x′, y) ≥ ρ(0, x) + ρ(x, y) − ρ(x′, x),

and clearly

ρ(x′, x) ≤ \frac{2θ}{1 − |x|^{2}} .

Finally, we mention a last evident fact:

(3)  A(N(x, ϵ)) ∼ πϵ^{2} as ϵ → 0.

3. Proof of the theorem

Let A ∈ B(X_{Γ}), p ≥ 1 and t > 0. Define a^{A}(p, t) : X_{Γ} → IR_{+} by

a^{A}(p, t) = ∫ ... ∫_{0<t_{1}<...<t_{p}<t} ∏_{ν=1}^{p} 1_{A} ∘ φ_{Γ}^{t_{ν}} dt_{1} … dt_{p}.

Then

(S_{t}(1_{A}))^{p} = p!a^{A}(p, t),

and

a^{A}(p + 1, t)(ω) = ∫_{0}^{t} 1_{A}(φ_{Γ}^{s}ω)a^{A}(p, t − s)(φ_{Γ}^{s}ω)ds.

Set

\overline{a}^{A}(p, t) = ∫_{A} a^{A}(p, t)dm_{Γ},

and, for λ > 0,

u^{A}(p, λ) = ∫_{0}^{∞} a^{A}(p, t)e^{−λt}dt
\overline{u}^{A}(p, λ) = ∫_{0}^{∞} \overline{a}^{A}(p, t)e^{−λt}dt.

In case A = Δ(x, ϵ) = N_{ρ}(x, ϵ) × TT we shall omit the index A. Also note that in this case (see [A-S] or the proof of the Geometric Lemma below)

\overline{a}^{Δ}(1, t) ∼ 8πA(N)^{2}a_{Γ}(t).

Our goal is the

Main Lemma.

(4)  ∃M > 0 ∋ ∀t > 0, p ≥ 1 \overline{a}^{Δ}(p, t) ≤ M^{p}a_{Γ}(t)^{p}

and

(5)  \overline{u}^{Δ}(p, λ) ∼ \frac{1}{λ}\overline{u}(λ)^{p}m_{Γ}(Δ)^{p+1} as λ → 0 ∀ p ∈ IN

where \overline{u}(λ) = \frac{1}{m_{Γ}(Δ)^{2}} ∫_{0}^{∞} m_{Γ}(Δ ∩ φ_{Γ}^{−t}Δ)e^{−λt}dt.

We first show how Theorem AD is obtained from the Main Lemma.

Proof of Theorem AD. Since a_{Γ}(t) is regularly varying, it has a representation a_{Γ}(t) = t^{α}L(t) for some α > 0 and some slowly varying function L. An easy calculation using the Laplace transform in (5) shows for p ≥ 1

∫_{Δ}(S_{t}(1_{Δ}))^{p}dm_{Γ} ∼ m_{Γ}(Δ)^{p+1}p!\frac{Γ(1 + α)^{p}}{Γ(1 + pα)}a_{Γ}(t)^{p}

as t → ∞. Since the bound in (4) is uniform over p we have

∫_{Δ} e^{zS_{t}(1_{Δ})a_{Γ}^{−1}(t)}dm_{Γ} = ∑_{p=0}^{∞} \frac{1}{p!} ∫_{Δ}(zS_{t}(1_{Δ})a_{Γ}^{−1}(t))^{p}dm_{Γ}

→ m_{Γ}(Δ) ∑_{p=0}^{∞} z^{p}m_{Γ}(Δ)^{p} \frac{Γ(1 + α)^{p}}{Γ(1 + pα)} = m_{Γ}(Δ)E(exp(zY_{α})).

The proof of the Main Lemma follows from the following two facts:

Geometric Lemma. Let x ∈ H. There is a function η(ϵ) → 0 as ϵ → 0 such that for every ϵ > 0, p ≥ 1 and all t sufficiently large

\overline{a}(p + 1, t) = e^{±η(ϵ)}8πm_{Γ}(Δ) ∫_{0}^{t} \overline{a}(p, t − s)a_{Γ}(ds),

where Δ = Δ(x, ϵ).

Probabilistic Lemma. Let ϵ_{0} > 0, and let Δ = Δ(x, ϵ_{0}). Then for A ∈ B(X_{Γ}) ∩ Δ and p ≥ 1

∫_{A} u^{A}(p, λ)dm_{Γ} ∼ \frac{m_{Γ}(A)^{p+1}}{m_{Γ}(Δ)^{p+1}} ∫_{Δ} u^{Γ}(p, λ)dm_{Γ}

as λ → 0.

Proof of the Main Lemma. (4) follows immediately from an iterated application of the Geometric Lemma.

In particular, choosing A = N(x, ϵ′) × TT for ϵ′ small,

\overline{u}^{A}(p + 1, λ) = e^{±η(ϵ′)}8πm_{Γ}(A) ∫_{0}^{∞} ∫_{0}^{t} \overline{a}^{A}(p, t − s)e^{−λt}a_{Γ}(ds)dt
= e^{±η(ϵ′)}8πm_{Γ}(A) ∫_{0}^{∞} ∫_{0}^{∞} \overline{a}^{A}(p, r)e^{−λ(r+s)}a_{Γ}(ds)dr
= e^{±η(ϵ′)}8πm_{Γ}(A)\overline{u}^{A}(p, λ) ∫_{0}^{∞} e^{−λs}a_{Γ}(ds).

Iterating this estimate gives

\overline{u}^{A}(p, λ) = e^{±pη(ϵ′)}m_{Γ}(A)^{p}\overline{u}^{A}(0, λ)(8π ∫_{0}^{∞} e^{−λs}a_{Γ}(ds))^{p}.

By the Probabilistic Lemma,

\overline{u}(p, λ) ∼ \frac{m_{Γ}(Δ)^{p+1}}{m_{Γ}(A)^{p+1}} \overline{u}^{A}(p, λ)

= e^{±pη(ϵ′)} \frac{m_{Γ}(Δ)^{p+1}}{m_{Γ}(A)} \overline{u}^{A}(0, λ)(8π ∫_{0}^{∞} e^{−λs}a_{Γ}(ds))^{p}.

The lemma now follows from

∫_{0}^{∞} e^{−λt}a_{Γ}(dt) = λ ∫_{0}^{∞} a_{Γ}(t)e^{−λt}dt

∼ \frac{λ}{8πm_{Γ}(Δ)^{2}} ∫_{0}^{∞} ∫_{0}^{t} m(Δ ∩ ϕ_{Γ}^{−s}(Δ))dse^{−λt}dt

= \frac{1}{8πm_{Γ}(Δ)^{2}} ∫_{0}^{∞} m(Δ ∩ ϕ_{Γ}^{−s}(Δ))e^{−λs}ds

and

\overline{u}^{A}(0, λ) = ∫_{0}^{∞} \overline{a}(0, t)e^{−λt}dt = m_{Γ}(Δ) ∫_{0}^{∞} e^{−λt}dt.

4. Proof of the Geometric Lemma.

For \underline{γ} ∈ Γ^{p} (resp. \underline{t} ∈ IR^{p}) we denote its coordinates by γ_{k} (resp. t_{k}), k = 1, ..., p. For t ∈ IR_{+} define

I_{p}(t) = \{\underline{t} ∈ IR^{p} : 0 < t_{1} < ... < t_{p} < t\}.

Let ϵ > 0 be fixed and N = Δ × TT as before, where Δ = Δ(x, ϵ). We assume ϵ to be sufficiently small.

First observe that

\overline{a}(p, t) = ∫_{Δ} a(p, t)dm_{Γ}

= ∫_{Δ} ∫_{I_{p}(t)} ∏_{ν=1}^{p} 1_{Δ} ∘ φ_{Γ}^{t_{ν}} d\underline{t}dm_{Γ}

= ∑_{\underline{γ}∈Γ^{p}} ∫_{Δ} ∫_{I_{p}(t)} ∏_{ν=1}^{p} 1_{γ_{ν}Δ} ∘ φ^{t_{ν}} d\underline{t}dm

= ∑_{\underline{γ}∈Γ^{p}} ∫_{N} ∫_{0}^{1} ∫_{I_{p}(t)} ∏_{ν=1}^{p} 1_{γ_{ν}N×TT} ∘ φ^{t_{ν}} (z, θ)d\underline{t}dθdA(z).

Set

ψ_{p}(t, z) = ∑_{\underline{γ}∈Γ^{p}} ∫_{0}^{1} ∫_{I_{p}(t)} ∏_{ν=1}^{p} 1_{γ_{ν}N×TT} ∘ φ^{t_{ν}} (z, θ)d\underline{t}dθ

= ∑_{\underline{γ}∈Γ^{p}} ∫_{0}^{1} ∫_{I_{p}(t)} ∏_{ν=1}^{p} 1_{φ_{z}^{−1}γ_{ν}N}(tanh t_{ν}e^{iθ})d\underline{t}dθ

= ∫_{0}^{1} ∫_{I_{p}(t)} ∑_{\underline{γ}∈Γ^{p}} ∏_{ν=1}^{p} 1_{φ_{z}^{−1}γ_{ν}N}(tanh t_{ν}e^{iθ})d\underline{t}dθ, (6)

so \overline{a}(p, t) = ∫_{N} ψ_{p}(t, z)A(dz).

Next consider

Γ_{0} := \{\underline{γ} ∈ Γ^{p} : ∫_{0}^{1} ∫_{I_{p}(t)} ∏_{ν=1}^{p} 1_{φ_{z}^{−1}γ_{ν}N}(tanh t_{ν}e^{iθ})d\underline{t}dθ > 0\}.

Since ϵ is so small so that {γN}_{γ∈Γ} are disjoint and since φ_{z} is ρ-preserving, it follows that if (γ_{1}, …, γ_{p}) ∈ Γ_{0}, then

ρ(γ_{k+1}(x), z) ≥ ρ(γ_{k}(x), z) ∀ 1 ≤ k ≤ p − 1,

Denote Λ_{γ} = Λ(φ_{z}^{−1}γ(x), ϵ), the angle set subtended by φ_{z}^{−1}γN at 0, then by (1),

∥Λ_{γ}∥ ∼ 2ϵ(1 − |φ_{z}^{−1}γ(x)|^{2}) as γ → ∞, and ϵ → 0.

Set

Γ_{p}^{±} := \{\underline{γ} = (γ_{1}, ..., γ_{p}) ∈ Γ^{p} : ρ(γ_{k+1}(x), z) ≥ ρ(γ_{k}(x), z),

∥ arg φ_{z}^{−1}γ_{k}(x) − arg φ_{z}^{−1}γ_{k+1}(x)∥ ≤ \frac{∥Λ_{γ_{k}}∥ ± ∥Λ_{γ_{k+1}}∥}{2} ∀ 1 ≤ k ≤ p − 1\}.

We claim that

Γ_{p}^{−} ⊆ Γ_{0} ⊆ Γ_{p}^{+}.

Clearly, if (γ_{1}, …, γ_{p}) ∈ Γ_{0}, then

⋂_{k=1}^{p} Λ_{γ_{k}} ≠ ∅,

and hence for 1 ≤ k ≤ p − 1,

∥ arg φ_{z}^{−1}γ_{k}(x) − arg φ_{z}^{−1}γ_{k+1}(x)∥ ≤ \frac{∥Λ_{γ_{k}}∥ + ∥Λ_{γ_{k+1}}∥}{2}.

On the other hand, if \underline{γ} = (γ_{1}, …, γ_{p}) ∈ Γ_{p}^{−}, then all balls ϕ_{z}^{−1}γ_{ν}N lie in the shadow of the first ball (ν = 1), and hence γ ∈ Γ_{0}.

Setting, for p ≥ 1, \underline{t} = (t_{1}, ..., t_{p}), 0 < t_{1} < ... < t_{p},

Γ_{0}(\underline{t}) = \{\underline{γ} = (γ_{1}, ..., γ_{p}) ∈ Γ_{0} : |ρ(z, γ_{k}(x)) − t_{k}| ≤ ϵ ∀k\}
Γ_{p}^{±}(\underline{t}) = \{\underline{γ} = (γ_{1}, ..., γ_{p}) ∈ Γ_{p}^{±} : |ρ(z, γ_{k}(x)) − t_{k}| ≤ ϵ ∀k\}

we have from (6) that

ψ_{p}(t, z) = ∫_{0}^{1} ∫_{I_{p}(t)} ∑_{\underline{γ}∈Γ_{0}(\underline{t})} ∏_{ν=1}^{p} 1_{φ_{z}^{−1}γ_{ν}N}(tanh t_{ν}e^{iθ})d\underline{t}dθ.

It follows that

ψ_{p}^{−}(t, z) ≤ ψ_{p}(t, z) ≤ ψ_{p}^{+}(t, z),

where

ψ_{p}^{±}(t, z) = ∫_{0}^{1} ∫_{I_{p}(t)} ∑_{\underline{γ}∈Γ_{p}^{±}(\underline{t})} 1_{φ_{z}^{−1}γ_{p}N}(tanh t_{p}e^{iθ})d\underline{t}dθ.

DISTRIBUTIONAL LIMITS 9

For β ∈ Γ, let

Γ^{±}(β) ={γ ∈ Γ : ρ(z, γ(x)) ≥ ρ(z, β(x)) − ϵ,

| arg φ_{z}^{−1}γ(x) − arg φ_{z}^{−1}β(x)| < \frac{∥Λ_{β}∥ ± ∥Λ_{γ}∥}{2}\}.

It follows that for \underline{t}′ = (t_{1}, ..., t_{p−1}), \underline{t} = (t_{1}, ..., t_{p}), 0 < t_{1} < ... < t_{p}

Γ_{p}^{±}(\underline{t}) := \{\underline{γ} ∈ Γ^{p} : (γ_{1}, ..., γ_{p−1}) ∈ Γ^{±}(\underline{t}′), γ_{p} ∈ Γ^{±}(γ_{p−1}), |ρ(z, γ_{p}(x) − t_{p}| ≤ ϵ\}. Next

ψ_{p}^{±}(t, z) = ∫_{0}^{1} ∫_{I_{p}(t)} ∑_{\underline{γ}∈Γ_{p}^{±}(\underline{t})} 1_{φ_{z}^{−1}γ_{p}N}(tanh t_{p}e^{iθ})d\underline{t}dθ

= ∫_{I_{p−1}(t)} ∑_{\underline{γ}∈Γ_{p−1}^{±}(\underline{t}′)} ∫_{0}^{1} ∫_{tanh t_{p−1}}^{tanh t} ∑_{{γ∈Γ^{±}(γ_{p−1}) \atop |ρ(z,γ(x))−tanh^{−1} r|≤ϵ}} 1_{φ_{z}^{−1}γN}(re^{iθ})\frac{drdθ}{1 − r^{2}} d\underline{t}′

∼ ∫_{I_{p−1}(t)} ∑_{\underline{γ}∈Γ_{p−1}^{±}(\underline{t}′)} ∫_{N_{ρ}(0,t)\N_{ρ}(0,t_{p−1})} \frac{1 − |ω|^{2}}{4|ω|} ∑_{γ∈Γ^{±}(γ_{p−1})} 1_{φ_{z}^{−1}γN}(ω)dA(ω)d\underline{t}′

∼ A(N) ∫_{I_{p−1}(t)} ∑_{\underline{γ}∈Γ_{p−1}^{±}(\underline{t}′)} ∑_{{γ∈Γ^{±}(γ_{p−1}) \atop ρ(z,γ(x))≤t±ϵ}} e^{−ρ(z,γ(x))}d\underline{t}′

=: A(N)Φ_{p}^{±}(t, z). The inductive step on Φ_{p}^{±} is (with \underline{t}′′ = (t_{1}, ..., t_{p−2}))

Φ_{p}^{±}(t, z) = ∫_{I_{p−1}(t)} ∑_{\underline{γ}∈Γ_{p−1}^{±}(\underline{t}′)} ∑_{{γ∈Γ^{±}(γ_{p−1}) \atop ρ(z,γ(x))≤t±ϵ}} e^{−ρ(z,γ(x))}d\underline{t}′

= ∫_{I_{p−2}(t)} ∑_{\underline{γ}∈Γ_{p−2}^{±}(\underline{t}′′)}(∫_{t_{p−2}}^{t} ∑_{{β∈Γ^{±}(γ_{p−2}) \atop ρ(z,|β(x))−τ|≤ϵ}} ∑_{{γ∈Γ^{±}(β)) \atop ρ(z,γ(x))≤t±ϵ}} e^{−ρ(z,γ(x))}dτ)d\underline{t}′′

∼ 2ϵ ∫_{I_{p−2}(t)} ∑_{\underline{γ}∈Γ_{p−2}^{±}(\underline{t}′′)}( ∑_{{β∈Γ^{±}(γ_{p−2}) \atop ρ(z,β(x))≤t±ϵ}} ∑_{{γ∈Γ^{±}(β)) \atop ρ(z,γ(x))≤t±ϵ}} e^{−ρ(z,γ(x))})d\underline{t}′′.

Fixing β ∈ Γ, we have

∑_{{γ∈Γ^{±}(β)) \atop ρ(z,γ(x))≤t±ϵ}} e^{−ρ(z,γ(x))} = ∑_{{κ∈β^{−1}Γ^{±}(β)) \atop ρ(z,βκ(x))≤t±ϵ}} e^{−ρ(z,βκ(x))}

∼ e^{−ρ(z,β(x))} ∑_{{κ∈β^{−1}Γ^{±}(β)) \atop ρ(x,κ(x))≤t−ρ(z,β(x)±2ϵ}} e^{−ρ(x,κ(x))}

by (2).

Let β ∈ Γ and Ω(β) denote the interval in S^{1} such that for ξ ∈ Ω(β) the ray β^{−1}(0)ξ intersects β^{−1}N_{ρ}(β(x), ϵ). It is easily seen that

(7)  β^{−1}Γ^{±}(β) ∼ {γ ∈ Γ : arg γ(x) ∈ Ω(β)} where |Ω(β)| → θ(ϵ) as |β(x)| → 1,

where 2π|Ω(β)| denotes the arc length of Ω(β) and where (this can be deduced from cosh(ϵ) = 2/|ξ − η| where ξ, η are the endpoints of a geodesic tangent to the geodesic ball of radius ϵ and center 0)

(8)  θ(ϵ) ∼ 4ϵ as ϵ → 0.

It has been shown in [A-S], that for a suitable measure μ on H/Γ

\frac{1}{a_{Γ}(t)} ∫_{0}^{t} 1_{N_{ρ}(z,ϵ)×TT} (y, ·) ∘ φ^{−s}ds → μ(N)

weakly in L^{2}(TT). By standard arguments it follows from this that

μ(N)a_{Γ}(t)|I| ∼ ∫_{I} S_{t}(1_{Δ})dt

∼ ∑_{γ:arg(γ)∈I±ϵ;ρ(0,γ(0))≤t} \frac{1}{4}(1 − |γ(0)|^{2})μ(N).

Therefore

∑_{{κ∈β^{−1}Γ^{±}(β)) \atop ρ(x,κ(x))≤t−ρ(z,β(x))±2ϵ}} e^{−ρ(x,κ(x))} ∼ |Ω(β)|a_{Γ}(t − ρ(z, β(x)) ± 2ϵ).

∑_{{κ∈γ{−1 \atop p−2}Γ^{±}(γ_{p−2})) \atop ρ(x,κ(x))≤t−ρ(z,β(x)±2ϵ}} e^{−ρ(x,κ(x))} ∼ |Ω(γ_{p−2})|a_{Γ}(t − ρ(z, β(x)) ± 2ϵ). (9)

Using (3), (7)–(9) we obtain (with ρ_{0} = ρ(z, β(x)))

Φ_{p}^{±}(t, z)

= 2ϵ ∫_{I_{p−2}(t)} ∑_{\underline{γ}∈Γ_{p−2}^{±}(\underline{t}′)}( ∑_{{β∈Γ^{±}(γ_{p−2}) \atop ρ_{0}≤t±ϵ}} e^{−ρ_{0}}|Ω(β)|a_{Γ}(t − ρ_{0} ± 2ϵ))d\underline{t}′′

∼ 8ϵ^{2} ∫_{I_{p−2}(t)} ∑_{\underline{γ}∈Γ_{p−2}^{±}(\underline{t}′)}( ∑_{{β∈Γ^{±}(γ_{p−2}) \atop ρ_{0}≤t±ϵ}} e^{−ρ_{0}}a_{Γ}(t − ρ_{0} ± 2ϵ))d\underline{t}′′

∼ 2ϵ ∫_{I_{p−2}(t)} ∑_{\underline{γ}∈Γ_{p−2}^{±}(\underline{t}′)} ∑_{{κ∈γ{−1 \atop p−2}Γ^{±}(γ_{p−2}) \atop ρ(x,κ(x))≤t−ρ_{0}±ϵ}} e^{−ρ(x,κ(x))} ∑_{β′} e^{−ρ(x,γ_{p−2}^{−1}β′(x))}d\underline{t}′′

= 8ϵ^{2}m_{Γ}(Δ)^{−1} ∫_{0}^{t} \overline{a}(p − 1, t − s)a_{Γ}(ds),

DISTRIBUTIONAL LIMITS 11

where

∑_{β′} e^{−ρ(x,γ_{p−2}^{−1}β′(x))} = ∑_{{γ{−1 \atop p−2}β′∈γ{−1 \atop p−2}Γ^{±}(γ_{p−2}) \atop ρ(x,γ{−1 \atop p−2}β′(x))≤t−ρ(x,g{−1 \atop p−2}(z))±ϵ}} e^{−ρ(x,γ_{p−2}^{−1}β′(x))}.

The lemma follows from m_{Γ}(Δ) = A(N) ∼ πϵ^{2} (see (3)).

5. Proof of the Probabilistic Lemma.

To prove this, we first show for Δ := N_{ρ}(x, ϵ) × TT that

(10)  ∃ M_{p} ∋ ∫_{Δ} u^{Δ}(p, λ)^{2}dm_{Γ} ≤ M_{p}(∫_{Δ} u^{Δ}(p, λ)dm_{Γ})^{2} ∀ λ > 0.

To see this,

∫_{Δ} u^{Δ}(p, λ)^{2}dm_{Γ} = ∫_{Δ} ∫_{0}^{∞} ∫_{0}^{∞} a(p, s)a(p, t)e^{−λs}e^{−λt}dsdtdm_{Γ}
= ∫_{0}^{∞} ∫_{0}^{∞}(∫_{Δ} a(p, s)a(p, t)dm_{Γ})e^{−λs}e^{−λt}dsdt.

Using the Geometric Lemma we have ∫_{Δ} a(p, s)a(p, t)dm_{Γ} ≤ (∫_{Δ} a(p, s)^{2}dm_{Γ})^{\frac{1}{2}} (∫_{Δ} a(p, t)^{2}dm_{Γ})^{\frac{1}{2}}

= p!^{−2}\sqrt{∫_{Δ} S_{s}^{2p}dm_{Γ} ∫_{Δ} S_{t}^{2p}dm_{Γ}} = \frac{(2p)!}{p!^{2}} \sqrt{\overline{a}(2p, s)\overline{a}(2p, t)}

≤ M_{p}a_{Γ}(s)^{p}a_{Γ}(t)^{p} ≤ M′_{p} ∫_{Δ} a(p, s)dm_{Γ} ∫_{Δ} a(p, t)dm_{Γ}.

Substituting in the above gives (10).

It suffices to show that for A ∈ B, A ⊂ Δ,

(11)  \frac{u^{A}(p, λ)}{∫_{Δ} u^{Δ}(p, λ)dm_{Γ}} ⟶_{λ→0} \frac{m_{Γ}(A)^{p}}{m_{Γ}(Δ)^{p+1}} weakly in L^{2}(Δ).

We begin by showing this for A = Δ. Using (10), we get that for fixed p ≥ 1

(12) sup_{λ>0}∥\frac{u^{Δ}(p, λ)}{∫_{Δ} u^{Δ}(p, λ)dm_{Γ}} ∥_{L^{2}(Δ)} < ∞.

Given λ_{k} → 0, ∃ a subsequence λ′_{k} → 0 and h ∈ L^{2}(Δ) such that

\frac{u^{Δ}(p, λ′_{k})}{∫_{Δ} u^{Δ}(p, λ′_{k})dm_{Γ}} ⟶_{k→∞} h,

12 JON AARONSON AND MANFRED DENKER

and ∃ a further subsequence λ′′_{k} → 0 such that

|∫_{Δ}(\frac{u^{Δ}(p, λ′′_{k})}{∫_{Δ} u^{Δ}(p, λ′′_{k})dm_{Γ}} − h)(\frac{u^{Δ}(p, λ′′_{ℓ})}{∫_{Δ} u^{Δ}(p, λ′′_{ℓ})dm_{Γ}} − h)dm_{Γ}| < \frac{1}{2^{ℓ}} ∀ k < ℓ,

whence

\frac{1}{N} ∑_{k=1}^{N}(\frac{u^{Δ}(p, λ′′_{k})}{∫_{Δ} u^{Δ}(p, λ′′_{k})dm_{Γ}} − h) → 0 a.e. as N → ∞,

and

\frac{1}{N} ∑_{k=1}^{N} \frac{u^{Δ}(p, λ′′_{k})}{∫_{Δ} u^{Δ}(p, λ′′_{k})dm_{Γ}} → h a.e. as N → ∞.

The set on which this convergence takes place is clearly φ_{Γ}-invariant, as is h, whence the convergence is a.e. on X_{Γ}, and h is constant. Since, clearly ∫_{Δ} hdm_{Γ} = 1, we have that h = \frac{1}{m_{Γ}(Δ)}.

Now fix A ∈ B, A ⊂ Δ. By the ratio theorem

(13)  \frac{u^{A}(p, λ)}{u^{Δ}(p, λ)} ⟶_{λ→0} \frac{m_{Γ}(A)^{p}}{m_{Γ}(Δ)^{p}} a.e.

Also, we have, by (12) that

sup_{λ>0}∥\frac{u^{A}(p, λ)}{∫_{Δ} u^{Δ}(p, λ)dm_{Γ}} ∥_{L^{2}(Δ)} ≤ sup_{λ>0}∥\frac{u^{Δ}(p, λ)}{∫_{Δ} u^{Δ}(p, λ)dm_{Γ}} ∥_{L^{2}(Δ)} < ∞,

whence, as above, ∀ λ_{k} → 0, ∃ a subsequence λ′_{k} → 0 and h ∈ L^{2}(Δ) such that

\frac{1}{N} ∑_{k=1}^{N} \frac{u^{A}(p, λ′_{k})}{∫_{Δ} u^{Δ}(p, λ′_{k})dm_{Γ}} → h a.e. as N → ∞.

Note that λ′_{k} → 0 can be chosen so that in addition,

\frac{1}{N} ∑_{k=1}^{N} \frac{u^{Δ}(p, λ′_{k})}{∫_{Δ} u^{Δ}(p, λ′_{k})dm_{Γ}} → \frac{1}{m_{Γ}(Δ)} a.e. as N → ∞,

whence, by (13)

h = \frac{m_{Γ}(A)^{p}}{m_{Γ}(Δ)^{p+1}} .

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Aaronson: School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel. E-mail: aaro@math.tau.ac.il

Denker: Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, 37083 Göttingen, Germany E-mail: denker@namu01.gwdg.de