The Rate of Mixing for Diagonal Flows on Spaces of Affine Lattices

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THE RATE OF MIXING FOR DIAGONAL FLOWS ON SPACES OF AFFINE LATTICES

SAMUEL EDWARDS

1. Introduction

Results on quantitative mixing play an important role in several recent developments in homogeneous dynamics; cf., e.g., [1, 4, 5, 6]. In the present paper our aim is to prove precise results on the rate of mixing on the spaces ASL(d, Z)\ ASL(d, R). As far as we are aware, these cases have not been covered previously in the literature. We start with (and focus mainly on) the two-dimensional case; let G be the semidirect product ASL(2, R) := SL(2, R) n R ² , with multiplication

(M, v) · (M ⁰ , v ⁰ ) = (M M ⁰ , M v ⁰ + v).

Similarly, let Γ be the discrete subgroup ASL(2, Z) of G. We note that our central object of study, the homogeneous space Γ\G, can be identified with the space of affine unimodular lattices in R ² . The one-parameter subgroup of G consisting of elements (Φ ⁰ _t , 0), where Φ ⁰ _t =

e

^t²

0 0 e

^{− t}²

, acts on Γ\G by right multiplication; given an element x = Γ(M, v) ∈ Γ\G, we let Φ _t (x) = Γ(M, v)(Φ ⁰ _t , 0) = Γ(M Φ ⁰ _t , v).

This transformation, together with a Haar measure µ on G (normalized so that the induced measure on Γ\G is a probability measure), gives rise to the measure-preserving dynamical system (Γ\G, µ, Φ _t ). This dynamical system is mixing, that is to say; for all f, g ∈ L ² (Γ\G, µ),

t→∞ lim Z

Γ\G

f (g ◦ Φ t ) dµ = Z

Γ\G

f dµ Z

Γ\G

g dµ.

By imposing certain regularity conditions on the functions f and g, we can obtain bounds on the rate of mixing. We let S ^m,p (Γ\G) be the space of m times continuously differentiable functions on Γ\G such that the L ^p (Γ\G) norms of all derivatives of order less or equal to m are finite. The Sobolev norm on this space is denoted k · k _S

^m,p

_(Γ\G) (we give a more precise definition of this space and norm in Section 2). We can now state our main results:

Theorem 1. For f, g ∈ S ^2,∞ (Γ\G) such that for all M ∈ G ⁰ Z

[0,1]×[0,1]

f (M, ξ) dξ = 0,

the flow Φ _t is mixing with exponential rate. More precisely, for t ≥ 0, Z

Γ\G

f (x)g(Φ t (x)) dµ(x) = O(e ⁻

²^t

kf k _S

2,∞

(Γ\G) kgk _S

2,∞

(Γ\G) ), where the implied constant is absolute.

By using results from [7] concerning the rate of mixing of diagonal flows on SL(2, Z)\ SL(2, R), we extend this to all f, g in S ^2,∞ (Γ\G) (Corollaries 6 and 7). We then turn our attention to functions in L ² (Γ\G), and prove the following:

Theorem 2. For f, g ∈ S ^4,2 (Γ\G), the flow Φ t is mixing with exponential rate. More precisely, for t ≥ 1,

Z

Γ\G

f (x)g(Φ t (x)) dµ(x) = Z

Γ\G

f dµ Z

Γ\G

g dµ + O(te ⁻

^t²

kf k _S

4,2

(Γ\G) kgk _S

4,2

(Γ\G) ), where the implied constant is absolute.

1

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After developing the necessary Fourier decomposition, these are proved in Sections 4 and 5. Finally, in Section 6, an analogue of Theorem 1 is proved for ASL(d, Z)\ ASL(d, R), when d ≥ 3.

2. Preliminaries

For notational convenience we denote G ⁰ = SL(2, R) and Γ ⁰ = SL(2, Z). We view C ^m (Γ\G) as the space of m times continuously differentiable functions on G that are invariant under left multiplication by elements of Γ. For every element X ∈ g, g being the Lie algebra of G (sl(2, R)⊕R ² , with Lie bracket [(M, v), (N, w)] = (M N −N M, M w −N v)), the corresponding left-invariant differential operator is

[Xf ](M ) := ∂

∂t f (M exp(tX)) t=0

. A basis for the Lie algebra is given by the elements

X 1 = (( ^{0 1} _{0 0} ), 0) , X 2 = (( ^{0 0} _{1 0} ), 0) , X 3 = ( ^{1 0} _{0 −1} ), 0 , X ₄ = (0 2 , ( ¹ ₀ )) , X 5 = (0 2 , ( ⁰ ₁ )) . For f ∈ C ^m (Γ\G), we define

N _m f (M, v) := X

ord(D)≤m

|[Df ](M, v)|,

where D is a monomial in X ₁ , ..., X ₅ . A norm for f such that f ∈ C ^m (Γ\G) and N _m f ∈ L ^p (Γ\G) is given by

kf k _S

m,p

(Γ\G) := kN m f k _L

^p

_(Γ\G) .

We denote the space of f such that kf k _S

^m,p

_(Γ\G) < ∞ as S ^m,p (Γ\G). The space L ^p (Γ\G) is identified with the space L ^p (F , µ), where F is a suitable fundamental domain (a set containing at least one representative for every coset, and the set of repetitions having measure zero) for Γ in G. By choosing a fundamental domain, F ⁰ , for Γ ⁰ \ G ⁰ , the product space F = F ⁰ × J , J being a fundamental domain for the torus T ² = Z ² \R ² , becomes a fundamental domain for Γ\G. We note that the torus is the correct factor for the Cartesian product with F ⁰ , since f being left-Γ invariant implies f (M, v) = f ((I ₂ , m)(M, v)) = f (M, v + m), ∀M ∈ G ⁰ , ∀v ∈ R ² , ∀m ∈ Z ² . Using Iwasawa decomposition, we can give an explicit description of one such F ⁰ and µ. This is done in the following way: for a given M ∈ G ⁰ , there is a unique decomposition

M = a b c d

= 1 x 0 1

√y 0 0 ^√ ¹ _y

cos θ − sin θ sin θ cos θ

, x ∈ R, y ∈ R ⁺ , θ ∈ R/2πZ.

For a given matrix M ∈ G ⁰ , we denote the variables x, y, θ given by the Iwasawa decom- position of M as x(M ), y(M ), θ(M ). Conversely, for given x ∈ R, y ∈ R ⁺ , θ ∈ R/2πZ, we define

n(x) := 1 x 0 1

, a(y) := √y 0 0 ^√ ¹ _y

, κ(θ) := cos θ − sin θ sin θ cos θ

. A possible choice of a fundamental domain for Γ ⁰ \G ⁰ is

F ⁰ = {n(x)a(y)κ(θ) : x ² + y ² ≥ 1, |x| ≤ 1

2 , θ ∈ [0, π)}.

The Haar measure on G is dµ = dµ ⁰ dv, where dv is the Lebesgue measure on R ² and dµ ⁰ is the Haar measure on G ⁰ . The measure dµ ⁰ given in Iwasawa coordinates is then _π ³

2

dxdydθ

y

²

.

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3. Fourier Decomposition

The proofs of Theorems 1 and 2 use the Fourier decomposition of the given functions in their torus variables, and rely heavily on Lemmas 4 and 5 in [10]. The Fourier decomposition is as follows: for f ∈ C ² (Γ\G), let

f (M, k) = b Z

T

²

f (M, ξ)e(−k · ξ) dξ, where e(x) = e ^2πix , x ∈ R. Then

(1) f (M, ξ) = X

k∈Z

²

f (M, k)e(k · ξ). b

Since f ∈ C ² (Γ\G), for a fixed M ∈ G ⁰ the corresponding function on the torus, ξ 7→

f (M, ξ), ξ ∈ T ² , is in C ² (T ² ) and thus has an absolutely convergent Fourier series which is also uniformly convergent on compact subsets of G. Let b Z ² denote the set of primitive lattice points in Z ² , that is to say elements _k ^j ∈ Z ² such that gcd(j, k) = 1. This allows us to rewrite (1) as

(2) f (M, ξ) = b f (M, 0) +

∞

X

n=1

X

j k

∈b Z

²

f (M, n b _k ^j )e(n _k ^j · ξ).

We now make use of the following lemma (Lemma 4 in [10]):

Lemma 3. ∀ T ∈ Γ ⁰ , ∀M ∈ G ⁰ , ∀k ∈ Z ² , ∀f ∈ C(Γ\G), f (T M, k) = b b f (M, ^t T k).

Proof.

f (T M, k) = b Z

T

²

f (T M, ξ)e(−k · ξ) dξ = Z

T

²

f (T M, T ξ)e(−k · T ξ) dξ

= Z

T

²

f (M, ξ)e(− ^t T k · ξ) dξ = b f (M, ^t T k).

The second equality is due to T being an area-preserving diffeomorphism of T ² , the third is

due to Γ-left invariance of f .

Applying Lemma 3 to (2) now gives (3) f (M, ξ) = b f (M, 0) +

∞

X

n=1

X

j k

∈b Z

²

f ( b ^{∗ ∗} _{j k} M, ( _n ⁰ ))e(n ^j _k · ξ),

where ^{∗ ∗} _{j k} is any matrix in Γ ⁰ with lower entries _k ^j ∈ Z b ² . We define

(4) f e _n (M ) := b f (M, ( _n ⁰ )).

The following lemma (Lemma 5 in [10]) provides a bound on the growth of the functions e f n : Lemma 4. For any m ∈ Z ^≥0 , n ∈ Z ^≥1 and f ∈ S ^m,∞ (Γ\G), we have

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f e n a b c d

kf k _S

m,∞

(Γ\G)

n ^m (c ² + d ² )

^m²

, ∀ ^{a b} _{c d} ∈ SL(2, R), with the implied constant depending only on m.

Proof. We have exp(tX ₄ ) = (I ₂ , ( ^t ₀ )) and exp(tX ₅ ) = (I ₂ , ( ⁰ _t )). Hence, parametrizing G by ((( ^{a b} _{c d} ), ( ^v v

¹2

)) gives

[X 4 f ](M, v) = a ∂f

∂v 1

(M, v) + c ∂f

∂v 2

(M, v),

(4)

and

[X ₅ f ](M, v) = b ∂f

∂v ₁ (M, v) + d ∂f

∂v ₂ (M, v).

Since

f e n a b c d =

Z

Z\R

Z

Z\R

f (( ^{a b} _{c d} ), ( ^ξ _ξ

¹

2

))e(−nξ 2 ) dξ 2 dξ 1 , repeated integration by parts gives

(6) (2πinc) ^m · e f _n ^{a b} _{c d} = Z

Z\R

Z

Z\R

[X ₄ ^m f ](( ^{a b} _{c d} ), ( ^ξ _ξ

¹

2

))e(−nξ ₂ ) dξ ₂ dξ ₁ and

(7) (2πind) ^m · e f _n ^{a b} _{c d} = Z

Z\R

Z

Z\R

[X ₅ ^m f ](( ^{a b} _{c d} ), ( ^ξ _ξ

¹

2

))e(−nξ ₂ ) dξ ₂ dξ ₁ . Hence

max(|c| ^m , |d| ^m ) ·

f e _n ^{a b} _{c d}

≤ (2πn) ^−m kf k _S

m,∞

(Γ\G) ,

giving (5).

4. Proof of Theorem 1

We now use the Fourier decomposition and the bound on the growth of the coefficients given by Lemma 4 to prove Theorem 1.

Proof of Theorem 1.

Z

Γ\G

f (x)g(Φ _t (x)) dµ(x) = Z

F

⁰

Z

T

²

f (M, ξ)g(M Φ ⁰ _t , ξ) dξ dµ ⁰ (M ).

Expanding both f and g in their Fourier decompositions gives Z

F

Z

T

²

f (M, 0) + b X

k∈Z

²

k6=0

f (M, k)e(k · ξ) b

b g(M Φ ⁰ _t , 0) + X

m∈Z

²

m6=0

b g(M Φ ⁰ _t , m)e(m · ξ)

dξ dµ ⁰ (M ).

Since the two series are absolutely convergent, we may multiply out their product. Uniform convergence of the series on compact sets allows the exchange of order of summation and integration on the torus. The summands are then all zero, except for when k = −m, giving (8)

Z

F

⁰

f (M, 0) b b g(M Φ ⁰ _t , 0) dµ ⁰ (M ) + Z

F

⁰

X

k∈Z

²

k6=0

f (M, k) b b g(M Φ ⁰ _t , −k) dµ ⁰ (M ).

By assumption, b f (M, 0) = 0, so we need only concern ourselves with the rate of convergence to zero of

(9)

Z

F

⁰

X

k∈Z

²

k6=0

f (M, k) b b g(M Φ ⁰ _t , −k) dµ ⁰ (M ).

Using Lemma 3 and (4), we can reparametrize the sum (9) similarly as we did when going from (1) to (2), obtaining:

(10)

Z

F

⁰

∞

X

n=1

X

j k

∈b Z

²

f e n ( ^{∗ ∗} _{j k} M ) e g n (− ^{∗ ∗} _{j k} M Φ ⁰ _t ) dµ ⁰ (M ).

Now, define the subgroup Γ ⁰ _∞ := {( ^{1 z} _{0 1} ) : z ∈ Z}. Each element ^{∗ ∗} j k , _k ^j ∈ Z is the b representative of a different coset in Γ ⁰ _∞ \Γ ⁰ , and moreover every coset is represented (i.e. there is a bijection from b Z ² to Γ ⁰ _∞ \Γ ⁰ ). This means that by taking the union of our fundamental domain F ⁰ multiplied from the left by elements ^{∗ ∗} _{j k} , i.e. S _j

k

∈ˆ Z

²

∗ ∗

j k F ⁰ , we construct a

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fundamental domain for Γ ⁰ _∞ \G ⁰ . We note that e f and e g are left Γ ⁰ _∞ -invariant, and hence the function M 7→ e f n (M ) e g n (−M Φ ⁰ _t ) is as well, so its integral is the same over any fundamental domain for Γ ⁰ _∞ \G ⁰ . We choose to integrate over I = {n(x)a(y)κ(θ) : − ¹ ₂ ≤ x ≤ ¹ ₂ , y > 0, θ ∈ [0, 2π)}, a standard fundamental domain for Γ ⁰ _∞ \G ⁰ . Hence, by the triangle inequality and Lebesgue’s monotone convergence theorem, the absolute value of (10) is less than or equal to (11)

∞

X

n=1

X

j k

∈b Z

²

Z

∗ ∗ j k

F

⁰

| e f n (M ) e g n (−M Φ ⁰ _t )| dµ ⁰ (M ) =

∞

X

n=1

Z

I

| e f n (M ) e g n (−M Φ ⁰ _t )| dµ ⁰ (M ).

Lemma 4 is used to bound the contribution from a given n in (11) in the following manner: if M = ^{a b} _{c d} , then c ² + d ² = _{y(M )} ¹ , hence denoting y = y(M ) and θ = θ(M ) gives

(12) | e f n (M )| min kf k _S

^0,∞

_(Γ\G) , kf k _S

^2,∞

_(Γ\G) yn ⁻² kf k _S

2,∞

(Γ\G) min 1, yn ⁻² , and

| e g _n (−M Φ ⁰ _t )| min kgk _S

0,∞

(Γ\G) , kgk _S

2,∞

(Γ\G) yn ⁻² (e ^t sin ² θ + e ^−t cos ² θ) ⁻¹ (13)

kgk _S

2,∞

(Γ\G) min 1, yn ⁻² (e ^t sin ² θ + e ^−t cos ² θ) ⁻¹ .

We note that the implied constants in these estimates are absolute. Now, using (12) and (13), we bound

Z

I

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M ) by

kf k _S

2,∞

(Γ\G) kgk _S

2,∞

(Γ\G)

Z ∞ 0

Z 2π 0

Z

¹

2

−

¹₂

min 1, yn ⁻²

× min 1, yn ⁻² (e ^t sin ² θ + e ^−t cos ² θ) ⁻¹ dx dθ dy y ² . Substituting y = n ² v gives

kf k _S

2,∞

(Γ\G) kgk _S

2,∞

(Γ\G)

n ²

Z ∞ 0

Z _2π

0

min(1, v) min 1, v(e ^t sin ² θ + e ^−t cos ² θ) ⁻¹ dθ dv v ²

kf k _S

2,∞

(Γ\G) kgk _S

2,∞

(Γ\G)

n ²

Z ∞ 0

min(1, v) Z π/2

0

min 1, ve ^−t θ ⁻² dθ dv v ²

kf k _S

2,∞

(Γ\G) kgk _S

2,∞

(Γ\G)

n ²

Z ∞ 0

min(1, v) Z

√ ve

^−t

0

dθ + Z ∞

√ ve

^−t

ve ^−t θ ⁻² dθ

! dv v ²

kf k _S

2,∞

(Γ\G) kgk _S

2,∞

(Γ\G)

n ² e ⁻

²^t

Z ∞

0

min(1, v)v

¹²

dv v ² . This gives

Z

F

⁰

X

k∈Z

²

k6=0

f (M, k) b b g(M Φ ⁰ _t , −k) dµ ⁰ (M ) = O

e ⁻

²^t

kf k _S

2,∞

(Γ\G) kgk _S

2,∞

(Γ\G)

,

as required.

Remark 1. In order to extend the result of Theorem 1 to all f, g ∈ S ^2,∞ (Γ\G), we need to consider the term

(14)

Z

F

⁰

f (M, 0) b b g(M Φ ⁰ _t , 0) dµ ⁰ (M )

in (8). This is done by noting that for f, g ∈ S ^m,2 (Γ\G), b f (·, 0) and b g(·, 0) can be viewed as

functions in L ² (Γ ⁰ \G ⁰ ), i.e. they are Γ ⁰ -left invariant and have finite L ² (F ⁰ , µ ⁰ ) norm. Clearly,

the L ² (Γ ⁰ \G ⁰ ) norms are bounded by the S ^m,2 (Γ\G) norms of f and g respectively (and even

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by the L ² (Γ\G) norms). Recall also that µ ⁰ (F ⁰ ) = 1, so for f, g ∈ S ^m,∞ (Γ\G), the S ^m,2 (Γ\G) norms are bounded by the S ^m,∞ (Γ\G) norms. The invariance condition is proved thus: let γ ∈ Γ ⁰ . Then

f (γM, 0) = b Z

T

²

f (γM, ξ) dξ = Z

T

²

f ((γ, 0)(M, γ ⁻¹ ξ)) dξ

= Z

T

²

f (M, γ ⁻¹ ξ) dξ = Z

T

²

f (M, ξ) dξ = b f (M, 0).

The third equality holds due to Γ-left invariance of f and the fourth due to γ ⁻¹ being an area-preserving diffeomorphism of the torus. Naturally, the same also holds for b g(·, 0). By viewing b f (·, 0) and b g(·, 0) in this manner, we may use results regarding the rate of mixing on Γ ⁰ \G ⁰ to ascertain the rate at which (14) converges to

Z

F

⁰

f (M, 0) dµ b ⁰ (M ) Z

F

⁰

b g(M, 0) dµ ⁰ (M )

= Z

F

⁰

Z

T

²

f (M, ξ) dξ dµ ⁰ (M ) Z

F

⁰

Z

T

²

g(M, ξ) dξ dµ ⁰ (M )

= Z

Γ\G

f dµ Z

Γ\G

g dµ.

We denote the regular representation of G ⁰ on L ² (Γ ⁰ \G ⁰ ) by π, i.e. π(M )g(Γ ⁰ N ) = g(Γ ⁰ N M ), and let K(m) be the set of L ² (Γ ⁰ \G ⁰ ) functions g such that the map ψ g : R → L ² (Γ ⁰ \G ⁰ ), θ 7→

π(κ(θ))g is m-times Fr´ echet differentiable.

Proposition 5. f ∈ S ^m,2 (Γ\G) ⇒ b f (·, 0) ∈ K(m).

Proof. Assume f ∈ S ^m,2 (Γ\G). By differentiating through the integral, we have b f (·, 0) ∈ S ^m,2 (Γ ⁰ \G ⁰ ) (we discuss this in greater detail in the beginning of Section 5). Now let g be an arbitrary function in S ^1,2 (Γ ⁰ \G ⁰ ). It suffices to show that θ 7→ π(κ(θ))[X _κ ⁰ g] is the Fr´ echet derivative of ψ g , that is

h→0 lim

kψ _g (θ + h) − ψ g (θ) − hπ(κ(θ))[X _κ ⁰ g]k _L

²

_(Γ

⁰

_\G

⁰

₎

h = 0,

where X _κ ⁰ = X ₂ ⁰ − X ₁ ⁰ (once again, we give precise definitions in Section 5). The representation is unitary, so

kπ(κ(θ + h))g − π(κ(θ))g − hπ(κ(θ))[X _κ ⁰ g]k _L

2

(Γ

⁰

\G

⁰

)

h = 1

h kπ(κ(h))g − g − h[X _κ ⁰ g]k _L

2

(Γ

⁰

\G

⁰

) . In the proof of Lemma 8, we note that _∂ϕ ^∂ g(M κ(ϕ)) = [X _κ ⁰ g](M κ(ϕ)), so by the fundamental theorem of calculus, the above expression equals

1 h

Z

F

⁰

g(M ) + Z h

0

[X _κ ⁰ g](M κ(ϕ)) dϕ − g(M ) − h[X _κ ⁰ g](M )

2

dµ ⁰ (M )

¹₂

. Minkowski’s inequality for integrals gives

1 h

Z

F

⁰

Z h 0

[X _κ ⁰ g](M κ(ϕ)) − [X _κ ⁰ g](M )

dϕ

2

dµ ⁰ (M )

¹₂

≤ 1 h

Z h 0

Z

F

⁰

|[X _κ ⁰ g](M κ(ϕ)) − [X _κ ⁰ g](M )| ² dµ ⁰ (M )

¹₂

dϕ

≤ sup

ϕ∈[0,h]

kπ(κ(ϕ))[X _κ ⁰ g] − [X _κ ⁰ g]k _L

²

_(Γ

⁰

_\G

⁰

₎ .

This bound tends to zero as h → 0, since G ⁰ acts continuously on L ² (Γ ⁰ \G ⁰ ); let us recall the standard proof of this fact: by density of C _c (Γ ⁰ \G ⁰ ) in L ² (Γ ⁰ \G ⁰ ), for a given > 0 there is a function g

3

∈ C _c (Γ ⁰ \G ⁰ ) such that k[X _κ g ⁰ ]−g

3

k _L

2

(Γ

⁰

\G

⁰

) < ₃ . Since g

3

has compact support, it

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is uniformly continuous, and therefore there exists an h 0 > 0 such that for every ϕ with |ϕ| < h 0

we have |g

3

(M κ(ϕ)) − g(M )| < ₃ ∀M ∈ F ⁰ , and hence also kπ(κ(ϕ))g

3

− g

3

k _L

2

(Γ

⁰

\G

⁰

) < ₃ . Now

kπ(κ(ϕ))[X _κ ⁰ g] − [X _κ ⁰ g]k _L

²

_(Γ

⁰

\G

⁰

) ≤ kπ(κ(ϕ))[X _κ ⁰ g] − π(κ(ϕ))g

3

k _L

2

(Γ

⁰

\G

⁰

)

+kπ(κ(ϕ))g

3

− g

3

k _L

2

(Γ

⁰

\G

⁰

)

+kg

3

− [X _κ ⁰ g]k _L

2

(Γ

⁰

\G

⁰

)

< .

Now, Theorem 2 in [7] gives the following: for f, g ∈ K(m) and t ≥ 1,

Z

Γ

⁰

\G

⁰

f π(Φ t )g dµ ⁰ = Z

Γ

⁰

\G

⁰

f dµ ⁰ Z

Γ

⁰

\G

⁰

g dµ ⁰ + O (te ⁻

²^t

) ^α(m) kf k _S

m,2

(Γ

⁰

\G

⁰

) kgk _S

m,2

(Γ

⁰

\G

⁰

) , where α(2) = ⁴ ₅ , and α(m) = 1 for m ≥ 3. We note that the explicit constants (that depend on f and g) are not given in the statement of this theorem, but by studying the proof (in particular, pages 286-288), we can extract the constants as stated here. This, together with Theorem 1, gives the following corollaries:

Corollary 6. For f, g ∈ S ^2,∞ (Γ\G), t ≥ 1, Z

Γ\G

f (x)g(Φ _t (x)) dµ(x) = Z

Γ\G

f dµ Z

Γ\G

g dµ + O t

⁴⁵

e ⁻

^2t⁵

kf k _S

2,∞

(Γ\G) kgk _S

2,∞

(Γ\G) , where the implied constant is absolute.

Corollary 7. For f, g ∈ S ^3,∞ (Γ\G), t ≥ 1, Z

Γ\G

f (x)g(Φ _t (x)) dµ(x) = Z

Γ\G

f dµ Z

Γ\G

g dµ + O te ⁻

²^t

kf k _S

3,∞

(Γ\G) kgk _S

3,∞

(Γ\G) , where the implied constant is absolute.

5. Proof of Theorem 2

In order to prove Theorem 2, we make use of two bounds on the growth of e f _n and e g _n . Both of these are based on Sobolev inequalities; the first (Lemma 8) on a one-dimensional inequality, the second (Lemma 9, which can be seen as a version of Lemma 4 for S ^m,2 (Γ\G) functions) on a three-dimensional inequality. These inequalities require the calculation of derivatives (on G ⁰ ) of the Fourier coefficients. This is done as follows: the Lie algebra of G ⁰ , sl(2, R), has the basis X ₁ ⁰ = ( ^{0 1} _{0 0} ) , X ₂ ⁰ = ( ^{0 0} _{1 0} ) , X ₃ ⁰ = ^{1 0} _{0 −1} , and is embedded in g by X ⁰ 7→ (X ⁰ , 0). Hence, for f ∈ C ^m (Γ\G), D ⁰ a monomial in X ₁ ⁰ , X ₂ ⁰ , X ₃ ⁰ of order not greater than m, we have (15) [D ⁰ f e n ]( ^{a b} _{c d} ) =

Z

T

²

[Df ]( ^{a b} _{c d} , ( ^ξ _ξ

¹

2

))e(−nξ 2 ) dξ 1 dξ 2 = ] [Df ] _n ( ^{a b} _{c d} ),

where D is the operator corresponding to the embedding of D ⁰ in U (g), the universal enveloping algebra of g. We can now state the two lemmas that will be of use when ascertaining the rate of mixing for S ^4,2 (Γ\G) functions.

Lemma 8. For f ∈ S ^1,2 (Γ\G), x ∈ R, y ∈ R ⁺ , θ ∈ [0, 2π), the following inequality holds

| e f _n (n(x)a(y)κ(θ))| ≤ 2 s

Z 2π 0

F _n (n(x)a(y)κ(ϕ)) dϕ,

where F _n := | e f _n | ² + |^ [X ₁ f ] _n | ² + |^ [X ₂ f ] _n | ² .

(8)

Proof. The one-dimensional Sobolev inequality gives (by an abuse of notation, we let f e _n (x, y, θ) := e f _n (n(x)a(y)κ(θ)))

(16) | e f _n (x, y, θ)| ≤ s

2 Z 2π

0

| e f _n (x, y, ϕ)| ² + | ∂

∂ϕ f e _n (x, y, ϕ)| ²

dϕ.

Now,

∂

∂ϕ f e n (x, y, ϕ) = lim

h→0

f e n (n(x)a(y)κ(ϕ + h)) − e f n (n(x)a(y)κ(ϕ))

h ,

and

f e n (n(x)a(y)κ(ϕ + h)) = e f n (n(x)a(y)κ(ϕ)κ(h)) = e f n (n(x)a(y)κ(ϕ) exp(h(X ₂ ⁰ − X ₁ ⁰ ))).

Hence

∂

∂ϕ f e _n (x, y, ϕ) = [X ₂ ⁰ f e _n ](x, y, ϕ) − [X ₁ ⁰ f e _n ](x, y, ϕ).

By (15), we have [X ₁ ⁰ f ˜ n ] = ^ [X 1 f ] _n and [X ₂ ⁰ f ˜ n ] = ^ [X 2 f ] _n . Hence

(17) | ∂

∂ϕ f e n (x, y, ϕ)| ² ≤ 2|^ [X 1 f ] _n (x, y, ϕ)| ² + 2|^ [X 2 f ] _n (x, y, ϕ)| ² .

Substituting (17) into (16) gives the desired inequality. Lemma 9. For f ∈ S ^m+2,2 (Γ\G), M ∈ G ⁰ , we have

(18) | e f n (M )| min

1, y(M ) n ²

^m₂

max

y(M ), 1 y(M )

¹₂

kf k _S

m+2,2

(Γ\G) , with the implied constant depending only on m.

Proof. Identities (6) and (7) give the following inequalities (19) | e f _n ( ^{a b} _{c d} )| n ^−m |c| ^−m

Z

T

²

[X ₄ ^m f ]( ^{a b} _{c d} , ( ^ξ _ξ

¹

2

))e(−nξ ₂ ) dξ ₁ dξ ₂ ,

(20) | e f _n ( ^{a b} _{c d} )| n ^−m |d| ^−m Z

T

²

[X ₅ ^m f ]( ^{a b} _{c d} , ( ^ξ _ξ

¹

2

))e(−nξ ₂ ) dξ ₁ dξ ₂ . Define the function e F n,X

_i^m

(i = 4, 5) on G ⁰ by

F e n,X

_i^m

( ^{a b} _{c d} ) :=

Z

T

²

[X _i ^m f ]( ^{a b} _{c d} , ( ^ξ _ξ

¹

2

))e(−nξ 2 ) dξ 1 dξ 2 . We note that e F _n,X

^m

i

is not generally Γ ⁰ -left invariant (it is, however, Γ ⁰ _∞ -left invariant). A bounded neighbourhood, U , of the identity in G ⁰ is chosen, and we define the function h _M on U ; for M ∈ G ⁰ , U ∈ U , h M (U ) := e F n,X

_i^m

(M U ). For f ∈ S ^m+2,2 (Γ\G), Sobolev’s inequality (for S ^2,2 (Ω), Ω being a bounded domain in R ³ , after using the appropriate coordinate change) gives

| e F _n,X

^m

i

(M )| kh _M k _S

2,2

(U ) , with the implied constant depending only on U . Now

kh _M k _S

2,2

(U ) = X

ordD

⁰

≤2

Z

U

|[D ⁰ F e _n,X

^m

i

](M U )| ² dµ ⁰ (U )

¹₂

.

Left invariance of the Haar measure gives kh _M k _S

2,2

(U ) = X

ordD

⁰

≤2

Z

M U

|[D ⁰ F e _n,X

^m

i

](N )| ² dµ ⁰ (N )

¹₂

.

(9)

By (15), we have

kh _M k _S

2,2

(U ) ≤ X

ordD≤2

Z

M U

Z

T

²

[DX _i ^m f ](N, ξ)e(−nξ 2 ) dξ

2

dµ ⁰ (N )

!

¹₂

. Hence

kh _M k _S

2,2

(U )

Z

M U

Z

T

²

N _m+2 ² dξdµ ⁰

¹₂

. For M ∈ G ⁰ we denote the integral over the torus of N _m+2 ² by I m+2 (M );

I m+2 (M ) :=

Z

T

²

N _m+2 ² (M, ξ) dξ.

Noting that since I _m+2 is Γ ⁰ -left invariant, the integral of I _m+2 over M U is bounded by the integral of I m+2 over Γ ⁰ \G ⁰ multiplied by the maximal cardinality of the fibers of the projection of M U onto Γ ⁰ \G ⁰ . More explicitly formulated; let π _M be the projection of M U onto Γ ⁰ \G ⁰ , i.e.

π M : M U → Γ ⁰ \G ⁰ , N 7→ Γ ⁰ N.

Define p(M ) by

p(M ) := max

x∈Γ

⁰

\G

⁰

|π ⁻¹ _M ({x})|.

Then

Z

M U

I m+2 dµ ⁰ ≤ p(M ) Z

Γ

⁰

\G

⁰

I m+2 dµ ⁰ p(M )kf k ² _S

m+2,2

(Γ\G) .

As before, noting that c ² + d ² = _{y(M )} ¹ , and making use of (19) and (20) gives a pointwise bound on e f n :

| e f _n (M )| min

1, y(M ) n ²

^m₂

p(M )

¹²

kf k _S

m+2,2

(Γ\G) .

From Lemma 2.11 in [3], we acquire the following bound; p(M ) max(y(M ), _{y(M )} ¹ ), giving

(18).

Proof of Theorem 2. Let us first prove a slightly weaker version of Theorem 2, namely that for f, g ∈ S ^5,2 (Γ\G), we have

(21) Z

Γ\G

f (x)g(Φ _t (x)) dµ(x) = Z

Γ\G

f dµ Z

Γ\G

g dµ + O(te ⁻

^t²

kf k _S

5,2

(Γ\G) kgk _S

5,2

(Γ\G) ).

We proceed as in the proof of Theorem 1; utilizing the Fourier decompositions of f and g, as well as the result from [7] as stated in Remark 1, we reduce the problem to bounding

∞

X

n=1

Z

I

| e f n (M ) e g n (−M Φ ⁰ _t )| dµ ⁰ (M ).

We now split the domain of integration in the following way; define the sets A := {M ∈ I : y(M ) ≤ 1},

B := {M ∈ I : y(−M Φ ⁰ _t ) ≤ 1}, and

C := {M ∈ I : y(M ) > 1, y(−M Φ ⁰ _t ) > 1}.

(10)

Then

Z

I

| e f _n (M ) e g _n (−M Φ ⁰ _t )|dµ ⁰ (M ) ≤ Z

A

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M ) +

Z

B

| e f n (M ) e g n (−M Φ ⁰ _t )| dµ ⁰ (M ) +

Z

C

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M ).

Substituting N = −M Φ ⁰ _t gives Z

B

| e f n (M ) e g n (−M Φ ⁰ _t )| dµ ⁰ (M ) = Z

A

| e g n (N ) e f n (−N Φ ⁰ _−t )| dµ ⁰ (N ).

We now have two integrals over A and one over C to consider. The integrals over A are dealt with first; from Lemma 9 we have the bounds

| e f _n (M )| min

1, y(M ) n ²

³₂

max

y(M ), 1 y(M )

¹₂

kf k _S

5,2

(Γ\G) , and

| e g n (−M Φ ⁰ _t )| min

1, y(M ) αn ²

³₂

max y(M )

α , α

y(M )

¹₂

kgk _S

5,2

(Γ\G) , where

α = e ^t sin ² (θ(M )) + e ^−t cos ² (θ(M )).

Hence Z

A

| e f n (M ) e g n (−M Φ ⁰ _t )| dµ ⁰ (M ) (22)

kf k _S

5,2

(Γ\G) kgk _S

5,2

(Γ\G)

Z 1 0

Z 2π 0

Z

¹

2

−

¹

2

min(1, n ⁻² y)

³²

max(y, y ⁻¹ )

¹²

× min(1, n ⁻² yα ⁻¹ )

³²

max(yα ⁻¹ , αy ⁻¹ )

¹²

dx dθ dy y ² . Simplifying the right hand side of (22) gives

kf k _S

5,2

(Γ\G) kgk _S

5,2

(Γ\G)

n ³

Z ₁

0

Z _2π

0

y ⁻¹ min(1, yα ⁻¹ )

³²

max(yα ⁻¹ , αy ⁻¹ )

¹²

dθ dy

kf k _S

5,2

(Γ\G) kgk _S

5,2

(Γ\G)

n ³

Z 1 0

Z 2π 0

α ⁻

¹²

y ⁻

¹²

min(1, yα ⁻¹ ) max(1, αy ⁻¹ ) dθ dy

= kf k _S

5,2

(Γ\G) kgk _S

5,2

(Γ\G)

n ³

Z 1 0

y ⁻

¹²

dy Z 2π

0

α ⁻

¹²

dθ.

In the same manner, we get the bound Z

A

| e g _n (M ) e f _n (−M Φ ⁰ _−t )| dµ ⁰ (M ) kf k _S

5,2

(Γ\G) kgk _S

5,2

(Γ\G)

n ³

Z 1 0

y ⁻

¹²

dy Z 2π

0

β ⁻

¹²

dθ, where β = e ^t cos ² θ + e ^−t sin ² θ. Now,

Z 2π 0

β ⁻

¹²

dθ = Z 2π

0

α ⁻

¹²

dθ Z π/2

0

1 max(θe

^t²

, e ⁻

²^t

( ^π ₂ − θ)) dθ

= e

^t²

Z

^π

2(1+et)

0

1

π

2 − θ dθ + e ⁻

^t²

Z

^π

2

π 2(1+et)

θ ⁻¹ dθ = e

^t²

log(e ^−t + 1) + e ⁻

^t²

log(e ^t + 1) = O(te ⁻

^t²

).

Hence,

∞

X

n=1

Z

A∪B

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M ) = O(te ⁻

^t²

kf k _S

5,2

(Γ\G) kgk _S

5,2

(Γ\G) ).

(11)

To bound the integral over C, we use dyadic decomposition; for j ∈ Z ≥0 , let C _j := {M ∈ I : 2 ^j ≤ y(M ) ≤ 2 ^j+1 },

and

C _t,j ⁰ := {M ∈ I : 2 ^j ≤ y(−M Φ ⁰ _t ) ≤ 2 ^j+1 }.

Denoting the intersection as

C _t,j,k := C _j ∩ C _t,k ⁰ gives

∞

X

n=1

Z

C

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M ) = X

j,k≥0

∞

X

n=1

Z

C

_t,j,k

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M ).

By the Cauchy-Schwarz inequality (and a change of variables),

∞

X

n=1

Z

C

_t,j,k

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M ) (23)

≤ v u u t

∞

X

n=1

Z

C

_t,j,k

| e f n (M )| ² dµ ⁰ (M ) v u u t

∞

X

n=1

Z

C

−t,k,j

| e g n (M )| ² dµ ⁰ (M ).

Lemma 8 gives

∞

X

n=1

Z

C

_t,j,k

| e f n (M )| ² dµ ⁰ (M ) Z

C

_t,j,k

Z 2π 0

∞

X

n=1

F n (x, y, ϕ) dϕ dθ dx dy y ² , and by Parseval’s identity,

Z

C

_t,j,k

Z 2π 0

I f (x, y, ϕ) dϕ dθ dx dy y ²

Z 2

^j+1

2

^j

|{θ ∈ [0, 2π) : 2 ^k ≤ y/α ≤ 2 ^k+1 }|

Z

¹

2

−

¹

2

Z 2π 0

I _f (x, y, ϕ) dϕ dx dy y ² , where

I _f :=

Z

T

²

(|f | ² + |[X ₁ f ]| ² + |[X ₂ f ]| ² ) dξ.

Now, for 2 ^j ≤ y ≤ 2 ^j+1 ,

{θ ∈ [0, 2π) : 2 ^k ≤ y/α ≤ 2 ^k+1 } ⊆ {θ ∈ [0, 2π) : 2 ^j−k−1 ≤ α ≤ 2 ^j−k+1 }, giving

|{θ ∈ [0, 2π) : 2 ^j−k−1 ≤ α ≤ 2 ^j−k+1 }|

Z 2

^j+1

2

^j

Z

¹

2

−

¹₂

Z 2π 0

I f (x, y, ϕ) dϕ dx dy y ² . Since e ^−t ≤ α ≤ e ^t , we have

|{θ ∈ [0, 2π) : 2 ^j−k−1 ≤ α ≤ 2 ^j−k+1 }| = 0

if e ^−t > 2 ^j−k+1 or e ^t < 2 ^j−k−1 , i.e. if 2 ^j−k ∈ [ / _2e ¹

t

, 2e ^t ]. For 2 ^j−k ∈ [ _2e ¹

t

, 2e ^t ], we have

|{θ ∈ [0, 2π) : 2 ^j−k−1 ≤ α ≤ 2 ^j−k+1 }| |{θ ∈ [0, ^π ₂ ) : 2 ^j−k−1 ≤ α ≤ 2 ^j−k+1 }|.

For θ ∈ [0, ^π ₂ ), we have ^e

^t

₅ ^θ

²

≤ α, so

|{θ ∈ [0, ^π ₂ ) : 2 ^j−k−1 ≤ α ≤ 2 ^j−k+1 }| ≤ |{θ ∈ [0, ^π ₂ ) : e ^t θ ²

5 ≤ 2 ^j−k+1 }| e ⁻

²^t

2

^j−k²

.

(12)

We note that we get the same bound when replacing t with −t, as α is replaced by β = e ^−t sin ² θ + e ^t cos ² θ, and

|{θ ∈ [0, 2π) : 2 ^j−k−1 ≤ β ≤ 2 ^j−k+1 }| = |{θ ∈ [π/2, 5π/2) : 2 ^j−k−1 ≤ β ≤ 2 ^j−k+1 }|

= |{θ ∈ [0, 2π) : 2 ^j−k−1 ≤ α ≤ 2 ^j−k+1 }|.

Returning to (23), we now have

∞

X

n=1

Z

C

_t,j,k

| e f n (M ) e g n (−M Φ ⁰ _t )| dµ ⁰ (M ) = 0, if 2 ^j−k ∈ [ / _2e ¹

_t

, 2e ^t ], and

∞

X

n=1

Z

C

_t,j,k

| e f _n (M ) e g _n (−M Φ ⁰ _t )|dµ ⁰ (M )

v u u

t e ⁻

²^t

2

^j−k²

Z 2

^j+1

2

^j

Z

¹

2

−

¹₂

Z 2π 0

I f (x, y, ϕ) dϕ dx dy y ²

× v u u

t e ⁻

²^t

2

^k−j²

Z 2

^k+1

2

^k

Z

¹

2

−

¹₂

Z 2π 0

I _g (x, y, ϕ) dϕ dx dy y ²

= e ⁻

^t²

s Z

C

_j

I _f (M ) dµ ⁰ (M ) s Z

C

_k

I g (M ) dµ ⁰ (M ) otherwise. We can now bound the sum of the integrals over C by

∞

X

n=1

Z

C

| e f n (M ) e g n (−M Φ ⁰ _t )| dµ ⁰ (M )

e ⁻

^t²

X

j,k≥0

|j−k|≤1+

^t

log 2

s Z

C

j

I f (M ) dµ ⁰ (M ) s Z

C

_k

I g (M ) dµ ⁰ (M )

e ⁻

^t²

X

j,k≥0

|j−k|≤3t

s Z

C

_j

I _f (M ) dµ ⁰ (M ) s Z

C

_k

I g (M ) dµ ⁰ (M )

e ⁻

^t²

X

d∈[−3t,3t]∩Z

∞

X

j=max(0,d)

s Z

C

j

I _f (M ) dµ ⁰ (M ) s Z

C

j−d

I _g (M ) dµ ⁰ (M ).

Once again, the Cauchy-Schwarz inequality (and taking the smallest possible lower limit of summation) gives

e ⁻

²^t

X

d∈[−3t,3t]∩Z

v u u t

∞

X

j=0

Z

C

_j

I _f (M ) dµ ⁰ (M ) v u u t

∞

X

j=0

Z

C

_j

I _g (M ) dµ ⁰ (M ) (24)

te ⁻

^t²

s Z

I\A

I _f (M ) dµ ⁰ (M ) s Z

I\A

I _g (M ) dµ ⁰ (M ).

By the definitions of I _f and I g , as well as noting that I \ A ⊂ F ⁰ ∪ F ⁰ κ(π), we conclude that the integrals in (24) are bounded by kf k ² _S

1,2

(Γ\G) and kgk ² _S

1,2

(Γ\G) , respectively, giving

∞

X

n=1

Z

C

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M ) = O(te ⁻

^t²

kf k _S

1,2

(Γ\G) kgk _S

1,2

(Γ\G) ),

(13)

completing the proof of (21).

We will next describe how to improve the above argument to get the desired result (that is, for S ^4,2 (Γ\G)). The bound on p(M ) used in Lemma 9 is very much a worst-case estimate, and on average greatly overestimates the value of p(M ) for M such that y(M ) < 1. Instead, we may bound p by the so-called invariant height function, Y _Γ

⁰

. This is done thus: let H be the upper-half plane

H := {x + iy : y > 0, x, y ∈ R}, together with the hyperbolic metric

ds ² = dx ² + dy ² y ² .

We let G ⁰ act on H by fractional linear transformations, i.e. for M = ^{a b} _{c d} ∈ G ⁰ , z ∈ H, let M · z = az + b

cz + d .

We note that for M = n(x)a(y)κ(θ), M · i = x + iy. In Lemma 9 we can choose U to be the set

U := {M ∈ G ⁰ : d(M · i, i) < 1}, where d is the distance induced by the given metric. This means that

p(M ) = sup

U ∈U

#{γ ∈ Γ ⁰ : d(γM U · i, M · i) < 1}.

We see that p is Γ ⁰ -left invariant, so we need only consider p on F ⁰ (F ⁰ as chosen in Section 2). Let F ⁰⁰ ⊂ H be the set {x(M) + iy(M) : M ∈ F ⁰ }. Since F ⁰ is a fundamental domain for Γ ⁰ \G ⁰ , and Γ ⁰ \G ⁰ / SO(2) ∼ = Γ ⁰ \H (see, e.g., [2] or [3]), we conclude that p(M) is less than or equal to the number of Γ ⁰ translates of F ⁰⁰ that a (hyperbolic) unit ball around x(M ) + iy(M ) intersects. For all M ∈ F ⁰ such that y(M ) is less or equal to 3, p(M ) is uniformly bounded, as the set of all w ∈ H which have distance not greater than one to {z ∈ F ⁰⁰ : Im z ≤ 3}

is compact and hence intersects only a finite number of Γ ⁰ -translates of F ⁰⁰ . For y(M ) larger than 3, the only translates of F ⁰⁰ that a unit ball around x(M ) + iy(M ) can intersect are translates of the type ^{1 k} _{0 1} F ⁰⁰ , k ∈ Z, i.e. integer translations of F ⁰⁰ along the x-axis. Hence, for M ∈ F ⁰ , p(M ) is asymptotically bounded by the Euclidean width along the x axis of a hyperbolic unit ball around x(M ) + iy(M ). For a point z ∈ H, this width is O(Im z), hence for all M ∈ G ⁰ ,

p(M ) sup

γ∈Γ

⁰

y(γM ) = sup

γ∈Γ

⁰

Im(γM · i).

For z ∈ H, we define

Y _Γ

⁰

(z) := sup

γ∈Γ

⁰

Im(γ · z).

By changing the estimate for p(M ) used in Lemma 9, now we obtain the bounds

| e f _n (M )| min

1, y(M ) n ²

Y _Γ

⁰

(M · i)

¹²

kf k _S

4,2

(Γ\G) ,

| e g n (−M Φ ⁰ _t )| min

1, y(M ) αn ²

max y(M )

α , α

y(M )

¹₂

kgk _S

4,2

(Γ\G) . We need only consider the integral over A, which now reads

Z

A

| e f _n (M ) e g _n (−M Φ ⁰ _t )| dµ ⁰ (M )

kf k _S

4,2

(Γ\G) kgk _S

4,2

(Γ\G)

Z 1 0

Z 2π 0

Z

¹

2

−

¹₂

min(1, n ⁻² y)Y Γ

⁰

(x + iy)

¹²

× min(1, n ⁻² yα ⁻¹ ) max(yα ⁻¹ , αy ⁻¹ )

¹²

dx dθ dy

y ² .

(14)

By Proposition 2.2 in [9],

Z

¹

2

−

¹₂

Y _Γ

⁰

(x + iy)

¹²

dx ≤ C, for all 0 < y < 1 and some positive constant C. This gives kf k _S

4,2

(Γ\G) kgk _S

4,2

(Γ\G)

n ²

Z 1 0

Z 2π 0

Z

¹

2

−

¹₂

y ⁻¹ min(1, yα ⁻¹ ) max(yα ⁻¹ , αy ⁻¹ )

¹²

Y _Γ

⁰

(x + iy)

¹²

dx dθ dy

kf k _S

4,2

(Γ\G) kgk _S

4,2

(Γ\G)

n ²

Z 1 0

Z 2π 0

α ⁻

¹²

y ⁻

¹²

min(1, yα ⁻¹ ) max(1, αy ⁻¹ ) dθ dy

kf k _S

4,2

(Γ\G) kgk _S

4,2

(Γ\G)

n ²

Z 2π 0

α ⁻

¹²

dθ kf k _S

4,2

(Γ\G) kgk _S

4,2

(Γ\G)

n ² te ⁻

²^t

.

6. Mixing in Higher Dimensions

We now let G = ASL(d, R) = SL(d, R) n R ^d , Γ = ASL(d, Z) = SL(d, Z) n Z ^d . As before, multiplication is

(M 1 , v 1 ) · (M 2 , v 2 ) = (M 1 M 2 , M 1 v 2 + v 1 ),

and we also reuse the notation G ⁰ = SL(d, R) and Γ ⁰ = SL(d, Z). As in the two-dimensional case, we can view Γ\G as the space of affine unimodular d-dimensional lattices. Let X _i denote the basis elements of the Lie algebra g = sl(d, R) ⊕ R ^d corresponding to the R ^d coordinates, i.e. X _i = (0, ^t (0, ..., 1, ..., 0)). The left-invariant differential operator corresponding to X _i is then

(25) [X i f ](M, v) = ∂

∂t f ((M, v) exp(tX i )) t=0

=

d

X

j=1

M ji

∂

∂v _j f (M, v),

where M and v are parametrized by {M _ji } _1≤j,i≤d and {v _j } _1≤j≤d respectively. We have the Iwasawa decomposition for G ⁰ : G ⁰ = N AK, where K = SO(d), N is the subgroup of matrices

n _d (x) =







1 x 12 · · · x _1d . .. ... .. .

. .. x _d−1,d 1







, x _jk ∈ R,

and A is the subgroup consisting of

a _d (y) =





 y ₁

. ..

y _d





 , y _j ∈ R ⁺ . Since A is a subgroup of G ⁰ , y _d = Q d−1

i=1 y ⁻¹ _i . The Haar measure on G ⁰ in these coordinates is, with M = n(x)a(y)k ([11], Section 2.1),

dµ ⁰ (M ) = 2 ^d−1 π ^d(d+1)/4 Q _d

j=1 Γ( ^j ₂ ) Q _d

j=2 ζ(j) ρ(y)dn(x)da(y)dk, where dk is the normalized Haar measure on SO(d), dn(x) = Q

1≤j<k≤d dx _jk and da(y) = Q d−1

j=1 y _j ⁻¹ dy j . The function ρ(y) is ρ(y) =

d

Y

j=1

y ^2j−d−1 _j =

d−1

Y

j=1

y _j ^2(j−d) .

We now state our result concerning the rate of mixing in higher dimensions:

(15)

Theorem 10. Let

Φ ⁰ _t =





 e ^λ

¹

^t

. ..

e ^λ

^d

^t





 , where P d

i=1 λ _i = 0. Given Γ\G 3 x = Γ(M, v), we define Φ t (x) := Γ(M, v) · (Φ ⁰ _t , 0 d ).

For f, g ∈ S ^d,∞ (Γ\G) such that R

T

^d

f dξ = 0 and t ≥ 0, Z

Γ\G

f (x)g(Φ _t (x)) dµ(x) = O(e ^−λ

^max

^t kf k _S

d,∞

(Γ\G) kgk _S

d,∞

(Γ\G) ), where λ max = max(λ 1 , . . . , λ d ).

The proof of Thereom 10 once again relies on bounds of Fourier coefficients; for a function f ∈ S ^m,∞ (Γ\G), m ≥ 2, we have the Fourier decomposition with respect to the torus variable

(26) f (M, v) = X

k∈Z

^d

f (M, k)e( b ^t kv), where

f (M, k) = b Z

T

^d

f (M, ξ)e(− ^t kξ) dξ.

As in the two-dimensional case, we note the following property of b f : for T ∈ Γ ⁰ , M ∈ G ⁰ , k ∈ Z ^d , we have (with a proof which is completely analogous to the proof of Lemma 3)

f (T M, k) = b b f (M, ^t T k).

This allows the reformulation of (26);

f (M, v) = b f (M, 0 d ) +

∞

X

n=1

X

k∈b Z

^d

f (( b

^t

^∗ _k ) M, ⁰

^d−1

_n )e(n ^t kv),

where b Z ^d is the set of d-dimensional primitive integer vectors, and (

^t

^∗ _k ) is any matrix in Γ ⁰ with dth row vector equal to ^t k (such a matrix always exists; cf. e.g., Theorem 31 in [8]). Let

f e n (M ) := b f (M, ⁰

^d−1

_n ).

This gives the following generalization of Lemma 4:

Lemma 11. For any f ∈ S ^m,∞ (Γ\G),

(27) | e f _n (M )| kf k _S

m,∞

(Γ\G)

n ^m kM _d,· k ^m , where kM _d,· k =

P d

j=1 M _d,j ²

¹₂

, and the implied constant depends only on d and m.

Proof. Since

f e _n (M ) = Z

T

^d

f (M, ξ)e(−nξ _d ) dξ, (25) and repeated integration by parts gives

(2πinM d,j ) ^m · e f n (M ) = Z

T

^d

[X _j ^m f ](M, ξ)e(−nξ d ) dξ.

Hence

1≤j≤d max (|M _d,j | ^m )| e f _n (M )| n ^−m kf k _S

m,∞

(Γ\G) ,

giving (27).

(16)

We note that e f n is Γ ⁰ _H -left invariant, where H = {T ∈ G ⁰ : ^t T e _d = e _d } =

A w

t 0 _d−1 1

: A ∈ SL(d − 1, R) , w ∈ R ^d−1

,

and Γ ⁰ _H := Γ ⁰ ∩ H . To integrate over Γ ⁰ _H \G ⁰ , we need a fundamental domain for the action of Γ ⁰ _H : let

F :=

n _d (x)a _d (y)k _d : x _ji ∈ (− ¹ ₂ , ¹ ₂ ], 0 < y _j+1 ≤ 2

√ 3 y _j (j = 1, ..., d − 2), k _d ∈ K

. Proposition 12. F contains a fundamental domain for Γ ⁰ _H \G ⁰ .

Proof. We need to show that for a given M = n d (x)a d (y)k d , there is a γ ∈ Γ ⁰ _H , and an M ⁰ ∈ F such that γM ⁰ = M . Let

S =







y 1 y 2 x 1,2 · · · y d−1 x 1,d−1

y ₂ · · · y _d−1 x _2,d−1 . .. .. .

y _d−1







∈ GL(d − 1, R), x =





 x _1,d

.. . x _d−1,d





 ∈ R ^d−1 .

This allows M to be written in block matrix form:

M =

S y d x

t 0 _d−1 y _d

k _d .

Since y

1 d−1

d S ∈ SL(d − 1, R), there is an A ∈ SL(d − 1, Z), and T = n d−1 (u)a _d−1 (v)k _d−1 ⁰ such that AT = y

1 d−1

d S and T ∈ S _d−1 , where S _d−1 is the Siegel set (see e.g. [11], Section 2.1) S _d−1 :=

n _d−1 (u)a _d−1 (v)k _d−1 ⁰ : u _ji ∈ (− ¹ ₂ , ¹ ₂ ], 0 < v j+1 ≤ 2

√

3 v j (j = 1, ..., d − 2), k ⁰ _d−1 ∈ SO(d − 1)

. Now, ^t u = ^t (u _1,d , ..., u _d−1,d ) ∈ (− ¹ ₂ , ¹ ₂ ] ^d−1 and w ∈ Z ^d−1 are chosen so that

x = Au + w.

Then

A w

t 0 _d−1 1

n _d−1 (u) u

t 0 _d−1 1

y ⁻

1 d−1

d a _d−1 (v) 0 _d−1

t 0 _d−1 y _d

! k ⁰ _d−1 0 _d−1

t 0 _d−1 1

k d = M.

Remark 2. By a similar discussion, using in particular the fact that S _d−1 is contained in a finite union of fundamental domains for SL(d − 1, Z)\ SL(d − 1, R), it follows that F is optimal in the sense that F is contained in a finite union of fundamental domains for Γ ⁰ _H \G ⁰ .

The Rate of Mixing for Diagonal Flows on Spaces of Affine Lattices