### Representation theory, discrete lattice subgroups,

### effective ergodic theorems, and applications

May 29, 2016

Geometric Analysis on Discrete Groups RIMS workshop, Kyoto

Amos Nevo, Technion

**Based on joint work with Alex Gorodnik, and on joint work with**
**Anish Ghosh and Alex Gorodnik**

### Plan

Talk I: Averaging operators in dynamical systems, operator norm estimates, and effective ergodic theorems for lattice subgroups

Talk II: Unitary representations, the automorphic representation of a lattice subgroup, counting lattice points and applications Talk III: An effective form for the duality principle for

homogeneous spaces and some of its applications :

equidistribution of lattice orbits, and Diophantine approximation on algebraic varieties

### Plan

Talk I: Averaging operators in dynamical systems, operator norm estimates, and effective ergodic theorems for lattice subgroups Talk II: Unitary representations, the automorphic representation of a lattice subgroup, counting lattice points and applications

Talk III: An effective form for the duality principle for homogeneous spaces and some of its applications :

equidistribution of lattice orbits, and Diophantine approximation on algebraic varieties

### Plan

Talk I: Averaging operators in dynamical systems, operator norm estimates, and effective ergodic theorems for lattice subgroups Talk II: Unitary representations, the automorphic representation of a lattice subgroup, counting lattice points and applications Talk III: An effective form for the duality principle for

homogeneous spaces and some of its applications :

equidistribution of lattice orbits, and Diophantine approximation on algebraic varieties

### Averaging in dynamical systems

A basic object of study in the context of classical Hamiltonian mechanics consists of a compact Riemannian manifoldM, the phase space, and adivergence-free vector fieldV onM.

The integral curves ofV give rise to a one-parameter group of volume-preserving transformationsau:M →M,u∈R, thetime evolution in phase space.

For a functionf :M →R, thetime averagesalong an orbit are
β_{t}f(x) = 1

t Z t

0

f(a_{u}x)du

Do the time averages converge ? If so, what is their limit ?

### Averaging in dynamical systems

A basic object of study in the context of classical Hamiltonian mechanics consists of a compact Riemannian manifoldM, the phase space, and adivergence-free vector fieldV onM.

The integral curves ofV give rise to a one-parameter group of volume-preserving transformationsau:M →M,u∈R, thetime evolution in phase space.

For a functionf :M →R, thetime averagesalong an orbit are
β_{t}f(x) = 1

t Z t

0

f(a_{u}x)du

Do the time averages converge ? If so, what is their limit ?

### Averaging in dynamical systems

A basic object of study in the context of classical Hamiltonian mechanics consists of a compact Riemannian manifoldM, the phase space, and adivergence-free vector fieldV onM.

The integral curves ofV give rise to a one-parameter group of volume-preserving transformationsau:M →M,u∈R, thetime evolution in phase space.

For a functionf :M →R, thetime averagesalong an orbit are
β_{t}f(x) = 1

t Z t

0

f(a_{u}x)du

Do the time averages converge ? If so, what is their limit ?

### Averaging in dynamical systems

For a functionf :M →R, thetime averagesalong an orbit are
β_{t}f(x) = 1

t Z t

0

f(a_{u}x)du

### Ergodicity and the ergodic hypothesis

For a domainD⊂M, the fraction of the time in[0,t]that the orbit
aux ofx spends inDis given by ^{1}_{t} Rt

0χ_{D}(aux)du.

Do thevisiting timesconverge to vol(D), namely the orbit spends time in the setDin proportion to its volume ? (we set volM =1).

Clearly, whenDis invariant under the flow, namely starting at x ∈Dthe orbitaux never leavesD, this is not the case. For the visiting times to converge to vol(D), it is necessary that invariant sets must be null or co-null.

Flows satisfying this condition are calledergodic flows. Bolzmann’s Ergodic Hypothesis : for an ergodic flow, thetime averages of an observablef converge to the space averageoff on phase space, namely toR

Mf dvol.

### Ergodicity and the ergodic hypothesis

For a domainD⊂M, the fraction of the time in[0,t]that the orbit
aux ofx spends inDis given by ^{1}_{t} Rt

0χ_{D}(aux)du.

Do thevisiting timesconverge to vol(D), namely the orbit spends time in the setDin proportion to its volume ? (we set volM =1).

Clearly, whenDis invariant under the flow, namely starting at x ∈Dthe orbitaux never leavesD, this is not the case. For the visiting times to converge to vol(D), it is necessary that invariant sets must be null or co-null.

Flows satisfying this condition are calledergodic flows. Bolzmann’s Ergodic Hypothesis : for an ergodic flow, thetime averages of an observablef converge to the space averageoff on phase space, namely toR

Mf dvol.

### Ergodicity and the ergodic hypothesis

For a domainD⊂M, the fraction of the time in[0,t]that the orbit
aux ofx spends inDis given by ^{1}_{t} Rt

0χ_{D}(aux)du.

Do thevisiting timesconverge to vol(D), namely the orbit spends time in the setDin proportion to its volume ? (we set volM =1).

Clearly, whenDis invariant under the flow, namely starting at x ∈Dthe orbitaux never leavesD, this is not the case.

For the visiting times to converge to vol(D), it is necessary that invariant sets must be null or co-null.

Flows satisfying this condition are calledergodic flows. Bolzmann’s Ergodic Hypothesis : for an ergodic flow, thetime averages of an observablef converge to the space averageoff on phase space, namely toR

Mf dvol.

### Ergodicity and the ergodic hypothesis

^{1}_{t} Rt

0χ_{D}(aux)du.

Clearly, whenDis invariant under the flow, namely starting at x ∈Dthe orbitaux never leavesD, this is not the case.

For the visiting times to converge to vol(D), it is necessary that invariant sets must be null or co-null.

Mf dvol.

### Ergodicity and the ergodic hypothesis

^{1}_{t} Rt

0χ_{D}(aux)du.

Clearly, whenDis invariant under the flow, namely starting at x ∈Dthe orbitaux never leavesD, this is not the case.

For the visiting times to converge to vol(D), it is necessary that invariant sets must be null or co-null.

Flows satisfying this condition are calledergodic flows.

Bolzmann’s Ergodic Hypothesis : for an ergodic flow, thetime averages of an observablef converge to the space averageoff on phase space, namely toR

Mf dvol.

### Ergodicity and the ergodic hypothesis

^{1}_{t} Rt

0χ_{D}(aux)du.

Flows satisfying this condition are calledergodic flows.

### The classical mean ergodic theorem

Consider any probability space(X, µ), with a one-parameter flow of measure-preserving transformationsau:X →X,u∈R.

A key observationis that the operatorsf 7→f ◦au=π_{X}(au)f are
unitary operators onL^{2}(X, µ), so thatπ_{X}(au)is a unitary group.
(Koopman, 1930)

This observation has influenced von-Neumann’s approach to the mean ergodic theorem(1932), which states :

thetime averages ^{1}_{t} Rt

0f(aux)duconverge to thespace average R

Xfdµ, inL^{2}-norm, for anyf ∈L^{2}(X, µ), if the flow is ergodic.
For the proof, von-Neumann utilized his recently established
spectral theorem for unitary operators.

### The classical mean ergodic theorem

Consider any probability space(X, µ), with a one-parameter flow of measure-preserving transformationsau:X →X,u∈R.

A key observationis that the operatorsf 7→f ◦au=π_{X}(au)f are
unitary operators onL^{2}(X, µ), so thatπ_{X}(au)is a unitary group.

(Koopman, 1930)

This observation has influenced von-Neumann’s approach to the mean ergodic theorem(1932), which states :

thetime averages ^{1}_{t} Rt

0f(aux)duconverge to thespace average R

Xfdµ, inL^{2}-norm, for anyf ∈L^{2}(X, µ), if the flow is ergodic.
For the proof, von-Neumann utilized his recently established
spectral theorem for unitary operators.

### The classical mean ergodic theorem

Consider any probability space(X, µ), with a one-parameter flow of measure-preserving transformationsau:X →X,u∈R.

A key observationis that the operatorsf 7→f ◦au=π_{X}(au)f are
unitary operators onL^{2}(X, µ), so thatπ_{X}(au)is a unitary group.

(Koopman, 1930)

This observation has influenced von-Neumann’s approach to the mean ergodic theorem(1932), which states :

thetime averages ^{1}_{t} Rt

0f(aux)duconverge to thespace average R

Xfdµ, inL^{2}-norm, for anyf ∈L^{2}(X, µ), if the flow is ergodic.
For the proof, von-Neumann utilized his recently established
spectral theorem for unitary operators.

### The classical mean ergodic theorem

A key observationis that the operatorsf 7→f ◦au=π_{X}(au)f are
unitary operators onL^{2}(X, µ), so thatπ_{X}(au)is a unitary group.

(Koopman, 1930)

thetime averages ^{1}_{t} Rt

0f(aux)duconverge to thespace average R

Xfdµ, inL^{2}-norm, for anyf ∈L^{2}(X, µ), if the flow is ergodic.

For the proof, von-Neumann utilized his recently established spectral theorem for unitary operators.

### The classical mean ergodic theorem

_{X}(au)f are
unitary operators onL^{2}(X, µ), so thatπ_{X}(au)is a unitary group.

(Koopman, 1930)

thetime averages ^{1}_{t} Rt

0f(aux)duconverge to thespace average R

Xfdµ, inL^{2}-norm, for anyf ∈L^{2}(X, µ), if the flow is ergodic.

For the proof, von-Neumann utilized his recently established spectral theorem for unitary operators.

### F. Riesz’s proof of the mean ergodic theorem, 1938

•LetHbe a Hilbert space,a_{u} :H → Ha one-parameter unitary
group,E_{I} the projection on the space of invariants,

•Apply the averaging operatorsβ_{t} =^{1}_{t} Rt

0auduto a vectorf of the formf =ash−h. Then :

β_{t}f =β_{t}(a_{s}h−h) = 1
t

Z t+s

s

a_{u}h du−
Z t

0

a_{u}h du

!

•so thatkβ_{t}fk ≤ ^{2s}_{t} khk −→0, ast → ∞. In addition 0=E_{I}f forf as
chosen, so that indeedβ_{t}f → E_{I}f in this case.

•To conclude the proof thatβ_{t}f → E_{I}f for everys, note first that iff is
invariant, thenβ_{t}f =f =E_{I}f for allt,

•and finally that the span of{a_{s}h−h;s∈R,h∈ H}is dense in the
orthogonal complement of the space of invariants.

### F. Riesz’s proof of the mean ergodic theorem, 1938

•LetHbe a Hilbert space,a_{u} :H → Ha one-parameter unitary
group,E_{I} the projection on the space of invariants,

•Apply the averaging operatorsβ_{t} =^{1}_{t} Rt

0auduto a vectorf of the formf =ash−h. Then :

β_{t}f =β_{t}(a_{s}h−h) = 1
t

Z t+s

s

a_{u}h du−
Z t

0

a_{u}h du

!

•so thatkβ_{t}fk ≤ ^{2s}_{t} khk −→0, ast → ∞. In addition 0=E_{I}f forf as
chosen, so that indeedβ_{t}f → E_{I}f in this case.

•To conclude the proof thatβ_{t}f → E_{I}f for everys, note first that iff is
invariant, thenβ_{t}f =f =E_{I}f for allt,

•and finally that the span of{a_{s}h−h;s∈R,h∈ H}is dense in the
orthogonal complement of the space of invariants.

### F. Riesz’s proof of the mean ergodic theorem, 1938

•LetHbe a Hilbert space,a_{u} :H → Ha one-parameter unitary
group,E_{I} the projection on the space of invariants,

•Apply the averaging operatorsβ_{t} =^{1}_{t} Rt

0auduto a vectorf of the formf =ash−h. Then :

β_{t}f =β_{t}(a_{s}h−h) = 1
t

Z t+s

s

a_{u}h du−
Z t

0

a_{u}h du

!

•so thatkβ_{t}fk ≤ ^{2s}_{t} khk −→0, ast → ∞. In addition 0=E_{I}f forf as
chosen, so that indeedβ_{t}f → E_{I}f in this case.

•To conclude the proof thatβ_{t}f → E_{I}f for everys, note first that iff is
invariant, thenβ_{t}f =f =E_{I}f for allt,

•and finally that the span of{a_{s}h−h;s∈R,h∈ H}is dense in the
orthogonal complement of the space of invariants.

### F. Riesz’s proof of the mean ergodic theorem, 1938

_{u} :H → Ha one-parameter unitary
group,E_{I} the projection on the space of invariants,

•Apply the averaging operatorsβ_{t} =^{1}_{t} Rt

0auduto a vectorf of the formf =ash−h. Then :

β_{t}f =β_{t}(a_{s}h−h) = 1
t

Z t+s

s

a_{u}h du−
Z t

0

a_{u}h du

!

_{t}fk ≤ ^{2s}_{t} khk −→0, ast → ∞. In addition 0=E_{I}f forf as
chosen, so that indeedβ_{t}f → E_{I}f in this case.

_{t}f → E_{I}f for everys, note first that iff is
invariant, thenβ_{t}f =f =E_{I}f for allt,

_{s}h−h;s∈R,h∈ H}is dense in the
orthogonal complement of the space of invariants.

### F. Riesz’s proof of the mean ergodic theorem, 1938

_{u} :H → Ha one-parameter unitary
group,E_{I} the projection on the space of invariants,

•Apply the averaging operatorsβ_{t} =^{1}_{t} Rt

0auduto a vectorf of the formf =ash−h. Then :

β_{t}f =β_{t}(a_{s}h−h) = 1
t

Z t+s

s

a_{u}h du−
Z t

0

a_{u}h du

!

_{t}fk ≤ ^{2s}_{t} khk −→0, ast → ∞. In addition 0=E_{I}f forf as
chosen, so that indeedβ_{t}f → E_{I}f in this case.

_{t}f → E_{I}f for everys, note first that iff is
invariant, thenβ_{t}f =f =E_{I}f for allt,

_{s}h−h;s∈R,h∈ H}is dense in the
orthogonal complement of the space of invariants.

### F. Riesz’s proof of the mean ergodic theorem, 1938

_{u} :H → Ha one-parameter unitary
group,E_{I} the projection on the space of invariants,

•Apply the averaging operatorsβ_{t} =^{1}_{t} Rt

0auduto a vectorf of the formf =ash−h. Then :

β_{t}f =β_{t}(a_{s}h−h) = 1
t

Z t+s

s

a_{u}h du−
Z t

0

a_{u}h du

!

_{t}fk ≤ ^{2s}_{t} khk −→0, ast → ∞. In addition 0=E_{I}f forf as
chosen, so that indeedβ_{t}f → E_{I}f in this case.

_{t}f → E_{I}f for everys, note first that iff is
invariant, thenβ_{t}f =f =E_{I}f for allt,

_{s}h−h;s∈R,h∈ H}is dense in the
orthogonal complement of the space of invariants.

### Meanable groups

von-Neumann was interested not only in averaging in dynamical systems, but also inaveraging on (semi-) groups.

The classical example is aBanach limit, namely a non-negative
linear functional on bounded sequences onN, which assigns to a
sequence{a_{n}}_{n≥0}and its shift{a_{n+1}}_{n≥0}the same value, and
agrees with the limit for convergent sequences.

von-Neumann (1940) considered groups which admit a

right-invariant mean, namely a translation-invariant non-negative linear functionalm(f)on bounded functions normalized so that m(1) =1, calling themmeanable groups,

von-Neumann established this class as a common generalization of compact groups and Abelian groups by proving the existence of Haar measure for compact groups, and the existence of invariant means (Banach limits) for general Abelian groups.

### Meanable groups

von-Neumann was interested not only in averaging in dynamical systems, but also inaveraging on (semi-) groups.

The classical example is aBanach limit, namely a non-negative
linear functional on bounded sequences onN, which assigns to a
sequence{a_{n}}_{n≥0}and its shift{a_{n+1}}_{n≥0}the same value, and
agrees with the limit for convergent sequences.

von-Neumann (1940) considered groups which admit a

right-invariant mean, namely a translation-invariant non-negative linear functionalm(f)on bounded functions normalized so that m(1) =1, calling themmeanable groups,

von-Neumann established this class as a common generalization of compact groups and Abelian groups by proving the existence of Haar measure for compact groups, and the existence of invariant means (Banach limits) for general Abelian groups.

### Meanable groups

von-Neumann was interested not only in averaging in dynamical systems, but also inaveraging on (semi-) groups.

The classical example is aBanach limit, namely a non-negative
linear functional on bounded sequences onN, which assigns to a
sequence{a_{n}}_{n≥0}and its shift{a_{n+1}}_{n≥0}the same value, and
agrees with the limit for convergent sequences.

von-Neumann (1940) considered groups which admit a

right-invariant mean, namely a translation-invariant non-negative linear functionalm(f)on bounded functions normalized so that m(1) =1, calling themmeanable groups,

von-Neumann established this class as a common generalization of compact groups and Abelian groups by proving the existence of Haar measure for compact groups, and the existence of invariant means (Banach limits) for general Abelian groups.

### Meanable groups

_{n}}_{n≥0}and its shift{a_{n+1}}_{n≥0}the same value, and
agrees with the limit for convergent sequences.

von-Neumann (1940) considered groups which admit a

### Asymptotic invariance

Clearly, the crucial property of the intervals[0,t]⊂Rin Riesz’s proof is that they areasymptotically invariant under translations,

namely the measure of[0,t]∆([0,t] +s)divided by the measure of[0,t]converges to zero, for any fixeds.

Given any lcsc groupG, Følner (1955) defined a family of sets Ft⊂Gof positive finite measure to be asymptotically invariant under right translations, if it satisfies for any fixedg ∈G

|F_{t}g∆Ft|

|F_{t}| −→0 ast→ ∞

Følner showed that the existence of an asymptotically invariant family is equivalent to the existence of an invariant mean, namely it characterizes meanable groups, subsequently renamed amenable groups.

### Asymptotic invariance

Clearly, the crucial property of the intervals[0,t]⊂Rin Riesz’s proof is that they areasymptotically invariant under translations, namely the measure of[0,t]∆([0,t] +s)divided by the measure of[0,t]converges to zero, for any fixeds.

Given any lcsc groupG, Følner (1955) defined a family of sets Ft⊂Gof positive finite measure to be asymptotically invariant under right translations, if it satisfies for any fixedg ∈G

|F_{t}g∆Ft|

|F_{t}| −→0 ast→ ∞

Følner showed that the existence of an asymptotically invariant family is equivalent to the existence of an invariant mean, namely it characterizes meanable groups, subsequently renamed amenable groups.

### Asymptotic invariance

Clearly, the crucial property of the intervals[0,t]⊂Rin Riesz’s proof is that they areasymptotically invariant under translations, namely the measure of[0,t]∆([0,t] +s)divided by the measure of[0,t]converges to zero, for any fixeds.

Given any lcsc groupG, Følner (1955) defined a family of sets Ft⊂Gof positive finite measure to be asymptotically invariant under right translations, if it satisfies for any fixedg ∈G

|F_{t}g∆Ft|

|F_{t}| −→0 ast→ ∞

Følner showed that the existence of an asymptotically invariant family is equivalent to the existence of an invariant mean, namely it characterizes meanable groups, subsequently renamed amenable groups.

### Asymptotic invariance

Clearly, the crucial property of the intervals[0,t]⊂Rin Riesz’s proof is that they areasymptotically invariant under translations, namely the measure of[0,t]∆([0,t] +s)divided by the measure of[0,t]converges to zero, for any fixeds.

|F_{t}g∆Ft|

|F_{t}| −→0 ast→ ∞

### Asymptotic invariance

|F_{t}g∆Ft|

|F_{t}| −→0 ast→ ∞

Følner showed that the existence of an asymptotically invariant family is equivalent to the existence of an invariant mean, namely

### Amenable groups

A straightforward application of Riesz’s argument shows that an
asymptotically invariant familysatisfies that the uniform averages
β_{t} on the setsFt converge to the projection on the space of
invariants.

Thusamenable groups satisfy the mean ergodic theorem. The main focus of ergodic theory has traditionally been on amenable groups, and asymptotically invariant sequences played a crucial role in many of the arguments.

We will mention an important consequence of asymptotic invariance below, but first let us introduce the general set-up of ergodic theorems and the averaging operators which will be our main subject.

### Amenable groups

A straightforward application of Riesz’s argument shows that an
asymptotically invariant familysatisfies that the uniform averages
β_{t} on the setsFt converge to the projection on the space of
invariants.

Thusamenable groups satisfy the mean ergodic theorem.

The main focus of ergodic theory has traditionally been on amenable groups, and asymptotically invariant sequences played a crucial role in many of the arguments.

We will mention an important consequence of asymptotic invariance below, but first let us introduce the general set-up of ergodic theorems and the averaging operators which will be our main subject.

### Amenable groups

A straightforward application of Riesz’s argument shows that an
asymptotically invariant familysatisfies that the uniform averages
β_{t} on the setsFt converge to the projection on the space of
invariants.

Thusamenable groups satisfy the mean ergodic theorem.

The main focus of ergodic theory has traditionally been on amenable groups, and asymptotically invariant sequences played a crucial role in many of the arguments.

We will mention an important consequence of asymptotic invariance below, but first let us introduce the general set-up of ergodic theorems and the averaging operators which will be our main subject.

### Amenable groups

_{t} on the setsFt converge to the projection on the space of
invariants.

Thusamenable groups satisfy the mean ergodic theorem.

The main focus of ergodic theory has traditionally been on amenable groups, and asymptotically invariant sequences played a crucial role in many of the arguments.

### Amenable groups

_{t} on the setsFt converge to the projection on the space of
invariants.

Thusamenable groups satisfy the mean ergodic theorem.

### Averaging operators and ergodic theorems for general groups

Ga locally compact second countable group, with left Haar
measurem_{G},

B_{t} ⊂Ga growing family of sets of positive finite measure,
(X, µ)an ergodic probability measure preserving action ofG.
Consider the Haar-uniform averagesβ_{t} supported onB_{t};
Basic problem :study the averaging operators

π_{X}(β_{t})f(x) = 1

|B_{t}|
Z

Bt

f(g^{−1}x)dm_{G}(g)
and their convergence properties

### Averaging operators and ergodic theorems for general groups

Ga locally compact second countable group, with left Haar
measurem_{G},

B_{t} ⊂Ga growing family of sets of positive finite measure,

(X, µ)an ergodic probability measure preserving action ofG.
Consider the Haar-uniform averagesβ_{t} supported onB_{t};
Basic problem :study the averaging operators

π_{X}(β_{t})f(x) = 1

|B_{t}|
Z

Bt

f(g^{−1}x)dm_{G}(g)
and their convergence properties

### Averaging operators and ergodic theorems for general groups

Ga locally compact second countable group, with left Haar
measurem_{G},

B_{t} ⊂Ga growing family of sets of positive finite measure,
(X, µ)an ergodic probability measure preserving action ofG.

Consider the Haar-uniform averagesβ_{t} supported onB_{t};
Basic problem :study the averaging operators

π_{X}(β_{t})f(x) = 1

|B_{t}|
Z

Bt

f(g^{−1}x)dm_{G}(g)
and their convergence properties

### Averaging operators and ergodic theorems for general groups

Ga locally compact second countable group, with left Haar
measurem_{G},

B_{t} ⊂Ga growing family of sets of positive finite measure,
(X, µ)an ergodic probability measure preserving action ofG.

Consider the Haar-uniform averagesβ_{t} supported onB_{t} ;

Basic problem :study the averaging operators
π_{X}(β_{t})f(x) = 1

|B_{t}|
Z

Bt

f(g^{−1}x)dm_{G}(g)
and their convergence properties

### Averaging operators and ergodic theorems for general groups

Ga locally compact second countable group, with left Haar
measurem_{G},

B_{t} ⊂Ga growing family of sets of positive finite measure,
(X, µ)an ergodic probability measure preserving action ofG.

Consider the Haar-uniform averagesβ_{t} supported onB_{t} ;
Basic problem :study the averaging operators

π_{X}(β_{t})f(x) = 1

|B_{t}|
Z

B

f(g^{−1}x)dm_{G}(g)

### Amenable group actions : norm of averaging operators

A significant consequence of the existence of asymptotically invariant family in an amenable groupGthe following spectral fact.

In every properly ergodic action ofGon a probability spaceX,
there exists anasymptotically invariant sequence of unit vectors
fk ∈L^{2}_{0}(with zero mean), namely a sequence such that

kπ_{X}(g)fk −fkk →0 for allg ∈G.

For countable groups this property is equivalent to the existence ofsmall asymptotically invariant sequence of sets inX and in fact characterizes countable amenable groups(del-Junco-Rosenblatt

’79, K. Schmidt ’81).

In particular, whenGis amenable all averaging operators satisfy
kπ_{X}(β)k_{L}2

0(X)=1, in every properly ergodic action.

Corollary : in properly ergodic actions of amenable groupsno rate of convergenceto the ergodic mean can be established, in the operator norm.

### Amenable group actions : norm of averaging operators

A significant consequence of the existence of asymptotically invariant family in an amenable groupGthe following spectral fact.

In every properly ergodic action ofGon a probability spaceX,
there exists anasymptotically invariant sequence of unit vectors
fk ∈L^{2}_{0}(with zero mean), namely a sequence such that

kπ_{X}(g)fk −fkk →0 for allg∈G.

For countable groups this property is equivalent to the existence ofsmall asymptotically invariant sequence of sets inX and in fact characterizes countable amenable groups(del-Junco-Rosenblatt

’79, K. Schmidt ’81).

In particular, whenGis amenable all averaging operators satisfy
kπ_{X}(β)k_{L}2

0(X)=1, in every properly ergodic action.

Corollary : in properly ergodic actions of amenable groupsno rate of convergenceto the ergodic mean can be established, in the operator norm.

### Amenable group actions : norm of averaging operators

A significant consequence of the existence of asymptotically invariant family in an amenable groupGthe following spectral fact.

In every properly ergodic action ofGon a probability spaceX,
there exists anasymptotically invariant sequence of unit vectors
fk ∈L^{2}_{0}(with zero mean), namely a sequence such that

kπ_{X}(g)fk −fkk →0 for allg∈G.

For countable groups this property is equivalent to the existence ofsmall asymptotically invariant sequence of sets inX and in fact characterizes countable amenable groups(del-Junco-Rosenblatt

’79, K. Schmidt ’81).

In particular, whenGis amenable all averaging operators satisfy
kπ_{X}(β)k_{L}2

0(X)=1, in every properly ergodic action.

Corollary : in properly ergodic actions of amenable groupsno rate of convergenceto the ergodic mean can be established, in the operator norm.

### Amenable group actions : norm of averaging operators

^{2}_{0}(with zero mean), namely a sequence such that

kπ_{X}(g)fk −fkk →0 for allg∈G.

’79, K. Schmidt ’81).

In particular, whenGis amenable all averaging operators satisfy
kπ_{X}(β)k_{L}2

0(X)=1, in every properly ergodic action.

### Amenable group actions : norm of averaging operators

^{2}_{0}(with zero mean), namely a sequence such that

kπ_{X}(g)fk −fkk →0 for allg∈G.

’79, K. Schmidt ’81).

In particular, whenGis amenable all averaging operators satisfy
kπ_{X}(β)k_{L}2

0(X)=1, in every properly ergodic action.

Corollary : in properly ergodic actions of amenable groupsno rate of convergenceto the ergodic mean can be established, in

### Non-amenable groups and spectral estimates

We conclude that whenGis non-amenable, at least in some
ergodic actions, at least some of the averaging operatorsπ_{X}(β)
arestrict contractionsonL^{2}_{0}(X).

It is natural to go back to von-Neumann’s original approach, and prove ergodic theorems via spectral methodsand operator theory.

already the simplest conclusion is useful in many situations : if
π_{X}(β)is a strict contraction, so thatkπ_{X}(β)k_{L}2

0(X)=α <1, then
the powersπ_{X}(β)^{k} =π_{X}(β^{∗k})satisfy

π_{X}(β^{∗k})−
Z

X

fdµ
L^{2}(X)

≤Cα^{k}.

We now turn to a systematic study of averaging operators which are strict contractions.

### Non-amenable groups and spectral estimates

We conclude that whenGis non-amenable, at least in some
ergodic actions, at least some of the averaging operatorsπ_{X}(β)
arestrict contractionsonL^{2}_{0}(X).

It is natural to go back to von-Neumann’s original approach, and prove ergodic theorems via spectral methodsand operator theory.

already the simplest conclusion is useful in many situations : if
π_{X}(β)is a strict contraction, so thatkπ_{X}(β)k_{L}2

0(X)=α <1, then
the powersπ_{X}(β)^{k} =π_{X}(β^{∗k})satisfy

π_{X}(β^{∗k})−
Z

X

fdµ
L^{2}(X)

≤Cα^{k}.

We now turn to a systematic study of averaging operators which are strict contractions.

### Non-amenable groups and spectral estimates

We conclude that whenGis non-amenable, at least in some
ergodic actions, at least some of the averaging operatorsπ_{X}(β)
arestrict contractionsonL^{2}_{0}(X).

It is natural to go back to von-Neumann’s original approach, and prove ergodic theorems via spectral methodsand operator theory.

already the simplest conclusion is useful in many situations : if
π_{X}(β)is a strict contraction, so thatkπ_{X}(β)k_{L}2

0(X)=α <1, then

the powersπ_{X}(β)^{k} =π_{X}(β^{∗k})satisfy

π_{X}(β^{∗k})−
Z

X

fdµ
L^{2}(X)

≤Cα^{k}.

We now turn to a systematic study of averaging operators which are strict contractions.

### Non-amenable groups and spectral estimates

_{X}(β)
arestrict contractionsonL^{2}_{0}(X).

_{X}(β)is a strict contraction, so thatkπ_{X}(β)k_{L}2

0(X)=α <1, then
the powersπ_{X}(β)^{k} =π_{X}(β^{∗k})satisfy

π_{X}(β^{∗k})−
Z

X

fdµ
L^{2}(X)

≤Cα^{k}.

We now turn to a systematic study of averaging operators which are strict contractions.

### Non-amenable groups and spectral estimates

_{X}(β)
arestrict contractionsonL^{2}_{0}(X).

_{X}(β)is a strict contraction, so thatkπ_{X}(β)k_{L}2

0(X)=α <1, then
the powersπ_{X}(β)^{k} =π_{X}(β^{∗k})satisfy

π_{X}(β^{∗k})−
Z

X

fdµ
L^{2}(X)

≤Cα^{k}.

### Spectral gaps

Definition :An ergodicG-action has aspectral gapinL^{2}(X)if one of
the following two equivalent conditions hold.

There does not exist a sequence of functions with zero mean
and unitL^{2}-norm, which is asymptoticallyG-invariant, namely for
everyg∈G,kπ_{X}(g)fk −fkk →0.

For every generating probability measureβonG

π_{X}(β)f−
Z

X

fdµ

<(1−η)kfk

for allf ∈L^{2}(X)and a fixedη(β)>0.

Hereβis generating if the support ofβ^{∗}∗βgenerates a dense
subgroup ofG.

### Spectral gaps

Definition :An ergodicG-action has aspectral gapinL^{2}(X)if one of
the following two equivalent conditions hold.

There does not exist a sequence of functions with zero mean
and unitL^{2}-norm, which is asymptoticallyG-invariant, namely for
everyg∈G,kπ_{X}(g)fk −fkk →0.

For every generating probability measureβonG

π_{X}(β)f−
Z

X

fdµ

<(1−η)kfk

for allf ∈L^{2}(X)and a fixedη(β)>0.

Hereβis generating if the support ofβ^{∗}∗βgenerates a dense
subgroup ofG.

### Spectral gaps

Definition :An ergodicG-action has aspectral gapinL^{2}(X)if one of
the following two equivalent conditions hold.

There does not exist a sequence of functions with zero mean
and unitL^{2}-norm, which is asymptoticallyG-invariant, namely for
everyg∈G,kπ_{X}(g)fk −fkk →0.

For every generating probability measureβonG

π_{X}(β)f−
Z

X

fdµ

<(1−η)kfk

for allf ∈L^{2}(X)and a fixedη(β)>0.

Hereβis generating if the support ofβ^{∗}∗βgenerates a dense
subgroup ofG.

### Spectral estimates associated with a spectral gap

Note that the spectral gap assumption implies only that

π_{X}(β_{t})_{L}0
2

<1 for eachtseparately(whenB_{t}^{−1}Btgenerates a
dense subgroup).

applications in dynamics naturally require showing that the family
of averaging operatorsπ_{X}(β_{t})converges to 0(in operator norm,
or at least strongly), whenm_{G}(B_{t})→ ∞ast→ ∞.

This problem, however, is completely open, in general. Let us demonstrate this point in the simplest cases, and formulate some natural conjectures.

LetΓbe countable and finitely generated,d the left-invariant metric associated with a finite symmetric generating set, andBn

the balls of of radiusnand centerew.r.t.d. Letβ_{n}be the
uniform measure onBn.

### Spectral estimates associated with a spectral gap

Note that the spectral gap assumption implies only that

π_{X}(β_{t})_{L}0
2

<1 for eachtseparately(whenB_{t}^{−1}Btgenerates a
dense subgroup).

applications in dynamics naturally require showing that the family
of averaging operatorsπ_{X}(β_{t})converges to 0(in operator norm,
or at least strongly), whenm_{G}(B_{t})→ ∞ast→ ∞.

This problem, however, is completely open, in general. Let us demonstrate this point in the simplest cases, and formulate some natural conjectures.

LetΓbe countable and finitely generated,d the left-invariant metric associated with a finite symmetric generating set, andBn

the balls of of radiusnand centerew.r.t.d. Letβ_{n}be the
uniform measure onBn.

### Spectral estimates associated with a spectral gap

Note that the spectral gap assumption implies only that

π_{X}(β_{t})_{L}0
2

<1 for eachtseparately(whenB_{t}^{−1}Btgenerates a
dense subgroup).

applications in dynamics naturally require showing that the family
of averaging operatorsπ_{X}(β_{t})converges to 0(in operator norm,
or at least strongly), whenm_{G}(B_{t})→ ∞ast→ ∞.

This problem, however, is completely open, in general. Let us demonstrate this point in the simplest cases, and formulate some natural conjectures.

LetΓbe countable and finitely generated,d the left-invariant metric associated with a finite symmetric generating set, andBn

the balls of of radiusnand centerew.r.t.d. Letβ_{n}be the
uniform measure onBn.

### Spectral estimates associated with a spectral gap

Note that the spectral gap assumption implies only that

π_{X}(β_{t})_{L}0
2

<1 for eachtseparately(whenB_{t}^{−1}Btgenerates a
dense subgroup).

_{X}(β_{t})converges to 0(in operator norm,
or at least strongly), whenm_{G}(B_{t})→ ∞ast→ ∞.

the balls of of radiusnand centerew.r.t.d. Letβ_{n}be the
uniform measure onBn.

### Spectral gap : some conjectures

Let(X, µ)be an ergodic action ofΓwith a spectral gap, and
assumeB_{1}is generating, so thatkπ_{X}(β_{1})k_{L}2

0(X)<1.

Conjecture I.kπ_{X}(β_{n})k_{L}2

0(X)→0.

Conjecture II.kπ_{X}(β_{n})k_{L}2

0(X)≤Cα^{n}, for someα <1

.

Note that both statements can be formulated for an arbitrary unitary representation ofΓwhich has a spectral gap. Remarkably, both are open even for the case of the regular representation ofΓ.

Conjecture III.kλΓ(β_{n})k_{`}2(Γ)→0

.

Conjecture IV.kλΓ(β_{n})k_{`}2(Γ)≤Cα^{n}, for someα <1

### Spectral gap : some conjectures

Let(X, µ)be an ergodic action ofΓwith a spectral gap, and
assumeB_{1}is generating, so thatkπ_{X}(β_{1})k_{L}2

0(X)<1.

Conjecture I.kπ_{X}(β_{n})k_{L}2

0(X)→0.

Conjecture II.kπ_{X}(β_{n})k_{L}2

0(X)≤Cα^{n}, for someα <1

.

Note that both statements can be formulated for an arbitrary unitary representation ofΓwhich has a spectral gap. Remarkably, both are open even for the case of the regular representation ofΓ.

Conjecture III.kλΓ(β_{n})k_{`}2(Γ)→0

.

Conjecture IV.kλΓ(β_{n})k_{`}2(Γ)≤Cα^{n}, for someα <1

.

### Spectral gap : some conjectures

Let(X, µ)be an ergodic action ofΓwith a spectral gap, and
assumeB_{1}is generating, so thatkπ_{X}(β_{1})k_{L}2

0(X)<1.

Conjecture I.kπ_{X}(β_{n})k_{L}2

0(X)→0.

Conjecture II.kπ_{X}(β_{n})k_{L}2

0(X)≤Cα^{n}, for someα <1.

Note that both statements can be formulated for an arbitrary unitary representation ofΓwhich has a spectral gap. Remarkably, both are open even for the case of the regular representation ofΓ.

Conjecture III.kλΓ(β_{n})k_{`}2(Γ)→0

.

Conjecture IV.kλΓ(β_{n})k_{`}2(Γ)≤Cα^{n}, for someα <1

### Spectral gap : some conjectures

_{1}is generating, so thatkπ_{X}(β_{1})k_{L}2

0(X)<1.

Conjecture I.kπ_{X}(β_{n})k_{L}2

0(X)→0.

Conjecture II.kπ_{X}(β_{n})k_{L}2

0(X)≤Cα^{n}, for someα <1.

Note that both statements can be formulated for an arbitrary unitary representation ofΓwhich has a spectral gap.

Remarkably, both are open even for the case of the regular representation ofΓ.

Conjecture III.kλΓ(β_{n})k_{`}2(Γ)→0

.

Conjecture IV.kλΓ(β_{n})k_{`}2(Γ)≤Cα^{n}, for someα <1

.

### Spectral gap : some conjectures

_{1}is generating, so thatkπ_{X}(β_{1})k_{L}2

0(X)<1.

Conjecture I.kπ_{X}(β_{n})k_{L}2

0(X)→0.

Conjecture II.kπ_{X}(β_{n})k_{L}2

0(X)≤Cα^{n}, for someα <1.

Note that both statements can be formulated for an arbitrary unitary representation ofΓwhich has a spectral gap.

Remarkably, both are open even for the case of the regular representation ofΓ.

Conjecture III.kλΓ(β_{n})k_{`}2(Γ)→0.

Conjecture IV.kλΓ(β_{n})k_{`}2(Γ)≤Cα^{n}, for someα <1

### Spectral gap : some conjectures

_{1}is generating, so thatkπ_{X}(β_{1})k_{L}2

0(X)<1.

Conjecture I.kπ_{X}(β_{n})k_{L}2

0(X)→0.

Conjecture II.kπ_{X}(β_{n})k_{L}2

0(X)≤Cα^{n}, for someα <1.

Note that both statements can be formulated for an arbitrary unitary representation ofΓwhich has a spectral gap.

Remarkably, both are open even for the case of the regular representation ofΓ.

Conjecture III.kλΓ(β_{n})k_{`}2(Γ)→0.

Conjecture IV.kλ_{Γ}(β_{n})k_{`}2(Γ)≤Cα^{n}, for someα <1.

### Some comments

The case of the regular representation has received considerable attention over the years.

For non-elementary word hyperbolic groups (and certain other classes of groups), a result much stronger than Conjecture IV has been established.

Namely, the property ofrapid decayof the convolution norms
kλΓ(β_{n})kholds, which implies that exponential decay innholds,
with the best possible rate, namelyCn^{k}|B_{n}|^{−1/2}.

But forΓ =SL3(Z)for example, even the weakest statement, namely Conjecture III (and certainly Conjecture I) seem to be completely open forany choice of word metric.

### Some comments

The case of the regular representation has received considerable attention over the years.

For non-elementary word hyperbolic groups (and certain other classes of groups), a result much stronger than Conjecture IV has been established.

Namely, the property ofrapid decayof the convolution norms
kλΓ(β_{n})kholds, which implies that exponential decay innholds,
with the best possible rate, namelyCn^{k}|B_{n}|^{−1/2}.

But forΓ =SL3(Z)for example, even the weakest statement, namely Conjecture III (and certainly Conjecture I) seem to be completely open forany choice of word metric.

### Some comments

The case of the regular representation has received considerable attention over the years.

For non-elementary word hyperbolic groups (and certain other classes of groups), a result much stronger than Conjecture IV has been established.

Namely, the property ofrapid decayof the convolution norms
kλΓ(β_{n})kholds, which implies that exponential decay innholds,
with the best possible rate, namelyCn^{k}|B_{n}|^{−1/2}.

But forΓ =SL3(Z)for example, even the weakest statement, namely Conjecture III (and certainly Conjecture I) seem to be completely open forany choice of word metric.

### Some comments

The case of the regular representation has received considerable attention over the years.

_{n})kholds, which implies that exponential decay innholds,
with the best possible rate, namelyCn^{k}|B_{n}|^{−1/2}.

### Kazhdan’s property T

A most remarkable class of non-amenable groups was unveiled by Kazhdan in 1967.

Ghas propertyT if and only if in every ergodic action it has a spectral gap. (The equivalence to the original definition was proved by Connes-Weiss ’80, K. Schmidt ’81).

In fact, an even more remarkable property holds, namely the following uniform operator norm estimate.

Ghas propertyT if and only if for every absolutely continuous
generating measureβ there existsα(β)<1, such that in every
ergodic action ofGonX, the following uniform operator norm
estimate holds :kπ_{X}(β)k_{L}2

0(X)≤α(β).

### Kazhdan’s property T

A most remarkable class of non-amenable groups was unveiled by Kazhdan in 1967.

Ghas propertyT if and only if in every ergodic action it has a spectral gap. (The equivalence to the original definition was proved by Connes-Weiss ’80, K. Schmidt ’81).

In fact, an even more remarkable property holds, namely the following uniform operator norm estimate.

Ghas propertyT if and only if for every absolutely continuous
generating measureβ there existsα(β)<1, such that in every
ergodic action ofGonX, the following uniform operator norm
estimate holds :kπ_{X}(β)k_{L}2

0(X)≤α(β).

### Kazhdan’s property T

A most remarkable class of non-amenable groups was unveiled by Kazhdan in 1967.

Ghas propertyT if and only if in every ergodic action it has a spectral gap. (The equivalence to the original definition was proved by Connes-Weiss ’80, K. Schmidt ’81).

In fact, an even more remarkable property holds, namely the following uniform operator norm estimate.

Ghas propertyT if and only if for every absolutely continuous
generating measureβ there existsα(β)<1, such that in every
ergodic action ofGonX, the following uniform operator norm
estimate holds :kπ_{X}(β)k_{L}2

0(X)≤α(β).

### Kazhdan’s property T

A most remarkable class of non-amenable groups was unveiled by Kazhdan in 1967.

_{X}(β)k_{L}2

0(X)≤α(β).