THE POINCARÉ SERIES OF ℂ \ ℤ

Jon Aaronson, Manfred Denker

Abstract. We show that the Poincaré series of the Fuchsian group of deck transformations of ℂ \ ℤ diverges logarithmically. This is because ℂ \ ℤ is a ℤ-cover of the three horned sphere, whence its geodesic flow has a good section which behaves like a random walk on ℝ with Cauchy distributed jump distribution and has logarithmic asymptotic type.

§0 Introduction

Let H := {z ∈ ℂ : |z| < 1} denote unit disc, and let Möb(H) denote the group of Möbius transformations (i.e. bianalytic diffeomorphisms of H). These have the form z ↦ λ\frac{z−α}{1−\overline{α}z} where |λ| = 1 and α ∈ H.

A Fuchsian group is a discrete subgroup of Möb(H). To any torsion free Fuchsian group Γ, there corresponds a hyperbolic Riemann surface which is obtained by endowing H/Γ := { Γ(x) := {γ(x) : γ ∈ Γ} : x ∈ H} with the canonical complex structure.

It is known (see [Ahl]) that any hyperbolic Riemann surface is of this form, the (torsion free) Fuchsian group being unique up to inner conjugacy in Möb(H).

The Poincaré series ([Be]) of the Fuchsian group Γ ⊂ Möb(H) at the point x ∈ H is the function

𝔓_{Γ}(x; s) := ∑_{γ∈Γ}(1 − |γ(x)|)^{s} ≤ ∞ (s > 0).

It is known (see §1) that 𝔓_{Γ}(x; s) < ∞ ∀ s > 1, and the Fuchsian group Γ ⊂ Möb(H) is called of divergence type if 𝔓_{Γ}(x; s) → ∞ as s → 1^{+} for some (and hence all) x ∈ H (see [Ho1] and [T]). For divergence type groups Γ, 𝔓_{Γ}(x; s) ∼ 𝔓_{Γ}(0; s) := 𝔓_{Γ}(s) as s → 1^{+} (see §1).

A Fuchsian group Γ ⊂ Möb(H) which is a lattice (in the sense that its homogeneous space Möb(H)/Γ has finite Haar measure, equivalently H/Γ has finite hyperbolic area) is always of divergence type and indeed 𝔓_{Γ}(s) ∝ \frac{1}{s−1} as s → 1^{+} (i.e. ∃ lim_{s→1^{+}}(s−1)𝔓_{Γ}(s) ∈ ℝ_{+}). This follows from the ergodic theorem for the geodesic flow on Möb(H)/Γ ([Ho1] and [T], see §1).

The Fuchsian group Γ is of divergence type iff its Riemann surface H/Γ has no Green’s function ([Myr] see also [T]).

There are Fuchsian groups Γ of divergence type which are not lattices:

Typeset by AMS-TEX

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

If 𝔇 ⊂ S^{2} := ℂ ∪ {∞} is a domain and |S^{2} \ 𝔇| ≥ 2, then 𝔇 is hyperbolic and has a Green’s function iff log - cap (S^{2} \ 𝔇) > 0 ([Ahl]); the hyperbolic area of 𝔇 being finite iff |S^{2} \ 𝔇| < ∞ ([T]). It follows that Γ(ℂ \ ℤ) is of divergence type but is not a lattice.

In this paper we prove

Theorem.

𝔓_{Γ(ℂ\ℤ)}(s) ∝ log \frac{1}{s − 1} as s → 1^{+}.

The method of proof uses the ergodic theory of hyperbolic geodesic flows.

The geodesic flow may be defined on Möb(H)/Γ by φ_{Γ}^{t}(Γg) := Γgγ^{t} where γ^{t}(z) := \frac{z+tanh(t/2)}{1+tanh(t/2)z} (the equivalent geometric definition is in §1). It evidently preserves Haar measure on Möb(H)/Γ.

It is ergodic on Möb(H)/Γ iff Γ is of divergence type (see [Ho1] and [T]).

It was shown in [A-S] that the geodesic flow is rationally ergodic with a return sequence a_{Γ}(t) satisfying 𝔓_{Γ}(1 + s) ∼ s ∫_{0}^{∞} a_{Γ}(t)e^{−st}dt as s → 0.

To prove the theorem, we show that a_{Γ(ℂ\ℤ)}(t) ∝ log t as t → ∞.

The Riemann surface ℂ \ ℤ appears as a ℤ-cover of the so called three-horned sphere ℂ\{0, 1} by means of the covering map z ↦ e^{2πiz}, the group of deck transformations being {z ↦ z + n : n ∈ ℤ}.

This means that Γ(ℂ \ ℤ) is a normal subgroup of the lattice Γ(ℂ \ {0, 1}) with quotient Γ(ℂ \ ℤ)/Γ(ℂ \ {0, 1}) ≅ ℤ, and our theorem is obtained in the context of the study of normal subgroups of lattices with Abelian quotients.

Definitions.

1) Let d ≥ 1, Γ, Γ̃ be Fuchsian groups and suppose that Γ ◁ Γ̃. We’ll say that Γ has index d in Γ̃ if [Γ̃ : Γ] ≅ ℤ^{d}.

2) We’ll say that a Fuchsian group Γ has lattice index d if it has index d in some lattice.

The Fuchsian group Γ has index d in Γ̃ iff H/Γ is a ℤ^{d}-cover of H/Γ̃.

It is shown in [Re1] that if Γ has index d in cocompact Γ̃ (i.e. where H/Γ̃ is compact), then Γ is of convergence type (i.e. not of divergence type) for d ≥ 3 and

𝔓_{Γ}(s) ≍ \{ \begin{matrix}\sqrt{\frac{1}{s−1}} d = 1, \\ log \frac{1}{s−1} d = 2. \end{matrix}

Here for a(s), b(s) ∈ ℝ_{+} (s > 1), a(s) ≍ b(s) as s → 1^{+} means that

0 < lim inf_{s→1^{+}} \frac{a(s)}{b(s)}, lim sup_{s→1^{+}} \frac{a(s)}{b(s)} < ∞.

The ≍ was improved to ∝ in [Ad-Su], [Ph-Sa], [La], [Po-Sh].

We identify the asymptotic types of the geodesic flows on certain abelian covers of surfaces with finite volume.

The method is to find a good section (defined in §2) for the geodesic flow and compute its asymptotic type using a local limit theorem. Good sections have the property that the flow has the same asymptotic type as the section (see §2).

In §3, we consider Abelian covers of compact surfaces obtaining the results advertised above. Here the good section is as in [G], [Re1] and we use the local limit theorem in [G-H].

In §4 and §5 we prove the theorem by considering normal subgroups of the lattice Γ(ℂ \ {0, 1}) of deck transformations for the covering map of the the 3-horned sphere. This lattice is not cocompact, the good sections are computed explicitly and we use the local limit theorem in [A-D].

In addition to proving the theorem, we also reprove the Lyons-McKean result ([Ly-Mc] see also [Mc-S] and [Re2]) that the subgroup having index 2 is of convergence type.

§1 hyperbolic geodesic flows

The Poincaré plane or hyperbolic space is H equipped with the arclength

ds(u, v) := \frac{2\sqrt{du^{2} + dv^{2}}}{1 − u^{2} − v^{2}} , and the area dA(u, v) := \frac{4dudv}{(1 − u^{2} − v^{2})^{2}} .

This metric gives constant Gaussian curvature −1 (the metric used in [A-S] has curvature −4). The hyperbolic distance between x, y ∈ H is defined 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 in particular, ρ(0, x) = 2 tanh^{−1} |x|, whence 1 − |x| = 1 − tanh \frac{ρ(0,x)}{2} ∼ 2e^{−ρ(0,x)} as |x| → 1 and the Poincaré series of the Fuchsian group Γ ⊂ Möb(H) satisfies 𝔓_{Γ}(x; s) ≍ ∑_{γ∈Γ} e^{−sρ(x,γ(x))} as s → 1 in general, and

𝔓_{Γ}(x; s) ∼ 2 ∑_{γ∈Γ} e^{−sρ(x,γ(x))} as s → 1

for Γ of divergence type.

Note that 𝔓_{Γ}(x; s) ≍ 𝔓_{Γ}(y; s) as s → 1 ∀ x, y ∈ H.

The isometries of (H, ρ) are precisely the Möbius transformations Möb(H) and their complex congugates.

If g is an isometry of H, then A ∘ g ≡ A.

The geodesics in H are arcs in H with the property that the ds-length of any of their segments is the hyperbolic distance between the endpoints of the segment. The geodesics turn out to be diameters of H, and circles orthogonal to ∂H.

The space of line elements of H is H × 𝕋 ≅ Möb(H) by γ ↦ (γ(0), arg γ′(0)), the measure dm(x, θ) = dA(x)dθ on H × 𝕋 corresponding to Haar measure on Möb(H).

The geodesic flow transformations φ^{t} are defined on H × 𝕋 as follows. To each line element ω there corresponds a unique directed geodesic passing through x(ω) whose directed tangent at x(ω) makes an angle θ(ω) with the radius (0, 1).

If t > 0, the point x(φ^{t}ω) is the unique point on the geodesic at distance t from x(ω) in the direction of the geodesic, and if t < 0, the point x(φ^{t}ω) is the unique point on the geodesic at distance −t against the direction of the geodesic.

The angle θ(φ^{t}ω) is the angle made by the directed tangent to the geodesic at the point x(φ^{t}ω) with the radius (0, 1).

There is an important involution χ : H ×𝕋 → H ×𝕋, of direction reversal: x(χω) = x(ω) and θ(χω) = θ(ω) + π.

The isometries act on H × 𝕋 (as differentiable maps) by

g(ω) = (g(x(ω)), θ(ω) + arg g′(x(ω))

and it is not hard to see that χg = gχ and φ^{t}g = gφ^{t}.

Both the geodesic flow, the involution and the isometries preserve the measure dm(x, θ) = dA(x)dθ on H × 𝕋.

Let Γ be a Fuchsian group. The space of line elements of H/Γ is X_{Γ} := (H/Γ) × TT = (H × TT)/Γ and the geodesic flow transformations on X_{Γ} are defined by

φ_{Γ}^{t}Γ(ω) = Γφ^{t}(ω).

Let π_{Γ} : H → H/Γ, \overline{π}_{Γ} : H × 𝕋 → X_{Γ} be the projections π_{Γ}(z) = Γz, \overline{π}_{Γ}(ω) = Γω, and let F be a fundamental domain for Γ in H, e.g.

F^{o} := {x ∈ H : ρ(y, x) < ρ(γ(y), x) ∀ γ ∈ Γ \ {e}}, y ∈ H,

then π_{Γ} and \overline{π}_{Γ} are 1-1 on F and F ×𝕋, and so the measures A_{|F} and m_{|F} induce measures A_{Γ} and m_{Γ} on H/Γ and X_{Γ} = H/Γ × 𝕋 respectively.

Theorem (E.Hopf, M.Tsuji).

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

The geodesic flow φ_{Γ} is conservative iff the Fuchsian group is of divergence type.

We consider here the asymptotic Poincaré series

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

where N_{Γ}(x, y; t) := #{γ ∈ Γ : ρ(x, γ(y)) ≤ t}.

It is shown in [A-S] that any conservative geodesic flow φ_{Γ} is rationally ergodic with return sequence given by a_{Γ}(t) := a_{Γ}(0, 0; t); that for bounded sets A ∈ B

∫_{0}^{t} m(A ∩ φ_{Γ}^{−s}A)ds ∼ m(A)^{2}a_{Γ}(t) as t → ∞,

whence a_{Γ}(x, y; s) ∼ a_{Γ}(t) ∀ x, y ∈ H in the same fundamental domain.

For surfaces H/Γ of finite volume, a_{Γ}(t) ∝ t as can be deduced from the ergodic theorem. The Poincaré series 𝔓_{Γ}(s) (of the divergence type group Γ) can be considered as a

Laplace transform:

𝔓_{Γ}(s) ∼ 2 ∫_{0}^{∞} e^{−su}N_{Γ}(0, 0; du) = ∫_{0}^{∞} e^{−(s−1)u}a_{Γ}(du)

where a_{Γ}(du) := e^{−u}N_{Γ}(du) (and a_{Γ}(t) = ∫_{0}^{t} a_{Γ}(du)).

§2 asymptotic type of flows and good sections

Suppose that T is a conservative, ergodic, measure preserving transformation of the standard, σ-finite measure space (X, B, m) and suppose that h : X → ℝ_{+} is measurable. The special flow over T with height function h is defined on

X_{h} := {(x, y) : x ∈ X : 0 ≤ y ≤ h(x)}

by

φ_{t}(x, y) = (T^{n}x, y + t − h_{n}(x)) h_{n}(x) ≤ y + t < h_{n+1}(x)

where

h_{n}(x) = \{ \begin{matrix}0 n = 0, \\ ∑_{k=0}^{n−1} h(T^{k}x) n ≥ 1, \\ −h_{|n|}(T^{−|n|}x) n ≤ −1. \end{matrix}

The special flow φ_{t} preserves the product measure μ defined on B(X_{h}) by

∫_{X_{h}} gdμ := ∫_{X}(∫_{0}^{h(x)} g(x, y)dy)dm(x).

The conservative, ergodic, measure preserving transformation T is a called a section of the flow φ_{t}. It can be seen that T is measure preserving if, and only if the special flow is measure preserving; but the finiteness of the measure preserved by the section has no connection with the finiteness of the measure preserved by the flow.

Recall from [Kak] that if (X, B, m, T) is a section for φ_{t} with height function h, and A ∈ B_{+}, then T_{A} (the transformation induced by T on A) is also a section for φ_{t}, with the height function

h̃_{A}(x) := ∑_{k=1}^{φ_{A}^{T}(x)−1} h(T^{k}x)

where φ_{A}^{T} : A → ℕ is the first return time function under T, so that T_{A}x = T^{φ_{A}^{T}(x)}x.

We’ll be interested in section transformations which visit sets of finite measure at comparable rates to the flow.

Accordingly, we’ll consider sections for which

(∗)  0 < a := lim inf_{n→∞} \frac{h_{n}}{n} ≤ b := lim sup_{n→∞} \frac{h_{n}}{n} < ∞ a.e.

We’ll call a section of type (∗) good if a = b.

If T is an ergodic probability preserving transformation, and the product measure μ is finite, then h is integrable and h_{n} ∼ an a.e. by Birkhoff’s theorem where a = ∫_{X} hdm. It follows that any ergodic finite measure preserving flow has a good section. We’ll find good sections for certain infinite measure preserving flows.

The good sections concerned will be skew products. Let (X, B, m, T) be an ergodic probability preserving transformation, let h : X → ℝ_{+} be integrable, and let (X_{h}, B(X_{h}), μ, φ) be the special flow over T with height function h. By the ergodic theorem, h_{n} ∼ cn where c = ∫_{X} hdm ∈ ℝ_{+} and so T is a good section for (X_{h}, B(X_{h}), μ, φ) .

Now let G be a locally compact, second countable topological group and let Ψ : X → G be measurable. Define the skew product T_{Ψ} : X × G → X × G by T_{Ψ}(x, a) = (T x, Ψ(x)a) and define h̃ : X × G → ℝ_{+} by h̃(x, a) := h(x).

It follows that h̃_{n}(x, a) = h_{n}(x) ∼ cn and so T_{Ψ} is a good section for the special flow over T_{Ψ} with height function h̃.

Recall from [Fe] that the measurable function A : ℝ_{+} → ℝ_{+} is regularly varying at ∞ if ∀ k > 0, ∃ lim_{t→∞} \frac{A(kt)}{A(t)} ∈ ℝ_{+} (and regularly varying at 0 if t ↦ A(1/t) is regularly varying at ∞).

The limits are always of form lim \frac{A(kt)}{A(t)} = k^{α} for some constant α ∈ ℝ called the index (of regular variation).

In this section, we prove is that if a flow φ_{t} has a good section which is rationally ergodic with regularly varying return sequence, then the flow is also rationally ergodic, and has proportional asymptotic type (see proposition 2.2 below).

For f : X_{h} → ℝ and t > 0, set

S_{t}(f) := ∫_{0}^{t} f ∘ φ_{s}ds,

and for g : X → ℝ and n ≥ 1, set

S_{n}^{T} (g) := ∑_{k=0}^{n−1} g ∘ T^{k}.

Lemma 2.1. Suppose that h ≥ c > 0 and let A = B × I where I = [a, b] ⊂ [0, c], and B ∈ B satisfies m(B) = 1 and \frac{h_{n}}{n} → κ ∈ ℝ_{+} uniformly on B; then for each ϵ > 0, ∃ t_{ϵ} such that for a.e. (x, y) ∈ A and ∀ t > t_{ϵ},

|I|S_{[(1−ϵ)t]}^{T}(1_{B})(x) − 2c ≤ S_{κt}(1_{A})(x, y) ≤ |I|S_{[(1+ϵ)t]}^{T}(1_{B})(x) + 2c.

Proof. For x ∈ B and t > 0, let k_{t}(x) ∈ ℤ_{+} be such that

h_{k_{t}(x)}(x) ≤ t < h_{k_{t}(x)+1}(x).

For x ∈ B:

S_{h_{n}(x)}(1_{A})(x, 0) = ∫_{0}^{h_{n}(x)} 1_{A} ∘ φ_{s}(x, 0)ds

= ∑_{k=0}^{n−1} ∫_{h_{k}(x)}^{h_{k+1}(x)} 1_{A} ∘ φ_{s}(x, 0)ds

= ∑_{k=0}^{n−1} ∫_{h_{k}(x)}^{h_{k+1}(x)} 1_{A}(T^{k}x, s − h_{k}(x))ds

= ∑_{k=0}^{n−1} ∫_{0}^{h(T^{k}x)} 1_{A}(T^{k}x, s)ds

= |I|S_{n}^{T} (1_{B})(x).

If (x, y) ∈ A then x ∈ B and 0 ≤ y ≤ c, so

S_{t}(1_{A})(x, y) := ∫_{0}^{t} 1_{A} ∘ φ_{s}(x, 0)ds ± c
= S_{h_{k_{t}(x)}}(1_{A})(x, 0) ± 2c
= |I|S_{k_{t}(x)}^{T}(1_{B})(x) ± 2c.

By assumption, \frac{h_{n}}{n} → κ uniformly on B, whence \frac{k_{t}}{t} → \frac{1}{κ} uniformly on B, and

∀ ϵ > 0 ∃ t_{ϵ} ∋ k_{κt}(x) = (1 ± ϵ)t ∀ t > t_{ϵ}, x ∈ B.

Thus, for t > t_{ϵ}, (x, y) ∈ A,

S_{κt}(1_{A})(x, y) = |I|S_{k_{κt}(x)}^{T}(1_{B})(x) ± 2c = |I|S_{[(1±ϵ)t]}^{T}(1_{B})(x) ± 2c

uniformly on A. □

Proposition 2.2. Suppose that T is a good section for φ and is rationally ergodic with α-regularly varying return sequence a_{n}(T),

then φ is rationally ergodic, and

a_{n}(φ) ∼ κ^{−α}a_{n}(T)

where \frac{h_{n}}{n} → κ ∈ ℝ_{+} a.e.

Proof.

Let B(T) denote

the collection of sets A ∈ B of positive finite measure with the property that ∃ M > 1 such that

∫_{A}(S_{n}(1_{A}))^{2}dm ≤ M(∫_{A} S_{n}(1_{A})dm)^{2},

and recall that there is a return sequence a_{n}(T) such that

∑_{k=0}^{n−1} m(B ∩ T^{−k}C) ∼ m(B)m(C)a_{n}(T) ∀ A ∈ B(T), B, C ∈ B ∩ A.

It follows that if A ∈ B(T), then (B ∩ A)_{+} ⊂ B(T). Also, if A ∈ B(T), then ⋃_{k=0}^{n} T^{−k}A ∈ B(T) ∀ n ≥ 1.

Thus, by Egorov’s theorem, ∃ B ∈ B(T) such that m(B) = 1 and \frac{h_{n}}{n} → κ ∈ ℝ_{+} uniformly on B. Setting A = B × [0, c], and using lemma 2.1 and regular variation of a_{n}(T),

∫_{A} S_{κt}(1_{A})^{2}dμ = ∫_{B} ∫_{0}^{c} S_{κt}(1_{A})^{2}(x, y)dydm(x)
≤ ∫_{B} ∫_{0}^{c}(cS_{(1+ϵ)t}^{T}(1_{B})(x, 0) + 2c)^{2}dydm(x)
= c^{3} ∫_{B} S_{(1+ϵ)t}^{T}(1_{B})(x, 0)^{2}dm(x) + O(a_{(1+ϵ)t}(T))
≤ Mc^{3}m(B)^{2}a_{(1+ϵ)t}(T)^{2} + O(a_{(1+ϵ)t}(T))
≤ M′a_{(1−ϵ)t}(T)^{2}
≤ M”(∫_{B} ∫_{0}^{c}(cS_{(1−ϵ)t}^{T}(1_{B})(x, 0) − 2c)dydm(x))^{2}

≤ M”(∫_{B} ∫_{0}^{c} S_{κt}(1_{A})(x, y))^{2}dydm(x)

proving rational ergodicity of φ. To get the asymptotic type of φ,

μ(A)^{2}a_{κt}(φ) ∼ ∫_{A} S_{κt}(1_{A})dμ = ∫_{B} ∫_{0}^{c} S_{κt}(1_{A})(x, y)dydm(x)

= ∫_{B} ∫_{0}^{c}(cS_{(1±ϵ)t}^{T}(1_{B})(x, 0) ± 2c)dydm(x)

= c^{2}m(B)^{2}a_{(1±ϵ)t}(T)(1 + o(1))

= μ(A)^{2}(\frac{1 ± ϵ}{κ})^{α}a_{κt}(T)(1 + o(1)).

□ §3 geodesic flows on compact hyperbolic surfaces and their Abelian covers

In this section, we reprove

Theorem 3.1 ([Ad-Su], [Ph-Sa], [La], [Po-Sh]).

If Γ has index d in cocompact Γ̃ then Γ is of convergence type for d ≥ 3 and

a_{Γ}(t) ∝ \{ \begin{matrix}\sqrt{t} d = 1, \\ log t d = 2. \end{matrix}

We shall use the Bowen-Ruelle theorem ([Bo-Ru]) on the special representation of the geodesic flow on a compact, hyperbolic surface by a special flow over a subshift of finite type.

Let M be a compact, hyperbolic surface, let φ_{M} : T M → T M denote the geodesic flow on T M and let χ : T M → T M be the involution of direction reversal.

By Bowen’s theorem, there is a subshift of finite type (Σ, T), a Gibb’s measure m ∈ P(Σ), and a Hölder continuous function h : Σ → ℝ_{+} such that (Σ_{h}, Φ), the special flow of (Σ, T, m) under h ”represents” φ_{M} in the sense that

∃ π : Σ_{h} → T M a Hölder continuous measure theoretic isomorphism such that πΦ = φ_{M}π.

By Rees’s refinement, (Σ, T, m) and π can be chosen so that χ(πΣ) = πΣ.

Now, as in [Re1] and [G] suppose that for some d ≥ 1, V is a ℤ^{d}-cover of M that is V is a complete hyperbolic surface equipped with a covering map p : V → M so that ∃ γ : ℤ^{d} → Möb(V) such that if y ∈ V and p(y) = x ∈ M then p^{−1}{x} = {γ_{n}y : n ∈ ℤ^{d}}.

Since πΣ is a section for φ_{M} with height function h ∘ π^{−1}, we have that p^{−1}πΣ ≅ Σ × ℤ^{d} is a section for φ_{V} with height function h ∘ π^{−1} ∘ p. The section transformation T̃ : p^{−1}πΣ → p^{−1}πΣ satisfies p ∘ T̃ = T ∘ p and T̃ ∘ γ_{n} = γ_{n} ∘ T̃ (n ∈ ℤ^{d}), whence ∃ ψ : Σ → ℤ^{d} Hölder continuous such that Φ̃ is the special flow over (Σ × ℤ^{d}, T_{ψ}) with height function h̃(x, n) = h(x) and π̃ : (Σ×ℤ^{d})h̃ → V is defined by π̃(x, n, t) := φ_{V}^{t} γ_{n}π(x), then π̃ ∘ Φ̃ = φ_{V} ∘ π̃.

It is important to note that ψχ = −ψ whence the distribution of ψ is symmetric about 0.

Evidently T_{ψ} is a good section for φ_{M} and so to prove theorem 3.1, it suffices by proposition 1.2 to establish

Proposition 3.2.

For d = 1, 2, T_{ψ} is rationally ergodic for d = 1, 2 with return sequence given by

a_{n}(T_{ψ}) ∝ \{ \begin{matrix}\sqrt{n} d = 1, \\ log n d = 2. \end{matrix}

Proof.

As in [G-H], We may assume that T is a unilateral subshift of finite type and ψ is Hölder continuous with ∫_{X} ψdm = 0. Let P_{T} be the Frobenius-Perron operator of T, let P_{t}(f) = P_{T} (e^{i⟨t,ψ⟩}f) (t ∈ 𝕋^{d}) and let λ(t) be the maximal eigenvalue of P_{t} for |t| small.

When d = 1, the symmetry of the distribution of ψ about 0 implies that T_{ψ} is conservative (see [At]). The geodesic flow is also conservative (since T_{ψ} is a section) whence ergodic by the Hopf-Tsuji theorem and so T_{ψ} is ergodic. It follows that ψ is not cohomologous to a constant, whence by [G-H], λ(t) = 1 − ct^{2} + o(t^{2}) as t → 0 for some c > 0. Theorem 7.3 in [A-D] now shows that T_{ψ} is rationally ergodic with return sequence a_{n}(T_{ψ}) ∝ \sqrt{n}.

Now let d = 2. As in §3 of [A-D] let

𝔔 := {t ∈ ℝ^{2} : e^{i⟨t,ψ⟩} is cohomologous to a constant},

then 𝔔 is a closed subgroup of ℝ^{2} (proposition 3.8 in [A-D]).

If 𝔔 is discrete, then e^{i⟨t,ψ⟩} is not cohomologous to a constant for arbitrarily small |t| (t ∈ 𝕋^{2}). It follows from the above that λ(x) = 1 − x^{t}Ax + o(∥x∥^{2}) as x → 0 for some A ∈ GL(2, ℝ); and again by theorem 7.3 of [A-D], T_{ψ} is rationally ergodic with return sequence a_{n}(T_{ψ}) ∝ log n.

To finish the proof, we prove that 𝔔 is discrete. It is necessary to eliminate the other possibilities for 𝔔. If 𝔔 = ℝ^{2} then (using symmmetry of ψ) e^{i⟨t,ψ⟩} is a coboundary ∀ t ∈ ℝ^{2} and by [Ha-Ok-Os] ψ is a coboundary whence T_{ψ} is conservative. So is the geodesic flow, which is ergodic (as before by the Hopf-Tsuji theorem) whence T_{ψ} is ergodic contradicting ψ being a coboundary.

If 𝔔 ≠ ℝ^{2} is not discrete, then (again using symmmetry of ψ) by proposition 3.9 of [A-D], ∃ a, b ∈ ℝ^{2} \{0} such that ⟨a, b⟩ = 0, and Hölder continuous functions g, ϕ : X → ℝ such that ψ = (g ∘ T − g)a + ϕb. It follows that ∫_{X} ϕdm = 0 whence (again by [At]) T_{ϕ} is conservative hence (by conservativity and hence ergodicity of the geodesic flow) ergodic, contradicting ψ = (g ∘ T − g)a + ϕb.

The only remaining possibility is that 𝔔 is discrete. □

§4 a section for the geodesic flow on the 3-horned sphere

Let ℝ^{2+} := {z ∈ ℂ : Im z > 0} denote the upper half plane which is conformal to H by z ↦ \frac{z−i}{z+i}. The group of Möbius transformations is given by Möb(ℝ^{2+}) = P SL(2, ℝ) with

the action z ↦ \frac{az+b}{cz+d} where ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℝ).

The 3-horned sphere ℂ \ {0, 1} is conformal to the Riemann surface ℝ^{2+}/Γ(2) where

Γ(2) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ) : ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) mod 2\}.

A fundamental domain for the action of Γ(2) is given by

F := \{z ∈ H : |Re z| < 1, |z ± \frac{1}{2}| > \frac{1}{2}\}.

The 3-horned sphere is \overline{F} under the boundary identifications: \{Re z = 1\} = ϕ(\{Re z = −1\}) where ϕ(z) = z + 2, and \{|z − \frac{1}{2}| = \frac{1}{2}\} = ψ(\{|z + \frac{1}{2}| = \frac{1}{2}\}) where ψ(z) = \frac{z}{2z+1}.

As the fundamental group for the 3-horned sphere is the free group on two generators, it follows from the nature of the identifications that

Γ(2) = F (φ, ψ).

The geodesic flow is defined on \overline{F} × 𝕋. The set

X := {(z, θ) ∈ ∂F × TT : z + ϵe^{2πiθ} ∈ F ∀ ϵ > 0 small}

is a Poincaré section for the geodesic flow on \overline{F} × 𝕋.

The section map τ : X → X is given by

τ(ω) = \{ \begin{matrix}(ϕ(x), θ) π_{+}(ω) < −1, \\ (ψ(x), θ) − 1 < π_{+}(ω) < 0, \\ (ψ^{−1}(x), θ) 0 < π_{+}(ω) < 1, \\ (ϕ^{−1}(x), θ) π_{+}(ω) > 1, \end{matrix}

where (x, θ) = φ_{t_{ω}}(ω) and t_{ω} = inf{t > 0 : φ_{t}(ω) ∈ ∂F }. Here, φ_{t} : H × 𝕋 → H × 𝕋 is the geodesic flow, and π_{+}(ω) := lim_{t→∞} x(φ_{t}ω) ∈ ℝ ∪ {∞}.

We note that this section is infinite measure preserving, and cannot be a section of type (∗) for the geodesic flow on the 3-horned sphere, which has finite area. A good section will be obtained in the sequel by inducing on a set of finite measure.

We’ll be interested in the factor τ_{0} : ℝ → ℝ defined by

τ_{0}(x) = \{ \begin{matrix}ϕ(x) = x + 2 & x < −1, \\ ψ(x) = \frac{x}{2x+1} & −1 < x < 0, \\ ψ^{−1}(x) = \frac{x}{1−2x} & 0 < x < 1, \\ ϕ^{−1}(x) = x − 2 & x > 1 \end{matrix}

and satisfying τ_{0} ∘ π_{+} = π_{+} ∘ τ.

Note that τ_{0} is an even function, and that τ_{0}(−1/x) = −1/τ_{0}(x). We use these relations to get some simplifications.

Define η : (0, 1) × {−1, +1}^{2} → ℝ by

η(x, δ, ϵ) := ϵx^{δ},

and define T : (0, 1) × {−1, +1}^{2} → (0, 1) × {−1, +1}^{2} by

T := η^{−1} ∘ τ_{0} ∘ η.

Defining π^{+} : X → (0, 1) × {−1, +1}^{2} by π^{+} = η^{−1} ∘ π_{+} we have that π^{+} ∘ τ = T ∘ π_{+}.

Proposition 4.1.

T (x, δ, ϵ) = (R(x), δL(x), K(x)ϵ),

where

R(x) = \{ \begin{matrix}\frac{x}{1−2x} 0 < x < \frac{1}{3}, \\ \frac{1}{x} − 2 \frac{1}{3} < x < \frac{1}{2}, \\ 2 − \frac{1}{x} \frac{1}{2} < x < 1, \end{matrix}

and

L(x) = 1 − 2 · 1_{( \frac{1}{3} ,1)}(x), & K(x) = 1_{(0, \frac{1}{2} )}(x) − 1_{( \frac{1}{2} ,1)}(x).

The proof of proposition 4.1 is a routine calculation which is left for the reader.

We’ll induce later on [\frac{1}{5}, \frac{2}{3}] × \{−1, +1\}^{2} since τ_{π_{+}^{−1}[ \frac{1}{5} , \frac{2}{3} ]×\{−1,+1\}^{2}} has an absolutely continuous invariant probability and is therefore a good section for the geodesic flow in the 3-horned sphere.

§5 the ℤ^{d}-covers of the 3-horned sphere and their sections

Since Γ(2) = F (φ, ψ), any γ ∈ Γ(2) is of form

γ = φ^{a_{1}}ψ^{b_{1}} … φ^{a_{n}}ψ^{b_{n}}

for some n ∈ ℕ and a_{1}, …, a_{n}.b_{1}, …, b_{n} ∈ ℤ.

Thus we can define a homomorphism Υ = (Υ_{a}, Υ_{b}) : Γ(2) → ℤ^{2} by

Υ(φ^{a_{1}}ψ^{b_{1}} … φ^{a_{n}}ψ^{b_{n}}) = (∑_{k=1}^{n} a_{k}, ∑_{k=1}^{n} b_{k}).

The Fuchsian group of the ℤ-cover of the 3-horned sphere will be 𝔎_{a} :=Ker Υ_{a}, and the Fuchsian group of the ℤ^{2}-cover of the 3-horned sphere will be 𝔎_{a,b} :=Ker Υ.

Indeed, a fundamental domain for the action of 𝔎_{a} is

F̂_{a} := ( ⋃_{n∈ℤ} φ^{n}\overline{F} )^{o},

and a fundamental domain for the action of 𝔎_{a,b} is

F̂ := ( ⋃_{m,n∈ℤ} ψ^{m}φ^{n}\overline{F} )^{o},

Let \overline{Ψ} = (\overline{Ψ}_{a}, \overline{Ψ}_{b}) : ℝ → ℤ^{2} be defined by

\overline{Ψ}(x) = \{ \begin{matrix}(−1, 0) x < −1, \\ (0, −1) − 1 < x < 0, \\ (0, 1) 0 < x < 1, \\ (1, 0) x > 1. \end{matrix}

The set X × ℤ^{2} is a Poincaré section for the geodesic flow on \overline{F̂} × 𝕋, and the section map τ\overline{Ψ} : X × ℤ^{2} → X × ℤ^{2} is given by

τ\overline{Ψ}(ω, n) = (τω, n + \overline{Ψ}(π_{+}(ω)).

Similarly, the set X × ℤ is a Poincaré section for the geodesic flow on \overline{F̂}_{a} × 𝕋, with section map τ_{\overline{Ψ}_{a}} : X × ℤ → X × ℤ given by

τ_{\overline{Ψ}_{a}}(ω, n) = (τω, n + \overline{Ψ}_{a}(π_{+}(ω)).

As mentioned above, these are not sections of type (∗). We’ll obtain a good section of form

(τ_{\overline{Ψ}_{a}})_{A×ℤ} resp. (τ\overline{Ψ})_{A×ℤ^{2}}

for some set A ∈ B with positive finite measure.

We’ll be interested in simpler factors of τ_{\overline{Ψ}_{a}} and τ\overline{Ψ}.

Define π̃_{d}^{+} : X × ℤ^{d} → (0, 1) × {−1, +1}^{2} × ℤ^{d} by π̃_{d}^{+}(x, n) := (π^{+}(x), n), and define Ψ = (Ψ_{a}, Ψ_{b}) : (0, 1) × {−1, +1}^{2} → ℤ^{2} by

Ψ_{a}(x, δ, ϵ) := ϵ\frac{1 − δ}{2} , & Ψ_{b}(x, δ, ϵ) := ϵ\frac{1 + δ}{2} .

Proposition 5.1.

π̃_{1}^{+} ∘ τ_{\overline{Ψ}_{a}} = T_{Ψ_{a}} ∘ π̃_{1}^{+}.
π̃_{2}^{+} ∘ τ\overline{Ψ} = T_{Ψ} ∘ π̃_{2}^{+}.

The proof of proposition 5.1 is a routine calculation which is left for the reader.

In order to facilitate production of good section, we now consider R : (0, 1) → (0, 1) as a shift.

Write (0, 1) ≅ {A, B, C}^{ℕ} where A = (0, 1/3), B = (1/3, 1/2), C = (1/2, 1), then R ≅ shift, and

L(x) = 2δ_{x_{1},A} − 1, & K(x) = 1 − 2δ_{x_{1},C}.

We have

T^{n}(x, δ, ϵ) = (R^{n}x, δL_{n}(x), ϵK_{n}(x))

where L_{0} = K_{0} = 1, and for n ≥ 1,

L_{n}(x) = ∏_{j=0}^{n−1} L(R^{j}x) = (−1)^{# {1≤k≤n:x_{k}≠A}},

and

K_{n}(x) = ∏_{j=0}^{n−1} K(R^{j}x) = (−1)^{# {1≤k≤n:x_{k}=C}}.

We’ll use the notation

[A_{1}, …, A_{n}] = ⋂_{k=1}^{n} R_{I}^{−(k−1)}A_{k}

where A_{1}, …, A_{n} ⊂ (0, 1).

We have that

U := (\frac{1}{5}, \frac{1}{3}) = [A, A^{c}], & W := (\frac{1}{2}, \frac{2}{3}) = [C, C^{c}],

whence

J := (\frac{1}{5}, \frac{2}{3}) = [A, A^{c}] ∪ B ∪ [C, C^{c}].

Define, for n ≥ 1,

U_{n} := [U, \underbrace{C, …, C}_{(n−1)-times}, B ∪ W], W_{n} := [W, \underbrace{A, …, A}_{(n−1)-times}, U ∪ B],

B_{1} := [B, J],

and for n ≥ 2,

B_{n}^{−} := [B, \underbrace{A, …, A}_{(n−1)-times}, U ∪ B], B_{n}^{+} := [B, \underbrace{C, …, C}_{(n−1)-times}, B ∪ W].

It can be checked that:-

1) α := {U_{n}, W_{n}, B_{1}, B_{n+1}^{−}, B_{n+1}^{+} : n ≥ 1} is a partition of J; that

(2)  φ_{J}^{R} = n on s_{n} := \{ \begin{matrix}U_{n} ∪ W_{n} ∪ B_{n}^{−} ∪ B_{n}^{+} & if n ≥ 2 \\ U_{1} ∪ W_{1} ∪ B_{1} & if n ≥ 1, \end{matrix}

where φ_{J}^{R} : J → ℕ is the first return time function under R, so that R_{J}x = R^{φ_{J}^{R}(x)}x; that

(3)  R_{J}U_{n} = B ∪ W, R_{J}W_{n} = U ∪ B (n ≥ 1);

and

(4)  R_{J}B_{1} = J, R_{J}B_{n}^{+} = B ∪ W, R_{J}B_{n}^{−} = U ∪ B (n ≥ 2)

showing that α is indeed a Markov partition for R_{J}, and R_{J}α = {U ∪ B, B ∪ W, J}.

THE POINCARÉ SERIES OF ℂ \ ℤ 15

Lemma 5.2. (R_{J}, α) is a mixing, almost onto Gibbs-Markov map. Proof.

Standard, see e.g. [A-D-U].

□ Next, we consider T_{J×{−1,1}^{2}} given by

T_{J×{−1,1}^{2}}(x, δ, ϵ) = T^{φ_{J}^{R}(x)}(x, δ, ϵ) = (R_{J}x, λ(x)δ, κ(x)ϵ)

where λ(x) := L_{φ_{J}^{R}(x)}(x), and κ(x) := K_{φ_{J}^{R}(x)}(x).

We have that for 2 ≤ k ≤ n = φ_{J}^{R},

L_{k} = \{ \begin{matrix}(−1)^{k−1} on U_{n}, \\ (−1)^{k} on B_{n}^{+}, \\ −1 on W_{n} ∪ B_{n}^{−}, \end{matrix} , & K_{k} = \{ \begin{matrix}(−1)^{k−1} on U_{n} ∪ B_{n}^{+}, \\ −1 on W_{n}, \\ 1 on B_{n}^{−}, \end{matrix}

In particular,

κ = \{ \begin{matrix}1 on U_{1} ∪ B_{1}, and B_{n}^{−}, \\ −1 on W_{n}, \\ (−1)^{n−1} on U_{n} ∪ B_{n}^{+} (n ≥ 2); \end{matrix} , & λ = \{ \begin{matrix}−1 & on W_{n} ∪ B_{n}^{−}, \\ (−1)^{n} & on B_{n}^{+}, \\ (−1)^{n−1} & on U_{n}. \end{matrix}

To get a Markov partition for T_{J×{−1,1}^{2}}, let

β := {A × {(δ, ϵ)} : A ∈ α, (δ, ϵ) ∈ {−1, 1}^{2}},

then (as can be checked)

T_{J×{−1,1}^{2}}β = {A × {(δ, ϵ)} : A ∈ R_{J}α, (δ, ϵ) ∈ {−1, 1}^{2}}

whence Lemma 5.3. (T_{J×{−1,1}^{2}}, β) is a mixing Gibbs-Markov map. Proof.

Standard, see [A-D]. □ Finally, to get our good sections, we calculate (T_{Ψ})_{J×{−1,1}^{2}×ℤ^{2}} given by

(T_{Ψ})_{J×{−1,1}^{2}×ℤ^{2}}(x, δ, ϵ, n) = (R_{J}x, λ(x)δ, κ(x)ϵ, n + Φ(x, δ, ϵ))

where

Φ(x, δ, ϵ) := ∑_{j=0}^{φ_{J} (x)−1} Ψ ∘ T^{j}(x, δ, ϵ), Ψ(x, δ, ϵ) := \frac{ϵ}{2}(1 − δ, 1 + δ).

16 JON AARONSON, MANFRED DENKER

We have

Φ(x, δ, ϵ) = ∑_{k=0}^{φ_{J} (x)−1} Ψ ∘ T^{k}(x, δ, ϵ)

= ∑_{k=0}^{φ_{J} (x)−1} Ψ(R^{k}x, δL_{k}(x), ϵK_{k}(x))

= \frac{1}{2} ∑_{k=0}^{φ_{J} (x)−1} ϵK_{k}(x)(1 − δL_{k}(x), 1 + δL_{k}(x)).

For x ∈ U_{1} ∪ B_{1} ∪ W_{1} = [φ_{J}^{R} = 1],

Φ(x, δ, ϵ) = Ψ(x, δ, ϵ) = \frac{ϵ}{2}(1 − δ, 1 + δ).

For φ_{J}^{R}(x) = n ≥ 2,

Φ(x, δ, ϵ) = \frac{1}{2} ∑_{k=0}^{n−1} ϵK_{k}(x)(1 − δL_{k}(x), 1 + δL_{k}(x))

= \frac{ϵ}{2}((1 − δ, 1 + δ) + ∑_{k=1}^{n−1} K_{k}(x)(1 − δL_{k}(x), 1 + δL_{k}(x)))

Now, we calculate

∑_{k=1}^{n−1} K_{k}(x)(1 − δL_{k}(x), 1 + δL_{k}(x)).

Lemma 5.4.

∑_{k=1}^{n−1} K_{k}(x)(1 − δL_{k}(x), 1 + δL_{k}(x)) = \{ \begin{matrix}δ(−a_{n−2}(−δ), a_{n−2}(δ)) x ∈ U_{n}, \\ (n − 1)(1 + δ, 1 − δ) x ∈ B_{n}^{−}, \\ δ(a_{n−2}(δ), −a_{n−2}(−δ)) x ∈ B_{n}^{+}, \\ −(n − 1)(1 + δ, 1 − δ) x ∈ W_{n} \end{matrix}

where

a_{n}(δ) := ∑_{k=0}^{n}(1 + δ(−1)^{k}) = \{ \begin{matrix}n + 2 n odd, \\ n + 1 + δ n even \end{matrix}

Proof. It’s not hard to show that

∑_{k=0}^{n}(−1)^{k}(1 + δ(−1)^{k}) = δa_{n}(δ)

and that

a_{n}(δ) = \{ \begin{matrix}n + 1 n odd, \\ n + 1 + δ n even. \end{matrix}

It follows that

∑_{k=1}^{n−1} K_{k}(x)(1 + δL_{k}(x))

= \{ \begin{matrix}∑_{k=1}^{n−1}(−1)^{k−1}(1 + (−1)^{k−1}δ) x ∈ U_{n}, \\ (n − 1)(1 − δ) x ∈ B_{n}^{−}, \\ ∑_{k=1}^{n−1}(−1)^{k−1}(1 + (−1)^{k}δ) x ∈ B_{n}^{+}, \\ −(n − 1)(1 − δ) x ∈ W_{n} \end{matrix}

= \{ \begin{matrix}δa_{n−2}(δ) x ∈ U_{n}, \\ (n − 1)(1 − δ) x ∈ B_{n}^{−}, \\ −δa_{n−2}(−δ) x ∈ B_{n}^{+}, \\ −(n − 1)(1 − δ) x ∈ W_{n}. \end{matrix}

The lemma follows from this. □

It follows that for φ_{J}^{R}(x) = n ≥ 2,

Φ(x, δ, ϵ) = \{ \begin{matrix}\frac{ϵ}{2}(1 − δ(1 + a_{n−2}(−δ)), 1 + δ(1 + a_{n−2}(δ))) x ∈ U_{n}, \\ \frac{ϵ}{2}(n + (n − 2)δ, n − (n − 2)δ) x ∈ B_{n}^{−}, \\ \frac{ϵ}{2}(1 + δ(a_{n−2}(δ) − 1), 1 − δ(a_{n−2}(−δ) − 1)) x ∈ B_{n}^{+}, \\ \frac{ϵ}{2}(−n + 2 − nδ, −n + 2 + nδ) x ∈ W_{n} \end{matrix}

whence

E(e^{i(sΦ_{a}+tΦ_{b})})
≈ E(1_{U×\{−1,1\}^{2}}e^{i(t−s) \frac{ϵδ}{2} φ_{J}^{R}}) + E(1_{B^{−}×\{−1,1\}^{2}}e^{i \frac{s+t+δ(s−t)}{2} ϵφ_{J}^{R}})
+ E(1_{B^{+}×\{−1,1\}^{2}}e^{i(s−t) \frac{ϵδ}{2} φ_{J}^{R}}) + E(1_{W ×\{−1,1\}^{2}}e^{−i \frac{s+t+δ(s−t)}{2} ϵφ_{J}^{R}})

as s, t → 0 and ∃ a, b, c > 0 such that ∀ (u, v) ∈ ℝ^{2} \ {0},

− log E(e^{it(uΦ_{a}+vΦ_{b})}) = (a|u − v| + b|u| + c|v|)|t|(1 + o(1))

as t → 0.

As in §3, let P_{T} be the Frobenius-Perron operator of T, let P_{t}(f) = P_{T} (e^{i⟨t,Φ⟩}f) (t ∈ 𝕋^{2}) and let λ(t) be the maximal eigenvalue of P_{t} for |t| small.

By theorem 5.1 of [A-D],

− log λ(t) = (a|u − v| + b|u| + c|v|)|t|(1 + o(1))

as t → 0.

As in §3, it follows that

𝔔 := {t ∈ ℝ^{2} : e^{i⟨t,ψ⟩} is cohomologous to a constant},

is discrete.

Theorem 7.3 in [A-D] now shows that T_{Φ} is totally dissipative and that T_{Φ_{a}} is rationally ergodic with return sequence a_{n}(T_{Φ_{a}}) ∝ log n.

Since these were good sections for the flows concerned, we have that the geodesic flow on the ℤ^{2} cover of the thrice punctured sphere is totally dissipative (confirming [Ly-Mc]); and that on the ℤ cover of the thrice punctured sphere is conservative, and rationally ergodic with return sequence ∝ log n.

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