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Polynomial actions of unitary operators and idempotent ultrafilters

Mariusz Lemańczyk (based on a joint work with Vitaly Bergelson and Stanisław Kasjan)

UMK Toruń

Heraklion, 3.06-7.06.2013

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Ultrafilters

An ultrafilter p on N is a family of subsets of N satisfying (i) ∅ / ∈ p, (ii) A ∈ p and A ⊂ B ⊂ N implies B ∈ p, (iii) A, B ∈ p implies A ∩ B ∈ p and (iv) if r ∈ N and N = A1 ∪ . . . ∪ Ar, then some Ai ∈ p. (In other words, an ultrafilter is a maximal filter.) The space of ultrafilters on N is denoted by βN (and is identified with the Stone-ˇ Cech compactification of N). Any element n ∈ N can be identified with the ultrafilter {A ⊂ N : n ∈ A} (principal ultrafilter). Topology: Given A ⊂ N, let A = {p ∈ βN : A ∈ p}. The family {A : A ⊂ N} forms a basis for the open sets of βN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Ultrafilters

An ultrafilter p on N is a family of subsets of N satisfying (i) ∅ / ∈ p, (ii) A ∈ p and A ⊂ B ⊂ N implies B ∈ p, (iii) A, B ∈ p implies A ∩ B ∈ p and (iv) if r ∈ N and N = A1 ∪ . . . ∪ Ar, then some Ai ∈ p. (In other words, an ultrafilter is a maximal filter.) The space of ultrafilters on N is denoted by βN (and is identified with the Stone-ˇ Cech compactification of N). Any element n ∈ N can be identified with the ultrafilter {A ⊂ N : n ∈ A} (principal ultrafilter). Topology: Given A ⊂ N, let A = {p ∈ βN : A ∈ p}. The family {A : A ⊂ N} forms a basis for the open sets of βN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Ultrafilters

An ultrafilter p on N is a family of subsets of N satisfying (i) ∅ / ∈ p, (ii) A ∈ p and A ⊂ B ⊂ N implies B ∈ p, (iii) A, B ∈ p implies A ∩ B ∈ p and (iv) if r ∈ N and N = A1 ∪ . . . ∪ Ar, then some Ai ∈ p. (In other words, an ultrafilter is a maximal filter.) The space of ultrafilters on N is denoted by βN (and is identified with the Stone-ˇ Cech compactification of N). Any element n ∈ N can be identified with the ultrafilter {A ⊂ N : n ∈ A} (principal ultrafilter). Topology: Given A ⊂ N, let A = {p ∈ βN : A ∈ p}. The family {A : A ⊂ N} forms a basis for the open sets of βN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Ultrafilters

An ultrafilter p on N is a family of subsets of N satisfying (i) ∅ / ∈ p, (ii) A ∈ p and A ⊂ B ⊂ N implies B ∈ p, (iii) A, B ∈ p implies A ∩ B ∈ p and (iv) if r ∈ N and N = A1 ∪ . . . ∪ Ar, then some Ai ∈ p. (In other words, an ultrafilter is a maximal filter.) The space of ultrafilters on N is denoted by βN (and is identified with the Stone-ˇ Cech compactification of N). Any element n ∈ N can be identified with the ultrafilter {A ⊂ N : n ∈ A} (principal ultrafilter). Topology: Given A ⊂ N, let A = {p ∈ βN : A ∈ p}. The family {A : A ⊂ N} forms a basis for the open sets of βN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Ultrafilters

Extension of addition on N to βN: Given p, q ∈ βN and A ⊂ N, set first A − n := {y ∈ N; y + n ∈ A} and then A ∈ p + q ⇔ {n ∈ N : A − n ∈ p} ∈ q. It makes (βN, +) a compact left-continuous semitopological semigroup, meaning that for each p ∈ βN the function λp(q) = p + q is continuous. Idempotents in βN: By Ellis’ lemma, any compact left-continuous semitopological semigroup has an idempotent. Note that if p ∈ βN is an idempotent (p ∈ E(βN)) then A ∈ p ⇔ A ∈ p + p ⇔ {n ∈ N : A − n ∈ p} ∈ p.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Ultrafilters

Extension of addition on N to βN: Given p, q ∈ βN and A ⊂ N, set first A − n := {y ∈ N; y + n ∈ A} and then A ∈ p + q ⇔ {n ∈ N : A − n ∈ p} ∈ q. It makes (βN, +) a compact left-continuous semitopological semigroup, meaning that for each p ∈ βN the function λp(q) = p + q is continuous. Idempotents in βN: By Ellis’ lemma, any compact left-continuous semitopological semigroup has an idempotent. Note that if p ∈ βN is an idempotent (p ∈ E(βN)) then A ∈ p ⇔ A ∈ p + p ⇔ {n ∈ N : A − n ∈ p} ∈ p.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Ultrafilters

Extension of addition on N to βN: Given p, q ∈ βN and A ⊂ N, set first A − n := {y ∈ N; y + n ∈ A} and then A ∈ p + q ⇔ {n ∈ N : A − n ∈ p} ∈ q. It makes (βN, +) a compact left-continuous semitopological semigroup, meaning that for each p ∈ βN the function λp(q) = p + q is continuous. Idempotents in βN: By Ellis’ lemma, any compact left-continuous semitopological semigroup has an idempotent. Note that if p ∈ βN is an idempotent (p ∈ E(βN)) then A ∈ p ⇔ A ∈ p + p ⇔ {n ∈ N : A − n ∈ p} ∈ p.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Ultrafilters

Extension of addition on N to βN: Given p, q ∈ βN and A ⊂ N, set first A − n := {y ∈ N; y + n ∈ A} and then A ∈ p + q ⇔ {n ∈ N : A − n ∈ p} ∈ q. It makes (βN, +) a compact left-continuous semitopological semigroup, meaning that for each p ∈ βN the function λp(q) = p + q is continuous. Idempotents in βN: By Ellis’ lemma, any compact left-continuous semitopological semigroup has an idempotent. Note that if p ∈ βN is an idempotent (p ∈ E(βN)) then A ∈ p ⇔ A ∈ p + p ⇔ {n ∈ N : A − n ∈ p} ∈ p.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Arithmetic sequences in idempotent ultrafilters

Fix k ∈ N and p ∈ E(βN). Since k−1

i=0 (kN + i) = N is a disjoint union, only for one

0 i < k we have kN + i ∈ p. However p + p = p, so B := {n ∈ N : (kN + i) − n ∈ p} ∈ p. It follows that (kN + i) ∩ B = ∅. Take n ∈ (kN + i) ∩ B. Then for some r ∈ N, n = kr + i and also (kN + i) − n ∈ p. It follows immediately that kN ∈ p. If p ∈ E(βN) then kN ∈ p for each k ∈ N.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Arithmetic sequences in idempotent ultrafilters

Fix k ∈ N and p ∈ E(βN). Since k−1

i=0 (kN + i) = N is a disjoint union, only for one

0 i < k we have kN + i ∈ p. However p + p = p, so B := {n ∈ N : (kN + i) − n ∈ p} ∈ p. It follows that (kN + i) ∩ B = ∅. Take n ∈ (kN + i) ∩ B. Then for some r ∈ N, n = kr + i and also (kN + i) − n ∈ p. It follows immediately that kN ∈ p. If p ∈ E(βN) then kN ∈ p for each k ∈ N.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Arithmetic sequences in idempotent ultrafilters

Fix k ∈ N and p ∈ E(βN). Since k−1

i=0 (kN + i) = N is a disjoint union, only for one

0 i < k we have kN + i ∈ p. However p + p = p, so B := {n ∈ N : (kN + i) − n ∈ p} ∈ p. It follows that (kN + i) ∩ B = ∅. Take n ∈ (kN + i) ∩ B. Then for some r ∈ N, n = kr + i and also (kN + i) − n ∈ p. It follows immediately that kN ∈ p. If p ∈ E(βN) then kN ∈ p for each k ∈ N.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Arithmetic sequences in idempotent ultrafilters

Fix k ∈ N and p ∈ E(βN). Since k−1

i=0 (kN + i) = N is a disjoint union, only for one

0 i < k we have kN + i ∈ p. However p + p = p, so B := {n ∈ N : (kN + i) − n ∈ p} ∈ p. It follows that (kN + i) ∩ B = ∅. Take n ∈ (kN + i) ∩ B. Then for some r ∈ N, n = kr + i and also (kN + i) − n ∈ p. It follows immediately that kN ∈ p. If p ∈ E(βN) then kN ∈ p for each k ∈ N.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

IP-sets

Let p ∈ E(βN). By Hindman’s theorem, each A ∈ p must contain a set of the form FS((ni)i1) := {ni1 +ni2 +. . .+nik : i1 < i2 < . . . < ik, k 1}. In ergodic theory and topological dynamics the sets of the form FS((ni)i1) are called IP-sets. On the other hand, given a sequence (ni)i1 we can find p ∈ E(βN) for which FS((ni)i1) ∈ p.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

IP-sets

Let p ∈ E(βN). By Hindman’s theorem, each A ∈ p must contain a set of the form FS((ni)i1) := {ni1 +ni2 +. . .+nik : i1 < i2 < . . . < ik, k 1}. In ergodic theory and topological dynamics the sets of the form FS((ni)i1) are called IP-sets. On the other hand, given a sequence (ni)i1 we can find p ∈ E(βN) for which FS((ni)i1) ∈ p.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

IP-sets

Let p ∈ E(βN). By Hindman’s theorem, each A ∈ p must contain a set of the form FS((ni)i1) := {ni1 +ni2 +. . .+nik : i1 < i2 < . . . < ik, k 1}. In ergodic theory and topological dynamics the sets of the form FS((ni)i1) are called IP-sets. On the other hand, given a sequence (ni)i1 we can find p ∈ E(βN) for which FS((ni)i1) ∈ p.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

IP-sets

Let p ∈ E(βN). By Hindman’s theorem, each A ∈ p must contain a set of the form FS((ni)i1) := {ni1 +ni2 +. . .+nik : i1 < i2 < . . . < ik, k 1}. In ergodic theory and topological dynamics the sets of the form FS((ni)i1) are called IP-sets. On the other hand, given a sequence (ni)i1 we can find p ∈ E(βN) for which FS((ni)i1) ∈ p.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits

Let X be a (compact) metric space and p ∈ βN. p-limits: Given a sequence (xn) ⊂ X, we will write p−limn∈N xn = x if for any neighbourhood U ∋ x, the set {n ∈ N : xn ∈ U} ∈ p. Recall that by the universality property of the Stone-ˇ Cech-compactification, whenever X is a compact Hausdorff space and x = (xn)n∈N : N → X then there exists a unique continuous extension βx : βN → X of x (whose value at p ∈ βN is given by the limit of the ultrafilter of subsets of X being the pushforward of p by x). Then p−limn∈N xn = βx(p). Whenever X is a compact metric space, p−limn∈N xn exists and is unique. If additionally p = p + p, then p− lim

n∈N xn = p− lim n∈N

• p− lim

m∈N xn+m

• .

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits

Let X be a (compact) metric space and p ∈ βN. p-limits: Given a sequence (xn) ⊂ X, we will write p−limn∈N xn = x if for any neighbourhood U ∋ x, the set {n ∈ N : xn ∈ U} ∈ p. Recall that by the universality property of the Stone-ˇ Cech-compactification, whenever X is a compact Hausdorff space and x = (xn)n∈N : N → X then there exists a unique continuous extension βx : βN → X of x (whose value at p ∈ βN is given by the limit of the ultrafilter of subsets of X being the pushforward of p by x). Then p−limn∈N xn = βx(p). Whenever X is a compact metric space, p−limn∈N xn exists and is unique. If additionally p = p + p, then p− lim

n∈N xn = p− lim n∈N

• p− lim

m∈N xn+m

• .

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits

Let X be a (compact) metric space and p ∈ βN. p-limits: Given a sequence (xn) ⊂ X, we will write p−limn∈N xn = x if for any neighbourhood U ∋ x, the set {n ∈ N : xn ∈ U} ∈ p. Recall that by the universality property of the Stone-ˇ Cech-compactification, whenever X is a compact Hausdorff space and x = (xn)n∈N : N → X then there exists a unique continuous extension βx : βN → X of x (whose value at p ∈ βN is given by the limit of the ultrafilter of subsets of X being the pushforward of p by x). Then p−limn∈N xn = βx(p). Whenever X is a compact metric space, p−limn∈N xn exists and is unique. If additionally p = p + p, then p− lim

n∈N xn = p− lim n∈N

• p− lim

m∈N xn+m

• .

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits

Let X be a (compact) metric space and p ∈ βN. p-limits: Given a sequence (xn) ⊂ X, we will write p−limn∈N xn = x if for any neighbourhood U ∋ x, the set {n ∈ N : xn ∈ U} ∈ p. Recall that by the universality property of the Stone-ˇ Cech-compactification, whenever X is a compact Hausdorff space and x = (xn)n∈N : N → X then there exists a unique continuous extension βx : βN → X of x (whose value at p ∈ βN is given by the limit of the ultrafilter of subsets of X being the pushforward of p by x). Then p−limn∈N xn = βx(p). Whenever X is a compact metric space, p−limn∈N xn exists and is unique. If additionally p = p + p, then p− lim

n∈N xn = p− lim n∈N

• p− lim

m∈N xn+m

• .

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits

Let X be a (compact) metric space and p ∈ βN. p-limits: Given a sequence (xn) ⊂ X, we will write p−limn∈N xn = x if for any neighbourhood U ∋ x, the set {n ∈ N : xn ∈ U} ∈ p. Recall that by the universality property of the Stone-ˇ Cech-compactification, whenever X is a compact Hausdorff space and x = (xn)n∈N : N → X then there exists a unique continuous extension βx : βN → X of x (whose value at p ∈ βN is given by the limit of the ultrafilter of subsets of X being the pushforward of p by x). Then p−limn∈N xn = βx(p). Whenever X is a compact metric space, p−limn∈N xn exists and is unique. If additionally p = p + p, then p− lim

n∈N xn = p− lim n∈N

• p− lim

m∈N xn+m

• .

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits continued

If Y is another compact metric space and f : X → Y is continuous then p− lim

n∈N f (xn) = f

• p− lim

n∈N xn

• .

Whenever p ∈ E(βN) and k 1, {n ∈ N : xn ∈ U} ∈ p ⇔ {n ∈ N : xn ∈ U} ∩ kN ∈ p ⇔ {n ∈ N : k|n and xn ∈ U} ∈ p. It follows that to check that x ∈ X is a p-limit it is enough to deal with numbers which are multiples of a fixed k 1. We write this as p− lim

n∈N xn = p−lim k|n xn.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits continued

If Y is another compact metric space and f : X → Y is continuous then p− lim

n∈N f (xn) = f

• p− lim

n∈N xn

• .

Whenever p ∈ E(βN) and k 1, {n ∈ N : xn ∈ U} ∈ p ⇔ {n ∈ N : xn ∈ U} ∩ kN ∈ p ⇔ {n ∈ N : k|n and xn ∈ U} ∈ p. It follows that to check that x ∈ X is a p-limit it is enough to deal with numbers which are multiples of a fixed k 1. We write this as p− lim

n∈N xn = p−lim k|n xn.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits continued

If Y is another compact metric space and f : X → Y is continuous then p− lim

n∈N f (xn) = f

• p− lim

n∈N xn

• .

Whenever p ∈ E(βN) and k 1, {n ∈ N : xn ∈ U} ∈ p ⇔ {n ∈ N : xn ∈ U} ∩ kN ∈ p ⇔ {n ∈ N : k|n and xn ∈ U} ∈ p. It follows that to check that x ∈ X is a p-limit it is enough to deal with numbers which are multiples of a fixed k 1. We write this as p− lim

n∈N xn = p−lim k|n xn.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and IP-limits

Let F denote the family of finite non-empty subsets of N. Given an increasing sequence (ni) of natural numbers, for each α ∈ F we set nα =

• i∈α

ni. IP-limits: Assume that (xn) ⊂ X and x ∈ X. Assume moreover that for each ε > 0 there exists N 1 such that for each α ∈ F satisfying min α N we have d(xnα, x) < ε. Then one says that x is the IP-limit (along (ni)1) of (xn) and we write IP − lim xnα = x.

1Note that a necessary condition for IP-convergence is that xni → x when

i → ∞.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and IP-limits

Let F denote the family of finite non-empty subsets of N. Given an increasing sequence (ni) of natural numbers, for each α ∈ F we set nα =

• i∈α

ni. IP-limits: Assume that (xn) ⊂ X and x ∈ X. Assume moreover that for each ε > 0 there exists N 1 such that for each α ∈ F satisfying min α N we have d(xnα, x) < ε. Then one says that x is the IP-limit (along (ni)1) of (xn) and we write IP − lim xnα = x.

1Note that a necessary condition for IP-convergence is that xni → x when

i → ∞.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and IP-limits

Let F denote the family of finite non-empty subsets of N. Given an increasing sequence (ni) of natural numbers, for each α ∈ F we set nα =

• i∈α

ni. IP-limits: Assume that (xn) ⊂ X and x ∈ X. Assume moreover that for each ε > 0 there exists N 1 such that for each α ∈ F satisfying min α N we have d(xnα, x) < ε. Then one says that x is the IP-limit (along (ni)1) of (xn) and we write IP − lim xnα = x.

1Note that a necessary condition for IP-convergence is that xni → x when

i → ∞.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and IP-limits

Let F denote the family of finite non-empty subsets of N. Given an increasing sequence (ni) of natural numbers, for each α ∈ F we set nα =

• i∈α

ni. IP-limits: Assume that (xn) ⊂ X and x ∈ X. Assume moreover that for each ε > 0 there exists N 1 such that for each α ∈ F satisfying min α N we have d(xnα, x) < ε. Then one says that x is the IP-limit (along (ni)1) of (xn) and we write IP − lim xnα = x.

1Note that a necessary condition for IP-convergence is that xni → x when

i → ∞.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and IP-limits

Lemma (Bergelson) (A) Assume that (ni) is an increasing sequence of natural numbers. Then there exists q ∈ E(βN) such that FS((ni)i1) ∈ q and such that whenever IP − lim xnα = x, we have x = q−limn∈N xn. (B) Assume that p ∈ E(βN) and p−limn∈N xn = x. Then there exists an IP-set FS((ni)i1) such that IP − lim xnα = x. Remark: Note that in (A), q ∈ E(βN) is universal for all IP-convergences along (ni)i1, while in (B), the choice of IP-set depends on (xn) and its p-limit.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and IP-limits

Lemma (Bergelson) (A) Assume that (ni) is an increasing sequence of natural numbers. Then there exists q ∈ E(βN) such that FS((ni)i1) ∈ q and such that whenever IP − lim xnα = x, we have x = q−limn∈N xn. (B) Assume that p ∈ E(βN) and p−limn∈N xn = x. Then there exists an IP-set FS((ni)i1) such that IP − lim xnα = x. Remark: Note that in (A), q ∈ E(βN) is universal for all IP-convergences along (ni)i1, while in (B), the choice of IP-set depends on (xn) and its p-limit.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits in semitopological semigroups

Let S be a compact metric Abelian semitopological semigroup with the group of invertible elements I(S) = ∅. We will constantly assume that (JCI) Multiplication is jointly continuous at each point (i, s) ∈ I(S) × S. Example: If H stands for a separable Hilbert space and U ∈ U(H) then for S we can take the closure of {Un : n ∈ Z} in the weak

• perator topology.

Lemma Assume that p ∈ E(βN). Assume that p−limn∈N sn = s, p−limn∈N tn = t. If t ∈ I(S) then t = 1 and p−limn∈N sntn = s.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits in semitopological semigroups

Let S be a compact metric Abelian semitopological semigroup with the group of invertible elements I(S) = ∅. We will constantly assume that (JCI) Multiplication is jointly continuous at each point (i, s) ∈ I(S) × S. Example: If H stands for a separable Hilbert space and U ∈ U(H) then for S we can take the closure of {Un : n ∈ Z} in the weak

• perator topology.

Lemma Assume that p ∈ E(βN). Assume that p−limn∈N sn = s, p−limn∈N tn = t. If t ∈ I(S) then t = 1 and p−limn∈N sntn = s.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits in semitopological semigroups

Let S be a compact metric Abelian semitopological semigroup with the group of invertible elements I(S) = ∅. We will constantly assume that (JCI) Multiplication is jointly continuous at each point (i, s) ∈ I(S) × S. Example: If H stands for a separable Hilbert space and U ∈ U(H) then for S we can take the closure of {Un : n ∈ Z} in the weak

• perator topology.

Lemma Assume that p ∈ E(βN). Assume that p−limn∈N sn = s, p−limn∈N tn = t. If t ∈ I(S) then t = 1 and p−limn∈N sntn = s.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits in semitopological semigroups

Let S be a compact metric Abelian semitopological semigroup with the group of invertible elements I(S) = ∅. We will constantly assume that (JCI) Multiplication is jointly continuous at each point (i, s) ∈ I(S) × S. Example: If H stands for a separable Hilbert space and U ∈ U(H) then for S we can take the closure of {Un : n ∈ Z} in the weak

• perator topology.

Lemma Assume that p ∈ E(βN). Assume that p−limn∈N sn = s, p−limn∈N tn = t. If t ∈ I(S) then t = 1 and p−limn∈N sntn = s.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Polynomials

P denotes the additive group of polynomials P ∈ Z[x] satisfying P(0) = 0. Given N 1, PN denotes the subgroup of P of polynomials whose degree is at most N. Lemma Let S satisfy (JCI) and s ∈ I(S). 1) For each p ∈ E(βN), we have p−limn∈N sn ∈ E(S). 2) Let P ∈ P be a polynomial of degree d 1. Assume that for j = 1, ..., d − 1 there exists rj 1 such that p − lim

n∈N srjnj = 1.

Then p−limn∈N sP(n) ∈ E(S).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Polynomials

P denotes the additive group of polynomials P ∈ Z[x] satisfying P(0) = 0. Given N 1, PN denotes the subgroup of P of polynomials whose degree is at most N. Lemma Let S satisfy (JCI) and s ∈ I(S). 1) For each p ∈ E(βN), we have p−limn∈N sn ∈ E(S). 2) Let P ∈ P be a polynomial of degree d 1. Assume that for j = 1, ..., d − 1 there exists rj 1 such that p − lim

n∈N srjnj = 1.

Then p−limn∈N sP(n) ∈ E(S).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Proof

Proof We have u := p − lim

n∈N sP(n) = p − lim n∈N(p − lim m∈N sP(n+m)) =

= p − lim

n∈N sP(n)(p − lim m∈N(s(Q(n,m)+P(m)),

where Q(x, y) ∈ Z[x, y] is divisible by xy (and the x- and y-degree

• f Q is d − 1). Set r = lcm(r1, ..., rd−1) and conclude:

u = p−lim

r|n sP(n)(p− lim m∈N(s(Q(n,m)·sP(m)) = p−lim r|n sP(n)(1·u) = u2.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits from an ordinary convergence

Problem: Assume that we have a countable collection of polynomials Pi ∈ P, i 1. Assume moreover that for some sequence (qn) ⊂ N such that for each i 1 we have sPi(qn) → ei in S, where ei ∈ E(S). Can we find p ∈ E(βN) such that p−limn∈N sPi(n) = ei for each i 1?

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

From ordinary convergence to IP- and p-limits

Proposition (Bergelson-Kasjan-L.) Fix N 1 and assume that 0 = Pi ∈ PN, i 1. Assume that S satisfies (JCI) and let s ∈ I(S). Let (qn) be an increasing sequence

• f natural numbers such that for some rj 1, j = 1, . . . , N − 1

srjqj

n → 1.

Denote r = lcm(r1, . . . , rN−1) and assume in addition that r|qn for n n0. Moreover, assume that for each i 1 sPi(qn) → ei ∈ E(S). Then there exists p ∈ E(βN) such that for each i 1 p− lim

n∈N sPi(n) = ei.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Unitary operators on separable Hilbert spaces

Assume that U ∈ U(H), r > 0, f ∈ H with f = r and p ∈ E(βN). The r-ball X := {g ∈ H : g r} equipped with a metric d induced by the weak topology is a U-invariant compact metric

• space. Set f ∗ := p−limn∈N Unf .

Then p−limn∈N Unf ∗ = f ∗ (any g ∈ H satisfying p−limn∈N Ung = g is called p-rigid and it has to be of the form f ∗). Moreover, f ∗ is a rigid vector (for each ε > 0, the set {n ∈ N : d(Unf , f ) < ε} is a member of p and hence is not

• empty. Therefore, we can find an increasing subsequence (ni)

such that d(Unif , f ) → 0 which is equivalent to Unif → f in H as the weak and the strong topologies are equivalent on spheres).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Unitary operators on separable Hilbert spaces

Assume that U ∈ U(H), r > 0, f ∈ H with f = r and p ∈ E(βN). The r-ball X := {g ∈ H : g r} equipped with a metric d induced by the weak topology is a U-invariant compact metric

• space. Set f ∗ := p−limn∈N Unf .

Then p−limn∈N Unf ∗ = f ∗ (any g ∈ H satisfying p−limn∈N Ung = g is called p-rigid and it has to be of the form f ∗). Moreover, f ∗ is a rigid vector (for each ε > 0, the set {n ∈ N : d(Unf , f ) < ε} is a member of p and hence is not

• empty. Therefore, we can find an increasing subsequence (ni)

such that d(Unif , f ) → 0 which is equivalent to Unif → f in H as the weak and the strong topologies are equivalent on spheres).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Unitary operators on separable Hilbert spaces

Assume that U ∈ U(H), r > 0, f ∈ H with f = r and p ∈ E(βN). The r-ball X := {g ∈ H : g r} equipped with a metric d induced by the weak topology is a U-invariant compact metric

• space. Set f ∗ := p−limn∈N Unf .

Then p−limn∈N Unf ∗ = f ∗ (any g ∈ H satisfying p−limn∈N Ung = g is called p-rigid and it has to be of the form f ∗). Moreover, f ∗ is a rigid vector (for each ε > 0, the set {n ∈ N : d(Unf , f ) < ε} is a member of p and hence is not

• empty. Therefore, we can find an increasing subsequence (ni)

such that d(Unif , f ) → 0 which is equivalent to Unif → f in H as the weak and the strong topologies are equivalent on spheres).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and the first decomposition theorem

Assume that p ∈ E(βN) and let W := p−limn∈N Un. Then W is an idempotent and hence an orthogonal projection on the subspace

• f p −rigid vectors.

Proposition (Bergelson) Assume that U ∈ U(H) and let p ∈ E(βN). Then H = Hr ⊕ Hm, where Hr = {f ∈ H : p−limn∈N Unf = f } and Hm = {f ∈ H : p−limn∈N Unf = 0}.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and the first decomposition theorem

Assume that p ∈ E(βN) and let W := p−limn∈N Un. Then W is an idempotent and hence an orthogonal projection on the subspace

• f p −rigid vectors.

Proposition (Bergelson) Assume that U ∈ U(H) and let p ∈ E(βN). Then H = Hr ⊕ Hm, where Hr = {f ∈ H : p−limn∈N Unf = f } and Hm = {f ∈ H : p−limn∈N Unf = 0}.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Polynomial extension

The following result represents a polynomial extension of the above result. Theorem (Bergelson) For each unitary operator U ∈ U(H), each p ∈ E(βN), and each polynomial P ∈ P, p− lim

n∈N UP(n) = projF,

where F is a closed, U-invariant subspace of H.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Polynomial extension

The following result represents a polynomial extension of the above result. Theorem (Bergelson) For each unitary operator U ∈ U(H), each p ∈ E(βN), and each polynomial P ∈ P, p− lim

n∈N UP(n) = projF,

where F is a closed, U-invariant subspace of H.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

A quick application: Khinchine-type theorem

Corollary (Bergelson, Furstenberg, McCutcheon) Given P ∈ P, for each ε > 0 and A ∈ B, the set Rε(A; P) := {n ∈ Z : µ(A ∩ T P(n)A) (µ(A))2 − ε} is an IP∗-set. IP∗-sets are subsets of integers that meet every IP-set. IP∗-sets are precisely those sets which are members of each idempotent p ∈ βN (they are syndetic but not vice-versa). The proof of Corollary then goes as follows. We have p−lim

n∈N µ(A∩T −P(n)A) = projF1A, 1A (projF1A, 1)2 = (µ(A))2.

Therefore, for each p ∈ E(βN), Rε(A; P) contains the ε-neighborhood of the p−limn∈N µ(A ∩ T −P(n)A), whence Rε(A; P) ∈ p. Thus Rε(A; P) is IP∗.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

A quick application: Khinchine-type theorem

Corollary (Bergelson, Furstenberg, McCutcheon) Given P ∈ P, for each ε > 0 and A ∈ B, the set Rε(A; P) := {n ∈ Z : µ(A ∩ T P(n)A) (µ(A))2 − ε} is an IP∗-set. IP∗-sets are subsets of integers that meet every IP-set. IP∗-sets are precisely those sets which are members of each idempotent p ∈ βN (they are syndetic but not vice-versa). The proof of Corollary then goes as follows. We have p−lim

n∈N µ(A∩T −P(n)A) = projF1A, 1A (projF1A, 1)2 = (µ(A))2.

Therefore, for each p ∈ E(βN), Rε(A; P) contains the ε-neighborhood of the p−limn∈N µ(A ∩ T −P(n)A), whence Rε(A; P) ∈ p. Thus Rε(A; P) is IP∗.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

A quick application: Khinchine-type theorem

Corollary (Bergelson, Furstenberg, McCutcheon) Given P ∈ P, for each ε > 0 and A ∈ B, the set Rε(A; P) := {n ∈ Z : µ(A ∩ T P(n)A) (µ(A))2 − ε} is an IP∗-set. IP∗-sets are subsets of integers that meet every IP-set. IP∗-sets are precisely those sets which are members of each idempotent p ∈ βN (they are syndetic but not vice-versa). The proof of Corollary then goes as follows. We have p−lim

n∈N µ(A∩T −P(n)A) = projF1A, 1A (projF1A, 1)2 = (µ(A))2.

Therefore, for each p ∈ E(βN), Rε(A; P) contains the ε-neighborhood of the p−limn∈N µ(A ∩ T −P(n)A), whence Rε(A; P) ∈ p. Thus Rε(A; P) is IP∗.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and mixing properties

Proposition (Bergelson) We have U is mildly mixing if and only if p−limn∈N Un = 0 for each p ∈ E(βN). U is weakly mixing if and only if p−limn∈N Un = 0 for some p ∈ E(βN). Comments: Mild mixing (Furstenberg, Weiss) means that U has no non-zero rigid vectors. As a matter of fact, U is weakly mixing if and only if p−limn∈N Un = 0 for each minimal idempotent p ∈ βN (Bergelson). An idempotent p ∈ βN is said to be minimal if it belongs to a minimal right ideal I of (βN, +).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and mixing properties

Proposition (Bergelson) We have U is mildly mixing if and only if p−limn∈N Un = 0 for each p ∈ E(βN). U is weakly mixing if and only if p−limn∈N Un = 0 for some p ∈ E(βN). Comments: Mild mixing (Furstenberg, Weiss) means that U has no non-zero rigid vectors. As a matter of fact, U is weakly mixing if and only if p−limn∈N Un = 0 for each minimal idempotent p ∈ βN (Bergelson). An idempotent p ∈ βN is said to be minimal if it belongs to a minimal right ideal I of (βN, +).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limits and mixing properties

Proposition (Bergelson) We have U is mildly mixing if and only if p−limn∈N Un = 0 for each p ∈ E(βN). U is weakly mixing if and only if p−limn∈N Un = 0 for some p ∈ E(βN). Comments: Mild mixing (Furstenberg, Weiss) means that U has no non-zero rigid vectors. As a matter of fact, U is weakly mixing if and only if p−limn∈N Un = 0 for each minimal idempotent p ∈ βN (Bergelson). An idempotent p ∈ βN is said to be minimal if it belongs to a minimal right ideal I of (βN, +).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Polynomial p-limits - first observations

Given P, Q ∈ P, we obtain two p-limits which are certain

• rthogonal projections. Are there relations between these two

projections (for example, P(x) = x and Q(x) = 2x)? (We consider the two extremal idempotents: 0 = 0H and Id = IdH.) The relation p−limn∈N Un = 0 does not imply, in general, neither p−limn∈N U2n = 0, nor, say, p−limn∈N Un2 = 0. On the

• ther hand, one can observe that p−limn∈N Un = Id implies

p−limn∈N Ukn = Id, for any k ∈ N, but this is consistent with both p−limn∈N Un2 = 0 and p−limn∈N Un2 = Id. One can ask whether for any independent family {P1, . . . , Pm} ⊂ P and any choice of Ei ∈ {0, Id}, i = 1, . . . , m, there exist U ∈ U(H) and p ∈ E(βN) such that p−limn∈N UPi(n) = Ei for each i = 1, . . . , m.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Polynomial p-limits - first observations

Given P, Q ∈ P, we obtain two p-limits which are certain

• rthogonal projections. Are there relations between these two

projections (for example, P(x) = x and Q(x) = 2x)? (We consider the two extremal idempotents: 0 = 0H and Id = IdH.) The relation p−limn∈N Un = 0 does not imply, in general, neither p−limn∈N U2n = 0, nor, say, p−limn∈N Un2 = 0. On the

• ther hand, one can observe that p−limn∈N Un = Id implies

p−limn∈N Ukn = Id, for any k ∈ N, but this is consistent with both p−limn∈N Un2 = 0 and p−limn∈N Un2 = Id. One can ask whether for any independent family {P1, . . . , Pm} ⊂ P and any choice of Ei ∈ {0, Id}, i = 1, . . . , m, there exist U ∈ U(H) and p ∈ E(βN) such that p−limn∈N UPi(n) = Ei for each i = 1, . . . , m.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Polynomial p-limits - first observations

Given P, Q ∈ P, we obtain two p-limits which are certain

• rthogonal projections. Are there relations between these two

projections (for example, P(x) = x and Q(x) = 2x)? (We consider the two extremal idempotents: 0 = 0H and Id = IdH.) The relation p−limn∈N Un = 0 does not imply, in general, neither p−limn∈N U2n = 0, nor, say, p−limn∈N Un2 = 0. On the

• ther hand, one can observe that p−limn∈N Un = Id implies

p−limn∈N Ukn = Id, for any k ∈ N, but this is consistent with both p−limn∈N Un2 = 0 and p−limn∈N Un2 = Id. One can ask whether for any independent family {P1, . . . , Pm} ⊂ P and any choice of Ei ∈ {0, Id}, i = 1, . . . , m, there exist U ∈ U(H) and p ∈ E(βN) such that p−limn∈N UPi(n) = Ei for each i = 1, . . . , m.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Rigidity and p-rigidity

For each U ∈ U(H) and p ∈ E(βN) we have {P ∈ P : p−limn∈N UP(n) = Id} is an additive subgroup of P. Our main aim will be to study this kind of groups2. This can be viewed as a contribution to the recently revived studies

• f the phenomenon of rigidity for weakly mixing operators:

Aaronson-Hosseini-L., Adams, Bergelson-del Junco-L.-Rosenblatt, Grivaux, Grivaux-Eisner.

2This problem is interesting only for unitary operators which are weakly

• mixing. Indeed, if U ∈ U(H) has discrete spectrum (that is, the space H is

spanned by the eigenvectors of U), we have p−limn∈N UP(n) = Id for each P ∈ P and each p ∈ E(βN).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Rigidity and p-rigidity

For each U ∈ U(H) and p ∈ E(βN) we have {P ∈ P : p−limn∈N UP(n) = Id} is an additive subgroup of P. Our main aim will be to study this kind of groups2. This can be viewed as a contribution to the recently revived studies

• f the phenomenon of rigidity for weakly mixing operators:

Aaronson-Hosseini-L., Adams, Bergelson-del Junco-L.-Rosenblatt, Grivaux, Grivaux-Eisner.

2This problem is interesting only for unitary operators which are weakly

• mixing. Indeed, if U ∈ U(H) has discrete spectrum (that is, the space H is

spanned by the eigenvectors of U), we have p−limn∈N UP(n) = Id for each P ∈ P and each p ∈ E(βN).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Rigidity and p-rigidity

For each U ∈ U(H) and p ∈ E(βN) we have {P ∈ P : p−limn∈N UP(n) = Id} is an additive subgroup of P. Our main aim will be to study this kind of groups2. This can be viewed as a contribution to the recently revived studies

• f the phenomenon of rigidity for weakly mixing operators:

Aaronson-Hosseini-L., Adams, Bergelson-del Junco-L.-Rosenblatt, Grivaux, Grivaux-Eisner.

2This problem is interesting only for unitary operators which are weakly

• mixing. Indeed, if U ∈ U(H) has discrete spectrum (that is, the space H is

spanned by the eigenvectors of U), we have p−limn∈N UP(n) = Id for each P ∈ P and each p ∈ E(βN).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-limit decomposition theorem

Theorem A (Bergelson-Kasjan-L.) For each N 1, each p ∈ E(βN) and each U ∈ U(H) there exists a unique decomposition (i) H =

k1 H(N) k

into U-invariant closed subspaces such that for each P ∈ PN and k 1 we have: (ii) p−limn∈N

• U|H(N)

k

P(n)

= 0 or Id, (iii) whenever k = l, there exists Q ∈ PN such that p−limn∈N

• U|H(N)

k

Q(n)

= p−limn∈N

• U|H(N)

l

Q(n)

. Furthermore, the decomposition (i) has the following property: (iv) For any k 1, if Q ∈ PN is such that p−limn∈N

• U|H(N)

k

sQ(n)

= 0 for each s ∈ N, then p−limn∈N

• U|H(N)

k

R(n)

= 0 for each R ∈ PN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Discussion...

If we have a decomposition (i) satisfying only (ii), then by adding up subspaces that are not distinguished by p-limits, we can achieve a decomposition satisfying (ii)+(iii). If a decomposition (i) satisfying (ii)+(iii) exists then it is unique. If 1 M < N and H =

l1 H(M) l

=

k1 H(N) k

are decompositions given by Theorem A for M and N respectively, then for each k 1 there exists a unique lk 1 such that H(N)

k

⊂ H(M)

lk

. Corollary The subspaces H(N)

k

in the decomposition (i) are spectral subspaces, i.e. if the spectral measure of y ∈ H is absolutely continuous with respect to the maximal spectral type of U|H(N)

k

then y ∈ H(N)

k

.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Discussion...

If we have a decomposition (i) satisfying only (ii), then by adding up subspaces that are not distinguished by p-limits, we can achieve a decomposition satisfying (ii)+(iii). If a decomposition (i) satisfying (ii)+(iii) exists then it is unique. If 1 M < N and H =

l1 H(M) l

=

k1 H(N) k

are decompositions given by Theorem A for M and N respectively, then for each k 1 there exists a unique lk 1 such that H(N)

k

⊂ H(M)

lk

. Corollary The subspaces H(N)

k

in the decomposition (i) are spectral subspaces, i.e. if the spectral measure of y ∈ H is absolutely continuous with respect to the maximal spectral type of U|H(N)

k

then y ∈ H(N)

k

.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Discussion...

If we have a decomposition (i) satisfying only (ii), then by adding up subspaces that are not distinguished by p-limits, we can achieve a decomposition satisfying (ii)+(iii). If a decomposition (i) satisfying (ii)+(iii) exists then it is unique. If 1 M < N and H =

l1 H(M) l

=

k1 H(N) k

are decompositions given by Theorem A for M and N respectively, then for each k 1 there exists a unique lk 1 such that H(N)

k

⊂ H(M)

lk

. Corollary The subspaces H(N)

k

in the decomposition (i) are spectral subspaces, i.e. if the spectral measure of y ∈ H is absolutely continuous with respect to the maximal spectral type of U|H(N)

k

then y ∈ H(N)

k

.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Discussion...

If we have a decomposition (i) satisfying only (ii), then by adding up subspaces that are not distinguished by p-limits, we can achieve a decomposition satisfying (ii)+(iii). If a decomposition (i) satisfying (ii)+(iii) exists then it is unique. If 1 M < N and H =

l1 H(M) l

=

k1 H(N) k

are decompositions given by Theorem A for M and N respectively, then for each k 1 there exists a unique lk 1 such that H(N)

k

⊂ H(M)

lk

. Corollary The subspaces H(N)

k

in the decomposition (i) are spectral subspaces, i.e. if the spectral measure of y ∈ H is absolutely continuous with respect to the maximal spectral type of U|H(N)

k

then y ∈ H(N)

k

.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Fourier transforms of continuous measures on the circle

Corollary Assume that σ is a continuous probability Borel measure on S1 and let p ∈ βN, p + p = p. (i) If p−limn∈N σ(ln + k) = 0 for each l 1 and k ∈ Z then p− lim

n∈N

σ(Q(n)) = 0 for each positive degree polynomial Q ∈ Z[x]. (ii) If, for some P ∈ PN, we have p−limn∈N σ(lP(n) + k) = 0 for each l 1 and k ∈ Z then p− lim

n∈N

σ(Q(n)) = 0 for each Q ∈ Z[x] of degree not smaller than the degree of P.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

A problem

Problem: We have been unable to decide whether the decomposition resulting from Theorem A exists for an arbitrary infinite family of polynomials in P. It might be that the answer is negative already for the family of all monomials.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Proof of Theorem A:

The main tools to prove Theorem A are p-orthogonal projection theorem and the following p-van der Corput lemma. Lemma (Bergelson-McCutcheon) Assume that (xn) ⊂ H is bounded. If p− lim

h∈N

• p− lim

n∈Nxn+h, xn

• = 0

then p−limn∈N xn = 0.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-rigidity groups

Motivated by the decomposition result given by Theorem A, we introduce the notion of N-rigidity group. Let N ∈ N. A subgroup G ⊂ PN is called an N-rigidity group if there exist p ∈ E(βN) and U ∈ U(H) such that G = {P ∈ PN : p− lim

n∈N UP(n) = Id}

and p−limn∈N UQ(n) = 0 for each Q ∈ PN \ G. The second main goal is to prove the following result. Theorem B (Bergelson-Kasjan-L.) Assume that G ⊂ PN is a subgroup with max{deg P : P ∈ G} = N. Then G is an N-rigidity group if and

• nly if G has finite index in PN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-rigidity groups

Motivated by the decomposition result given by Theorem A, we introduce the notion of N-rigidity group. Let N ∈ N. A subgroup G ⊂ PN is called an N-rigidity group if there exist p ∈ E(βN) and U ∈ U(H) such that G = {P ∈ PN : p− lim

n∈N UP(n) = Id}

and p−limn∈N UQ(n) = 0 for each Q ∈ PN \ G. The second main goal is to prove the following result. Theorem B (Bergelson-Kasjan-L.) Assume that G ⊂ PN is a subgroup with max{deg P : P ∈ G} = N. Then G is an N-rigidity group if and

• nly if G has finite index in PN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

p-rigidity groups

Motivated by the decomposition result given by Theorem A, we introduce the notion of N-rigidity group. Let N ∈ N. A subgroup G ⊂ PN is called an N-rigidity group if there exist p ∈ E(βN) and U ∈ U(H) such that G = {P ∈ PN : p− lim

n∈N UP(n) = Id}

and p−limn∈N UQ(n) = 0 for each Q ∈ PN \ G. The second main goal is to prove the following result. Theorem B (Bergelson-Kasjan-L.) Assume that G ⊂ PN is a subgroup with max{deg P : P ∈ G} = N. Then G is an N-rigidity group if and

• nly if G has finite index in PN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

More about the definition of p-rigidity groups

We fix N 1 and p ∈ E(βN). Assume that {0} = J ⊂ H is a closed U-invariant subspace such that (ii) holds on it. Then: G(p, N, U, J ) := {P ∈ PN : p−limn∈N (U|J )P(n) = Id} is a group and for all remaining Q, i.e. Q ∈ PN \ G(p, N, U, J ), the corresponding p-limit is 0. Let N′ := maxG(p,N,U,J ) deg P. We claim that on J , for each r = 1, . . . , N′, there exists (a unique) kr 1 such that p−limn∈N (U|J )jnr = 0 for j = 1, . . . , kr − 1 and p−limn∈N (U|J )krnr = Id. Indeed, otherwise, for some 1 j N′, p−limn∈N (U|J )lnj = 0 for all l 1. In the latter case however we have p−limn∈N (U|J )P(n) = 0 for all P ∈ Pj (see Theorem A).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

More about the definition of p-rigidity groups

We fix N 1 and p ∈ E(βN). Assume that {0} = J ⊂ H is a closed U-invariant subspace such that (ii) holds on it. Then: G(p, N, U, J ) := {P ∈ PN : p−limn∈N (U|J )P(n) = Id} is a group and for all remaining Q, i.e. Q ∈ PN \ G(p, N, U, J ), the corresponding p-limit is 0. Let N′ := maxG(p,N,U,J ) deg P. We claim that on J , for each r = 1, . . . , N′, there exists (a unique) kr 1 such that p−limn∈N (U|J )jnr = 0 for j = 1, . . . , kr − 1 and p−limn∈N (U|J )krnr = Id. Indeed, otherwise, for some 1 j N′, p−limn∈N (U|J )lnj = 0 for all l 1. In the latter case however we have p−limn∈N (U|J )P(n) = 0 for all P ∈ Pj (see Theorem A).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

More about the definition of p-rigidity groups

We fix N 1 and p ∈ E(βN). Assume that {0} = J ⊂ H is a closed U-invariant subspace such that (ii) holds on it. Then: G(p, N, U, J ) := {P ∈ PN : p−limn∈N (U|J )P(n) = Id} is a group and for all remaining Q, i.e. Q ∈ PN \ G(p, N, U, J ), the corresponding p-limit is 0. Let N′ := maxG(p,N,U,J ) deg P. We claim that on J , for each r = 1, . . . , N′, there exists (a unique) kr 1 such that p−limn∈N (U|J )jnr = 0 for j = 1, . . . , kr − 1 and p−limn∈N (U|J )krnr = Id. Indeed, otherwise, for some 1 j N′, p−limn∈N (U|J )lnj = 0 for all l 1. In the latter case however we have p−limn∈N (U|J )P(n) = 0 for all P ∈ Pj (see Theorem A).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

More about the definition of p-rigidity groups

We fix N 1 and p ∈ E(βN). Assume that {0} = J ⊂ H is a closed U-invariant subspace such that (ii) holds on it. Then: G(p, N, U, J ) := {P ∈ PN : p−limn∈N (U|J )P(n) = Id} is a group and for all remaining Q, i.e. Q ∈ PN \ G(p, N, U, J ), the corresponding p-limit is 0. Let N′ := maxG(p,N,U,J ) deg P. We claim that on J , for each r = 1, . . . , N′, there exists (a unique) kr 1 such that p−limn∈N (U|J )jnr = 0 for j = 1, . . . , kr − 1 and p−limn∈N (U|J )krnr = Id. Indeed, otherwise, for some 1 j N′, p−limn∈N (U|J )lnj = 0 for all l 1. In the latter case however we have p−limn∈N (U|J )P(n) = 0 for all P ∈ Pj (see Theorem A).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

N-rigidity groups continued

We assume that N′ = N (otherwise, the corresponding group has already been considered as G(p, N′, U, J )) and let P(x) = N

r=1 arxr ∈ Z[x].

Then there are 0 jr < kr, mr ∈ Z such that ar = jr + mrkr for r = 1, . . . , N such that p−lim

n∈N (U|J )P(n) = p−lim n∈N (U|J )j1n+j2n2+...+jNnN

. For k 1 we denote Zk = Z/kZ and πk stands for the natural homomorphism Z → Zk. It follows that

• G(p, N, U, J ) :=

   πk1 × . . . × πkN(j1, . . . , jN) ∈ Zk1 ⊕ . . . ⊕ ZkN : 0 js < ks, s = 1, . . . , N and p−limn∈N (U|J )j1n+...+jNnN = Id    is a subgroup of Zk1 ⊕ . . . ⊕ ZkN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

N-rigidity groups continued

We assume that N′ = N (otherwise, the corresponding group has already been considered as G(p, N′, U, J )) and let P(x) = N

r=1 arxr ∈ Z[x].

Then there are 0 jr < kr, mr ∈ Z such that ar = jr + mrkr for r = 1, . . . , N such that p−lim

n∈N (U|J )P(n) = p−lim n∈N (U|J )j1n+j2n2+...+jNnN

. For k 1 we denote Zk = Z/kZ and πk stands for the natural homomorphism Z → Zk. It follows that

• G(p, N, U, J ) :=

   πk1 × . . . × πkN(j1, . . . , jN) ∈ Zk1 ⊕ . . . ⊕ ZkN : 0 js < ks, s = 1, . . . , N and p−limn∈N (U|J )j1n+...+jNnN = Id    is a subgroup of Zk1 ⊕ . . . ⊕ ZkN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

N-rigidity groups continued

We assume that N′ = N (otherwise, the corresponding group has already been considered as G(p, N′, U, J )) and let P(x) = N

r=1 arxr ∈ Z[x].

Then there are 0 jr < kr, mr ∈ Z such that ar = jr + mrkr for r = 1, . . . , N such that p−lim

n∈N (U|J )P(n) = p−lim n∈N (U|J )j1n+j2n2+...+jNnN

. For k 1 we denote Zk = Z/kZ and πk stands for the natural homomorphism Z → Zk. It follows that

• G(p, N, U, J ) :=

   πk1 × . . . × πkN(j1, . . . , jN) ∈ Zk1 ⊕ . . . ⊕ ZkN : 0 js < ks, s = 1, . . . , N and p−limn∈N (U|J )j1n+...+jNnN = Id    is a subgroup of Zk1 ⊕ . . . ⊕ ZkN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Periodic N-rigidity groups

The group G(p, N, U, J ) is called a periodic N-rigidity group. The vector (k1, . . . , kN) ∈ ZN is called a period of G. If we identify PN with ZN then the relationship between the N-rigidity group and the periodic N-rigidity group can be written in a more suggestive form:

G(p, N, U, J ) = (πk1 × . . . × πkN)−1 (πk1×. . .×πkN(G(p, N, U, J ))).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Periodic N-rigidity groups

The group G(p, N, U, J ) is called a periodic N-rigidity group. The vector (k1, . . . , kN) ∈ ZN is called a period of G. If we identify PN with ZN then the relationship between the N-rigidity group and the periodic N-rigidity group can be written in a more suggestive form:

G(p, N, U, J ) = (πk1 × . . . × πkN)−1 (πk1×. . .×πkN(G(p, N, U, J ))).

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

A dual structure

We set V = Zk1 ⊕ . . . ⊕ ZkN. Then, up to some identification,

• V = Zk1 × . . . × ZkN.

Assume that we are given injective characters χj : Zkj → S1, j = 1, . . . , N. Let ξ : V × V → S1 be a Z-bilinear map (the Abelian group S1 is a Z-module) defined by the formula ξ((c1, . . . , cN), (d1, . . . , dN)) = χ1(c1d1) · . . . · χN(cNdN) for any (c1, . . . , cN) ∈ V , (d1, . . . , dN) ∈ V , where the product cjdj is the result of the multiplication in the ring Zkj.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

A dual structure

We set V = Zk1 ⊕ . . . ⊕ ZkN. Then, up to some identification,

• V = Zk1 × . . . × ZkN.

Assume that we are given injective characters χj : Zkj → S1, j = 1, . . . , N. Let ξ : V × V → S1 be a Z-bilinear map (the Abelian group S1 is a Z-module) defined by the formula ξ((c1, . . . , cN), (d1, . . . , dN)) = χ1(c1d1) · . . . · χN(cNdN) for any (c1, . . . , cN) ∈ V , (d1, . . . , dN) ∈ V , where the product cjdj is the result of the multiplication in the ring Zkj.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Classification result of peridic p-rigidity groups

Theorem C (Bergelson-Kasjan-L.) Assume that G ⊂ Zk1 ⊕ . . . ⊕ ZkN is a subgroup. Then the following assertions are equivalent: (a) G is an N-periodic rigidity group. (b) G satisfies the property (∗): For each r = 1, . . . , N (∗) (j1, . . . , jr−1, jr, jr+1, . . . , jN) ∈ G (j1, . . . , jr−1, j′

r, jr+1, . . . , jN) ∈

G = ⇒ jr = j′

r.

(c) G = K ⊥ for an algebraic coupling K of Zk1, . . . , ZkN. A subgroup K ⊂ Zk1 × . . . × ZkN is called an algebraic coupling if it has the full projection on each coordinate.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

“Proof”

(a)⇒(b) obvious; (b)⇔ (c) algebra; (c)⇒(a) construction (the most difficult part of the paper). It is given using weighted unitary operators over odometers. The cocycle, up to a limit distribution, take values in the coupling

• G ⊥ = K. We will control ordinary limits to apply the result on

finding “good” p ∈ E(βN) to obtain desired p-limits.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Come back to N-rigidity group

Proposition A subgroup G of ZN is an N-rigidity group if and only if there exists a sequence k = (k1, ..., kN) of natural numbers such that: (a) G enjoys the (∗)-property, that is, for any i = 1, . . . , N and any element g = (g1, ..., gN) ∈ G: if kj|gj for each j = i, then ki|gi, (b) G = π−1

k (πk(G)).

Corollary G ⊂ PN is an N-rigidity group if and only if G has finite index in PN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters

Come back to N-rigidity group

Proposition A subgroup G of ZN is an N-rigidity group if and only if there exists a sequence k = (k1, ..., kN) of natural numbers such that: (a) G enjoys the (∗)-property, that is, for any i = 1, . . . , N and any element g = (g1, ..., gN) ∈ G: if kj|gj for each j = i, then ki|gi, (b) G = π−1

k (πk(G)).

Corollary G ⊂ PN is an N-rigidity group if and only if G has finite index in PN.

Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters