Distribution of dense lattice orbits on homogeneous spaces June 6, - - PowerPoint PPT Presentation

Distribution of dense lattice orbits

• n homogeneous spaces

June 6, 2013 Ergodic Theory with connections to arithmetic University of Crete, Iraklion Amos Nevo, Technion based on joint work with Alex Gorodnik

Wrawick, ETDS 30

Plan

1

Classical ratio ergodic theorem

2

Kazhdan’s problem : groups of isometries

3

Arnold’s problem : homogeneous spaces

4

Equidistribution of matrices with entries in algebraic number field

5

General duality principle

Wrawick, ETDS 30

The classical ratio ergodic theorem

• Birkhoff’s pointwise ergodic theorem was generalized to any

non-singular Z-action (Hopf [1937], Hurewicz [1944], Chacon-Ornstein [1960]). Let us formulate one important special case.

Wrawick, ETDS 30

The classical ratio ergodic theorem

• Birkhoff’s pointwise ergodic theorem was generalized to any

non-singular Z-action (Hopf [1937], Hurewicz [1944], Chacon-Ornstein [1960]). Let us formulate one important special case.

• Assume that T : (X, λ) → (X, λ) preserves the measure and is

conservative and ergodic. Then the ratio theorem states that for u, v ∈ L1(X, λ) with

• v dλ = 0, the ratios

Rn[u, v] := n

k=0 T ku

n

k=0 T kv

converge almost everywhere as n → ∞ to the constant

• u dλ
• v dλ

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Absence of rate of convergence

• Initially, one might expect that it is possible to pick normalization

constants a(n) satisfying a(n)

n

→ 0, such that the averages

1 a(n)

n

k=0 u(T kx) will converge to a nontrivial limit.

Wrawick, ETDS 30

Absence of rate of convergence

• Initially, one might expect that it is possible to pick normalization

constants a(n) satisfying a(n)

n

→ 0, such that the averages

1 a(n)

n

k=0 u(T kx) will converge to a nontrivial limit.

• But in fact, J. Aaronson showed that if the action is ergodic and

conservative, then for any normalization constants a(n), either for every nonnegative u ∈ L1(X), u = 0 lim inf

n→∞

1 a(n)

n

• k=0

u(T kx) = 0 almost everywhere,

Wrawick, ETDS 30

Absence of rate of convergence

• Initially, one might expect that it is possible to pick normalization

constants a(n) satisfying a(n)

n

→ 0, such that the averages

1 a(n)

n

k=0 u(T kx) will converge to a nontrivial limit.

• But in fact, J. Aaronson showed that if the action is ergodic and

conservative, then for any normalization constants a(n), either for every nonnegative u ∈ L1(X), u = 0 lim inf

n→∞

1 a(n)

n

• k=0

u(T kx) = 0 almost everywhere,

• or there exists a subsequence nm such that for nonnegative

u ∈ L1(X), u = 0, lim

m→∞

1 a(nm)

nm

• k=0

φ(T kx) = ∞ almost everywhere.

Wrawick, ETDS 30

The ratio ergodic theorem for commuting transformations

• The generalization of the ratio ergodic theorem to actions of two or

more commuting transformations has proved to be a difficult

• challenge. It was resolved in final form only recently by Hochman

[2009], who obtained results directly generalizing the case of general non-singular Z-actions. Important partial results were obtained earlier by Feldman in 2007.

Wrawick, ETDS 30

The ratio ergodic theorem for commuting transformations

• The generalization of the ratio ergodic theorem to actions of two or

more commuting transformations has proved to be a difficult

• challenge. It was resolved in final form only recently by Hochman

[2009], who obtained results directly generalizing the case of general non-singular Z-actions. Important partial results were obtained earlier by Feldman in 2007.

• As always, one has to decide on a sequence of asymptotically

invariant sets in Zd to sum over. The crucial property the (symmetric) family must satisfy was shown by Hochman to be the Besicovich covering property. There are counterexamples to the ratio ergodic theorem when summing over general asymptotically invariant sequences due to Brunel and Krengel [1985].

Wrawick, ETDS 30

The ratio ergodic theorem for commuting transformations

• The generalization of the ratio ergodic theorem to actions of two or

more commuting transformations has proved to be a difficult

• challenge. It was resolved in final form only recently by Hochman

[2009], who obtained results directly generalizing the case of general non-singular Z-actions. Important partial results were obtained earlier by Feldman in 2007.

• As always, one has to decide on a sequence of asymptotically

invariant sets in Zd to sum over. The crucial property the (symmetric) family must satisfy was shown by Hochman to be the Besicovich covering property. There are counterexamples to the ratio ergodic theorem when summing over general asymptotically invariant sequences due to Brunel and Krengel [1985].

• As to non-amenable groups, let us now describe some recent

examples of general ratio ergodic theorems and also of some counterexamples.

Wrawick, ETDS 30

A ratio ergodic theorem for Gromov-hyperbolic groups

• A (weighted) ratio ergodic theorem for actions of Gromov-hyperbolic

groups preserving a σ-finite measure (X, µ), was established recently by Pollicott and Sharp [2011], as follows.

Wrawick, ETDS 30

A ratio ergodic theorem for Gromov-hyperbolic groups

• A (weighted) ratio ergodic theorem for actions of Gromov-hyperbolic

groups preserving a σ-finite measure (X, µ), was established recently by Pollicott and Sharp [2011], as follows.

• Let S be a symmetric generating set of such a group Γ, and let

Sn =

• γ ∈ Γ ; |γ|S = n
• , where |γ|S is the word length of γ w.r.t. S.

Wrawick, ETDS 30

A ratio ergodic theorem for Gromov-hyperbolic groups

• A (weighted) ratio ergodic theorem for actions of Gromov-hyperbolic

groups preserving a σ-finite measure (X, µ), was established recently by Pollicott and Sharp [2011], as follows.

• Let S be a symmetric generating set of such a group Γ, and let

Sn =

• γ ∈ Γ ; |γ|S = n
• , where |γ|S is the word length of γ w.r.t. S.
• Then there exists ρ = ρS > 1 such that for any f, g ∈ L1(X, µ) with

g > 0 : N

n=0 ρ−n |γ|S=n f(γx)

N

n=0 ρ−n |γ|S=n g(γx)

converges almost surely as N → ∞. Limit : ????

Wrawick, ETDS 30

A ratio ergodic theorem for Gromov-hyperbolic groups

• A (weighted) ratio ergodic theorem for actions of Gromov-hyperbolic

groups preserving a σ-finite measure (X, µ), was established recently by Pollicott and Sharp [2011], as follows.

• Let S be a symmetric generating set of such a group Γ, and let

Sn =

• γ ∈ Γ ; |γ|S = n
• , where |γ|S is the word length of γ w.r.t. S.
• Then there exists ρ = ρS > 1 such that for any f, g ∈ L1(X, µ) with

g > 0 : N

n=0 ρ−n |γ|S=n f(γx)

N

n=0 ρ−n |γ|S=n g(γx)

converges almost surely as N → ∞. Limit : ????

• When Γ is the fundamental group of a closed surface of constant

negative curvature with genus > 1, and the set of generators is the standard one, the limit is

• X fdµ/
• X gdµ provided that the action is

ergodic.

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Balls on the free group

• Consider the free group F = a1, . . . , ar, and let |g| denote its word

length with respect to the free generators.

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Balls on the free group

• Consider the free group F = a1, . . . , ar, and let |g| denote its word

length with respect to the free generators.

• Let (Tg)g∈F be a measure preserving action on a standard σ-finite

measure space (X, B, λ). For u, v ∈ L1(X, λ) with v > 0 let

Wrawick, ETDS 30

Balls on the free group

• Consider the free group F = a1, . . . , ar, and let |g| denote its word

length with respect to the free generators.

• Let (Tg)g∈F be a measure preserving action on a standard σ-finite

measure space (X, B, λ). For u, v ∈ L1(X, λ) with v > 0 let Rn[u, v] :=

• |g|≤n Tgu
• |g|≤n Tgv .

Wrawick, ETDS 30

Balls on the free group

• Consider the free group F = a1, . . . , ar, and let |g| denote its word

length with respect to the free generators.

• Let (Tg)g∈F be a measure preserving action on a standard σ-finite

measure space (X, B, λ). For u, v ∈ L1(X, λ) with v > 0 let Rn[u, v] :=

• |g|≤n Tgu
• |g|≤n Tgv .
• Does the sequence R2n[u, v] of ball ratios converge pointwise

almost everywhere as n → ∞ ?

Wrawick, ETDS 30

Balls on the free group

• Consider the free group F = a1, . . . , ar, and let |g| denote its word

length with respect to the free generators.

• Let (Tg)g∈F be a measure preserving action on a standard σ-finite

measure space (X, B, λ). For u, v ∈ L1(X, λ) with v > 0 let Rn[u, v] :=

• |g|≤n Tgu
• |g|≤n Tgv .
• Does the sequence R2n[u, v] of ball ratios converge pointwise

almost everywhere as n → ∞ ?

• This was recently answered negatively by Hochman [2012]. The

counterexamples constructed exploit the failure of the Besicovich property for balls, and are quite general.

Wrawick, ETDS 30

Dense groups of isometries : General formulation of Kazhdan’s problem

• Let (X, d) be an lcsc metric space, and let G = Isom(X) be its

group of isometries. Assume that the action of G on X is transitive, and let mX be the unique isometry-invariant Radon measure on X. We consider the action of a dense subgroup Γ of G on X.

Wrawick, ETDS 30

Dense groups of isometries : General formulation of Kazhdan’s problem

• Let (X, d) be an lcsc metric space, and let G = Isom(X) be its

group of isometries. Assume that the action of G on X is transitive, and let mX be the unique isometry-invariant Radon measure on X. We consider the action of a dense subgroup Γ of G on X.

• In particular, we can consider the case where (X, B, λ) is a locally

compact group with Haar measure, φ : Γ′ → X is a group homomorphism onto a dense subgroup Γ, and the action (Tγ′)γ′∈Γ′ is given by Tγ′(x) = φ(γ′)x.

Wrawick, ETDS 30

Dense groups of isometries : General formulation of Kazhdan’s problem

• Let (X, d) be an lcsc metric space, and let G = Isom(X) be its

group of isometries. Assume that the action of G on X is transitive, and let mX be the unique isometry-invariant Radon measure on X. We consider the action of a dense subgroup Γ of G on X.

• In particular, we can consider the case where (X, B, λ) is a locally

compact group with Haar measure, φ : Γ′ → X is a group homomorphism onto a dense subgroup Γ, and the action (Tγ′)γ′∈Γ′ is given by Tγ′(x) = φ(γ′)x.

• Fix two bounded open sets A1 and A2 with boundary of zero
• measure. Consider a countable dense subgroup Γ ⊂ G, and a family
• f sets Bt ⊂ Γ, for example, balls w.r.t. a left-invariant metric.

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• For each x ∈ X the orbit Γ · x is dense in X and we form the ratios
• γ ∈ Bt ; γ−1x ∈ A1
• |{γ ∈ Bt ; γ−1x ∈ A2}| .

Wrawick, ETDS 30

• For each x ∈ X the orbit Γ · x is dense in X and we form the ratios
• γ ∈ Bt ; γ−1x ∈ A1
• |{γ ∈ Bt ; γ−1x ∈ A2}| .
• Consider the problem of whether the ratios converge, and whether

in that case the limit is given by lim

t→∞

• γ∈Bt χA1(γ−1x)
• γ∈Bt χA2(γ−1x) = mX(A1)

mX(A2).

Wrawick, ETDS 30

• For each x ∈ X the orbit Γ · x is dense in X and we form the ratios
• γ ∈ Bt ; γ−1x ∈ A1
• |{γ ∈ Bt ; γ−1x ∈ A2}| .
• Consider the problem of whether the ratios converge, and whether

in that case the limit is given by lim

t→∞

• γ∈Bt χA1(γ−1x)
• γ∈Bt χA2(γ−1x) = mX(A1)

mX(A2).

• For isometric action it is natural to ask whether the ratios converge

in fact everywhere (instead of almost everywhere).

Wrawick, ETDS 30

• For each x ∈ X the orbit Γ · x is dense in X and we form the ratios
• γ ∈ Bt ; γ−1x ∈ A1
• |{γ ∈ Bt ; γ−1x ∈ A2}| .
• Consider the problem of whether the ratios converge, and whether

in that case the limit is given by lim

t→∞

• γ∈Bt χA1(γ−1x)
• γ∈Bt χA2(γ−1x) = mX(A1)

mX(A2).

• For isometric action it is natural to ask whether the ratios converge

in fact everywhere (instead of almost everywhere).

• This problem was studied by Kazhdan [1965], in the case of certain

dense 2-generator subsemigroups of Isom(R2) acting on the plane.

Wrawick, ETDS 30

Ratio equidistribution of dense subgroups of amenable groups

• Assuming one of the generators was an irrational rotation, ratio

equidistribution was established by Kazhdan for plane isometries. Here Bt were in fact taken as balls in the free semigroup, and not as balls w.r.t. the word metric. This amounts to considering weighted averages on Γ, the weights being given by convolution powers.

Wrawick, ETDS 30

Ratio equidistribution of dense subgroups of amenable groups

• Assuming one of the generators was an irrational rotation, ratio

equidistribution was established by Kazhdan for plane isometries. Here Bt were in fact taken as balls in the free semigroup, and not as balls w.r.t. the word metric. This amounts to considering weighted averages on Γ, the weights being given by convolution powers.

• This result was generalized by Guivarc’h [1978] who considered

weighted averages given by convolution powers on dense subsemigroups of Isom(Rn) acting on Rn (filling also a gap in the argument given by Kazhdan), and later on by Vorobets [2004]. An important further advance was recently obtained by Varju [2012]

Wrawick, ETDS 30

Ratio equidistribution of dense subgroups of amenable groups

• Assuming one of the generators was an irrational rotation, ratio

equidistribution was established by Kazhdan for plane isometries. Here Bt were in fact taken as balls in the free semigroup, and not as balls w.r.t. the word metric. This amounts to considering weighted averages on Γ, the weights being given by convolution powers.

• This result was generalized by Guivarc’h [1978] who considered

weighted averages given by convolution powers on dense subsemigroups of Isom(Rn) acting on Rn (filling also a gap in the argument given by Kazhdan), and later on by Vorobets [2004]. An important further advance was recently obtained by Varju [2012]

• Breuillard has obtained positive results when X is the Heisenberg

group [2005] and the averages are convolution powers on a dense subgroup, and also when X is any simply-connected nilpotent Lie group and the averages are uniform over balls in the Cayley graph of the group Γ [2010].

Wrawick, ETDS 30

Ratio ergodic theorems on homogeneous spaces

• More generally, consider a non-compact lcsc space X, with the

Γ-action preserving an infinite Radon measure µ. Let Γt ⊂ Γ be a growing family of finite sets.

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Ratio ergodic theorems on homogeneous spaces

• More generally, consider a non-compact lcsc space X, with the

Γ-action preserving an infinite Radon measure µ. Let Γt ⊂ Γ be a growing family of finite sets.

• Fix a choice of growth rate function V(t), and consider the
• perators, defined on a compactly supported test-function f : X → R

πX(λt)f(x) = 1 V(t)

• γ∈Γt

f(γ−1x) .

Wrawick, ETDS 30

Ratio ergodic theorems on homogeneous spaces

• More generally, consider a non-compact lcsc space X, with the

Γ-action preserving an infinite Radon measure µ. Let Γt ⊂ Γ be a growing family of finite sets.

• Fix a choice of growth rate function V(t), and consider the
• perators, defined on a compactly supported test-function f : X → R

πX(λt)f(x) = 1 V(t)

• γ∈Γt

f(γ−1x) .

• Of course, to show that ratios converge, it is not necessary to

establish the much stronger result that both the numerator and the denominator converge at a common rate and find an explicit expression for the rate.

Wrawick, ETDS 30

Ratio ergodic theorems on homogeneous spaces

• More generally, consider a non-compact lcsc space X, with the

Γ-action preserving an infinite Radon measure µ. Let Γt ⊂ Γ be a growing family of finite sets.

• Fix a choice of growth rate function V(t), and consider the
• perators, defined on a compactly supported test-function f : X → R

πX(λt)f(x) = 1 V(t)

• γ∈Γt

f(γ−1x) .

• Of course, to show that ratios converge, it is not necessary to

establish the much stronger result that both the numerator and the denominator converge at a common rate and find an explicit expression for the rate.

• Nevertheless, we will establish such a result for diverse family of

actions of certain non-amenable groups. Furthermore, the rate of convergence turns out to be crucial in several applications.

Wrawick, ETDS 30

Ratio ergodic theorems : some surprises

Anticipating the results described below, we note that actions of non-amenable groups preserving a σ-finite measure exhibit several new phenomena that do not have analogues in classical amenable ergodic theory, as follows.

Wrawick, ETDS 30

Ratio ergodic theorems : some surprises

Anticipating the results described below, we note that actions of non-amenable groups preserving a σ-finite measure exhibit several new phenomena that do not have analogues in classical amenable ergodic theory, as follows.

• 1. The operators πX(λt), as well as the ratios
• γ ∈ Bt ; γ−1x ∈ A1
• |{γ ∈ Bt ; γ−1x ∈ A2}|

may converge, but the measure νx appearing in the limiting expression νx(A1)

νx(A2) may be different than the invariant measure.

Wrawick, ETDS 30

Ratio ergodic theorems : some surprises

Anticipating the results described below, we note that actions of non-amenable groups preserving a σ-finite measure exhibit several new phenomena that do not have analogues in classical amenable ergodic theory, as follows.

• 1. The operators πX(λt), as well as the ratios
• γ ∈ Bt ; γ−1x ∈ A1
• |{γ ∈ Bt ; γ−1x ∈ A2}|

may converge, but the measure νx appearing in the limiting expression νx(A1)

νx(A2) may be different than the invariant measure.

• 2. This can happen even when the invariant measure is unique and

the action is isometric. Moreover, the limit measure νx may depend non-trivially on the initial point x.

Wrawick, ETDS 30

Ratio ergodic theorems : some surprises

Anticipating the results described below, we note that actions of non-amenable groups preserving a σ-finite measure exhibit several new phenomena that do not have analogues in classical amenable ergodic theory, as follows.

• 1. The operators πX(λt), as well as the ratios
• γ ∈ Bt ; γ−1x ∈ A1
• |{γ ∈ Bt ; γ−1x ∈ A2}|

may converge, but the measure νx appearing in the limiting expression νx(A1)

νx(A2) may be different than the invariant measure.

• 2. This can happen even when the invariant measure is unique and

the action is isometric. Moreover, the limit measure νx may depend non-trivially on the initial point x.

• 3. The limit measures may depend non-trivially on the family of sets

Bt which are taken as the support of the measures λt, even when the action is isometric.

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• 4. Under a spectral gap assumption, the operators πX(λt) may

converge with a uniform rate of convergence, valid for almost all

• points. In isometric actions, the rate can be uniform over all

points (for Hölder functions). This implies of course that equidistribution of orbits points, or their ratios, takes place at a uniform rate.

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• 4. Under a spectral gap assumption, the operators πX(λt) may

converge with a uniform rate of convergence, valid for almost all

• points. In isometric actions, the rate can be uniform over all

points (for Hölder functions). This implies of course that equidistribution of orbits points, or their ratios, takes place at a uniform rate.

• Some of the facts above were first exhibited by a pioneering work
• n Ledrappier [1999] on SL2(Z) acting on R2. A major generalization

was obtained by Gorodnik and (Barak) Weiss [2006], who analyzed the problem systematically. Other contributions were by Ledrappier-Pollicott, Maucourant, and Oh.

Wrawick, ETDS 30

• 4. Under a spectral gap assumption, the operators πX(λt) may

converge with a uniform rate of convergence, valid for almost all

• points. In isometric actions, the rate can be uniform over all

points (for Hölder functions). This implies of course that equidistribution of orbits points, or their ratios, takes place at a uniform rate.

• Some of the facts above were first exhibited by a pioneering work
• n Ledrappier [1999] on SL2(Z) acting on R2. A major generalization

was obtained by Gorodnik and (Barak) Weiss [2006], who analyzed the problem systematically. Other contributions were by Ledrappier-Pollicott, Maucourant, and Oh.

• We will demonstrate Facts 1- 4 below in the case of dense

subgroups acting isometrically by translations. Namely we establish pointwise everywhere convergence with a uniform rate, giving an effective form to the results of Gorodnik and (Barak) Weiss [2006].

Wrawick, ETDS 30

Dense groups of isometries : some examples

Let G = SL2(R), Γ = SL2(Z[ 1

p]). Let H(γ) = γ · |γ|p be the height

function given by the product of the Euclidean norm and the p-adic norm associated with the p-adic valuation.

Wrawick, ETDS 30

Dense groups of isometries : some examples

Let G = SL2(R), Γ = SL2(Z[ 1

p]). Let H(γ) = γ · |γ|p be the height

function given by the product of the Euclidean norm and the p-adic norm associated with the p-adic valuation.

• Let Γt = {γ ∈ Γ ; log H(γ) < t}.

Wrawick, ETDS 30

Dense groups of isometries : some examples

Let G = SL2(R), Γ = SL2(Z[ 1

p]). Let H(γ) = γ · |γ|p be the height

function given by the product of the Euclidean norm and the p-adic norm associated with the p-adic valuation.

• Let Γt = {γ ∈ Γ ; log H(γ) < t}.

Theorem A. There exist α ∈ R+ and β ∈ N such that for every Hölder function on G with exponent c and compact support and for every x ∈ G 1 tβ−1eαt

• γ∈Γt

f(γx) =

• G

f(g)dmG(g) H(gx)α + Of,x(t−θc) uniformly for x in compact sets, where mG is a Haar measure on G and θc > 0.

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effective ratio theorem

We therefore also have the following consequence for ratios :

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effective ratio theorem

We therefore also have the following consequence for ratios : if f1 and f2 are continuous and f2 ≥ 0 (and not identically zero), then for every x1, x2 ∈ G, lim

t→∞

• γ∈Γt f1(γx1)
• γ∈Γt f2(γx2) =
• G f1(g)H(gx1)−αdmG(g)
• G f2(g)H(gx2)−αdmG(g),

Wrawick, ETDS 30

effective ratio theorem

We therefore also have the following consequence for ratios : if f1 and f2 are continuous and f2 ≥ 0 (and not identically zero), then for every x1, x2 ∈ G, lim

t→∞

• γ∈Γt f1(γx1)
• γ∈Γt f2(γx2) =
• G f1(g)H(gx1)−αdmG(g)
• G f2(g)H(gx2)−αdmG(g),

if in addition f1 and f2 are Hölder continuous with exponent c, then for every x1, x2 ∈ X,

• γ∈Γ: H(γ)<t f1(γx1)
• γ∈Γ: H(γ)<t f2(γx2) =
• G f1(g)H(gx1)−αdmG(g)
• G f2(g)H(gx2)−αdmG(g) +Of1,f2,x1,x2(t−θc)

uniformly over x1, x2 in compact sets.

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• Thus the ratios converge for every point, with uniform rate, but the

limit is not the ratio of the integrals with respect to the isometry invariant measure, but with respect to a different measure.

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• Thus the ratios converge for every point, with uniform rate, but the

limit is not the ratio of the integrals with respect to the isometry invariant measure, but with respect to a different measure.

• We also remark that if f1 and f2 > 0 are bounded measurable

functions with bounded support, then the ratios converge to the stated limit at almost every point, with uniform rate.

Wrawick, ETDS 30

• Thus the ratios converge for every point, with uniform rate, but the

limit is not the ratio of the integrals with respect to the isometry invariant measure, but with respect to a different measure.

• We also remark that if f1 and f2 > 0 are bounded measurable

functions with bounded support, then the ratios converge to the stated limit at almost every point, with uniform rate.

• These results are consequences of general ergodic theorems on

homogeneous spaces based on the duality principle:

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• Thus the ratios converge for every point, with uniform rate, but the

limit is not the ratio of the integrals with respect to the isometry invariant measure, but with respect to a different measure.

• We also remark that if f1 and f2 > 0 are bounded measurable

functions with bounded support, then the ratios converge to the stated limit at almost every point, with uniform rate.

• These results are consequences of general ergodic theorems on

homogeneous spaces based on the duality principle:

• Letting L = SL2(R) × SL2(Qp), H = SL2(Qp), and Γ = SL2(Z[ 1

p]),

the principle asserts that the ergodic properties of the H-action on the homogeneous space L/Γ can be used to deduce the ergodic properties of the Γ-action on the homogeneous space L/H = SL2(R).

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• Thus the ratios converge for every point, with uniform rate, but the

limit is not the ratio of the integrals with respect to the isometry invariant measure, but with respect to a different measure.

• We also remark that if f1 and f2 > 0 are bounded measurable

functions with bounded support, then the ratios converge to the stated limit at almost every point, with uniform rate.

• These results are consequences of general ergodic theorems on

homogeneous spaces based on the duality principle:

• Letting L = SL2(R) × SL2(Qp), H = SL2(Qp), and Γ = SL2(Z[ 1

p]),

the principle asserts that the ergodic properties of the H-action on the homogeneous space L/Γ can be used to deduce the ergodic properties of the Γ-action on the homogeneous space L/H = SL2(R).

• Let us describe some further examples regarding the distribution of
• rbits of arithmetic groups on homogeneous algebraic varieties.

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Distribution of orbits in the de-Sitter space : Arnold’s problem

• Consider the groups of isometries of the d-dimensional de Sitter

space dSd. This space can be realised as a hypersurface in Rd+1 : x2

1 + · · · + x2 d − x2 d+1 = 1

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Distribution of orbits in the de-Sitter space : Arnold’s problem

• Consider the groups of isometries of the d-dimensional de Sitter

space dSd. This space can be realised as a hypersurface in Rd+1 : x2

1 + · · · + x2 d − x2 d+1 = 1

• The problem of distribution of orbits of lattice subgroups on the de

Sitter space was raised by V. Arnol’d [1996].

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Let Γ be a lattice subgroup of the orthogonal group SO(d, 1) and denote by · the standard Euclidean norm on the space of (d + 1)-dimensional matrices.

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Let Γ be a lattice subgroup of the orthogonal group SO(d, 1) and denote by · the standard Euclidean norm on the space of (d + 1)-dimensional matrices.

• Let Γt = {γ ∈ Γ ; log γ < t}.

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Let Γ be a lattice subgroup of the orthogonal group SO(d, 1) and denote by · the standard Euclidean norm on the space of (d + 1)-dimensional matrices.

• Let Γt = {γ ∈ Γ ; log γ < t}.

Theorem B. When d = 2, for every φ ∈ L1(dS2) with compact support lim

t→∞

1 t

• γ∈Γt

φ(γ−1x) =

• dSd φ dν

almost everywhere, (1) where ν denotes a (nonzero) invariant measure on dS2 (whose normalization depends on Γ).

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When d ≥ 3, for every continuous function φ with compact support and every x ∈ dSd with dense Γ-orbit, lim

t→∞

1 e(d−2)t

• γ∈Γt

φ(γ−1x) =

• dSd φ dνx,

(2) where the limit measure is dνx(y) =

dν(y) (1+x2)(d−2)/2(1+y2)(d−2)/2 .

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When d ≥ 3, for every continuous function φ with compact support and every x ∈ dSd with dense Γ-orbit, lim

t→∞

1 e(d−2)t

• γ∈Γt

φ(γ−1x) =

• dSd φ dνx,

(2) where the limit measure is dνx(y) =

dν(y) (1+x2)(d−2)/2(1+y2)(d−2)/2 .

Moreover, for sufficiently nice functions φ, and for almost every x ∈ dSd, convergence takes place exponentially fast : 1 e(d−2)t

• γ∈Γ: log γ≤t

φ(γ−1x) =

• dSd φ(y) dνx(y) + Oφ,x(e−δt) (3)

with δ = δ(φ, x) > 0.

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• The above equidistribution results have several interesting features,

as follows.

Wrawick, ETDS 30

• The above equidistribution results have several interesting features,

as follows.

• In the case d = 2, while the cardinality of the set

Γt = {γ ∈ Γ : log γ ≤ t} grows exponentially as const · et, it turns

• ut that only for a polynomial number of points γ does the orbit points

γx come back to a compact set. Nonetheless, the small fraction of points returning by time t becomes equidistributed in the compact set, with respect to the invariant measure as t → ∞.

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• The above equidistribution results have several interesting features,

as follows.

• In the case d = 2, while the cardinality of the set

Γt = {γ ∈ Γ : log γ ≤ t} grows exponentially as const · et, it turns

• ut that only for a polynomial number of points γ does the orbit points

γx come back to a compact set. Nonetheless, the small fraction of points returning by time t becomes equidistributed in the compact set, with respect to the invariant measure as t → ∞.

• When d ≥ 3, while the cardinality of the set {γ ∈ Γ : log γ ≤ t}

grows as const · e(d−1)t, only an exponentially small fraction of them satisfy that γx returns to a compact set. The set of returning points does become equidistributied in the compact set, but this time the limiting measures νx are not invariant under Γ, and furthermore depend nontrivially on x. The measures νx, x ∈ X, on the pseudo-Riemannian space X should be considered as analogues of the Patterson–Sullivan measures in this context.

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Equidistribution of matrices with entries in an algebraic number field

• Fix a non-square integer d > 0 and let us consider the action of the

dense subgroup Γ = SL2(Z[ √ d]) ⊂ SL2(R) on the upper half plane H by fractional linear transformations.

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Equidistribution of matrices with entries in an algebraic number field

• Fix a non-square integer d > 0 and let us consider the action of the

dense subgroup Γ = SL2(Z[ √ d]) ⊂ SL2(R) on the upper half plane H by fractional linear transformations.

• Set Γt = {γ ∈ Γ : log(γ2 + ¯

γ2) ≤ t} with · the standard Euclidean norm, and ¯ γ the Galois involution of the field Q( √ d).

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Equidistribution of matrices with entries in an algebraic number field

• Fix a non-square integer d > 0 and let us consider the action of the

dense subgroup Γ = SL2(Z[ √ d]) ⊂ SL2(R) on the upper half plane H by fractional linear transformations.

• Set Γt = {γ ∈ Γ : log(γ2 + ¯

γ2) ≤ t} with · the standard Euclidean norm, and ¯ γ the Galois involution of the field Q( √ d).

• Since Γ embeds diagonally in G = SL2(R) × SL2(R) as an

irreducible lattice, and H is a homogeneous space of the group G, duality applies in this case. The action of Γ is isometric, and the following fast equidistribution result holds :

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Equidistribution of matrices with entries in an algebraic number field

• Fix a non-square integer d > 0 and let us consider the action of the

dense subgroup Γ = SL2(Z[ √ d]) ⊂ SL2(R) on the upper half plane H by fractional linear transformations.

• Set Γt = {γ ∈ Γ : log(γ2 + ¯

γ2) ≤ t} with · the standard Euclidean norm, and ¯ γ the Galois involution of the field Q( √ d).

• Since Γ embeds diagonally in G = SL2(R) × SL2(R) as an

irreducible lattice, and H is a homogeneous space of the group G, duality applies in this case. The action of Γ is isometric, and the following fast equidistribution result holds : Theorem C. There exists δ > 0 such that for every φ ∈ C1(H2) with support contained in a compact set D, and for every x ∈ D 1 et

• γ∈Γt

φ(xγ) =

• H2 φ(z) dν(z) + OD(e−δt),

where ν is the G-invariant measure on H.

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The duality principle on homogeneous spaces

• The techniques we develop apply in the following general setting.

Let G ⊂ SLd(R) be connected and of finite index in an algebraic group, and Γ a discrete subgroup of G with finite covolume.

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The duality principle on homogeneous spaces

• The techniques we develop apply in the following general setting.

Let G ⊂ SLd(R) be connected and of finite index in an algebraic group, and Γ a discrete subgroup of G with finite covolume.

• Let X be an algebraic homogeneous space of G equipped with a

smooth measure on which Γ acts ergodically.

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The duality principle on homogeneous spaces

• The techniques we develop apply in the following general setting.

Let G ⊂ SLd(R) be connected and of finite index in an algebraic group, and Γ a discrete subgroup of G with finite covolume.

• Let X be an algebraic homogeneous space of G equipped with a

smooth measure on which Γ acts ergodically.

• We fix a proper homogeneous polynomial p : Md(R) → [0, ∞) and

consider the sets Γt = {γ ∈ Γ : log p(γ) ≤ t}.

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The duality principle on homogeneous spaces

• The techniques we develop apply in the following general setting.

Let G ⊂ SLd(R) be connected and of finite index in an algebraic group, and Γ a discrete subgroup of G with finite covolume.

• Let X be an algebraic homogeneous space of G equipped with a

smooth measure on which Γ acts ergodically.

• We fix a proper homogeneous polynomial p : Md(R) → [0, ∞) and

consider the sets Γt = {γ ∈ Γ : log p(γ) ≤ t}.

• Our goal is to describe the asymptotic distribution of orbits of Γ, or

in other words the asymptotic behavior of the sums

γ∈Γt φ(γ−1x) as

t → ∞ for a sufficiently rich collection of functions φ on X with compact support.

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Normalizing the ergodic sums

• In order to find the right normalisation for the sum

γ∈Γt φ(γ−1x),

• ne needs to compute what proportion of points in the orbit returns to

compact subsets Ω ⊂ X.

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Normalizing the ergodic sums

• In order to find the right normalisation for the sum

γ∈Γt φ(γ−1x),

• ne needs to compute what proportion of points in the orbit returns to

compact subsets Ω ⊂ X.

• Given a compact subset Ω of X with non-empty interior and x ∈ X,

we set a = lim sup

t→∞

log |Γ−1

t

x ∩ Ω| t .

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Normalizing the ergodic sums

• In order to find the right normalisation for the sum

γ∈Γt φ(γ−1x),

• ne needs to compute what proportion of points in the orbit returns to

compact subsets Ω ⊂ X.

• Given a compact subset Ω of X with non-empty interior and x ∈ X,

we set a = lim sup

t→∞

log |Γ−1

t

x ∩ Ω| t .

• One can check that this quantity is independent of the choices of Ω

and x. We will distinguish two cases according to whether a = 0, i.e., the return rates are at most subexponentially, and a > 0, i.e., the return rates are exponential.

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Theorem D : polynomial return rates

Assume that a = 0. Then there exists b ∈ Z>0 such that the averaging operator Atφ(x) = 1 tb

• γ∈Γt

φ(γ−1x) satisfy strong maximal inequality : For every p > 1, a compact D ⊂ X, and φ ∈ Lp(D),

• sup

t≥t0

|Atφ|

• Lp(D)

≤ const(p, D) · φLp(D) pointwise ergodic theorem : For every φ ∈ L1(X) with compact support, lim

t→∞ Atφ(x) =

• X

φ dν almost everywhere, where ν is a (nonzero) G-invariant measure on X.

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Theorem E : exponential return rates

When a > 0, we prove analogous results for functions lying in a suitable Sobolev space Lp

l (X). In this case it is also possible to

establish rates of convergences. Assume that a > 0. Then there exist b ∈ Z≥0 and l ∈ Z≥0 such that the following holds for the averaging operators Atφ(x) = 1 eattb

• γ∈Γt

φ(γ−1 · x) strong maximal inequality : For every p > 1, a compact D ⊂ X, and a nonnegative φ ∈ Lp

l (D),

• t≥t0

|Atφ|

• Lp(D)

≤ const(p, D) · φLp

l (D)

pointwise ergodic theorem : For every p > 1 and a nonnegative bounded φ ∈ Lp

l (X) with compact support,

lim

t→∞ Atφ(x) =

• X

φ dνx almost everywhere,

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where νx, x ∈ X, is a family of absolutely continuous measures on X with positive densities.

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where νx, x ∈ X, is a family of absolutely continuous measures on X with positive densities. quantitative pointwise ergodic theorem. For sufficiently nice functions φ Atφ(x) =

• X

φ dνx +

b

• i=1

ci(x, φ)t−i + Oφ,x

• e−δt

almost everywhere, with δ = δ(x, φ) > 0.

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where νx, x ∈ X, is a family of absolutely continuous measures on X with positive densities. quantitative pointwise ergodic theorem. For sufficiently nice functions φ Atφ(x) =

• X

φ dνx +

b

• i=1

ci(x, φ)t−i + Oφ,x

• e−δt

almost everywhere, with δ = δ(x, φ) > 0.

• If in the Theorem we additionally assume that the stabilizer of a

point x in G is semisimple, then the stated results can be further

• improved. One can replace the Sobolev norms by Lp norms.

Moreover, one can show for continuous functions φ with compact support, the asymptotic formula holds for all x ∈ X whose Γ-orbit in X is dense.

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Ideas of the proof

To indicate some ideas that lie behind the proof of Theorems D and E, the argument can be divided into two main steps: (I) compare the asymptotic behaviour of discrete averages

• γ∈Γ: log p(γ)≤t

φ(γ−1x) (4) with the asymptotic behavior of continuous averages

• g∈G: log p(g)≤t

φ(g−1x) dmG(g), (5) where mG is an invariant measure on G. (II) establish the asymptotics for the continuous averages.

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To achieve step (I), we think of G as a fiber bundle over X, and show that the fibers of this bundle become equidistributed on the space Γ\G. More explicitly, we identify X with the homogeneous space G/H where H is a subgroup of G. For u, v ∈ G, we set Ht[u, v] = {h ∈ H : log p(uhv) ≤ t}. Then we show that the discrete sum can be approximated by a finite linear combination of integrals

• Hti [ui,vi]

fi(yih) dmH(h) with suitably chosen ti ≈ t, ui, vi ∈ G, fi : Γ\G → R, and yi ∈ Γ\G, where mH is a left invariant measure on H.

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The asymptotic behaviour of these integrals can be analysed using either representation-theoretic techniques or the theory of unipotent flows (the approach developed by Gorodnik and (Barak) Weiss). In both cases we conclude that as t → ∞, 1 |Ht[u, v]|

• Ht[u,v]

f(yh)dmH(h) →

• Γ\G

f dmΓ\G, where mΓ\G denotes the normalised invariant measure on Γ\G. Finally, this allows to conclude that the above linear combination is approximately equal to the integral, which completes step (I).

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In step (II), we establish an asymptotic formula for integral v(t) =

• g∈G: log p(g)≤t

φ(g−1x) dm(g) where φ is a sufficiently nice function. For this, we consider its transform ψ(s) = ∞ t−sv(log t) dt which converges when Re(s) is sufficiently large. Using a suitable version of resolution of singularities we show that ψ(s) has a meromorphic continuation beyond the first pole. Then the asymptotic formula for v(t) follows via a Tauberian argument.

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