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 = A 1 ∪ . . . ∪ A r , then some A i ∈ p . (In other words, an ultrafilter is a maximal filter.) The space of ultrafilters on N is denoted by β N (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 ∈ β N : A ∈ p } . The family { A : A ⊂ N } forms a basis for the open sets of β N . 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 = A 1 ∪ . . . ∪ A r , then some A i ∈ p . (In other words, an ultrafilter is a maximal filter.) The space of ultrafilters on N is denoted by β N (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 ∈ β N : A ∈ p } . The family { A : A ⊂ N } forms a basis for the open sets of β N . 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 = A 1 ∪ . . . ∪ A r , then some A i ∈ p . (In other words, an ultrafilter is a maximal filter.) The space of ultrafilters on N is denoted by β N (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 ∈ β N : A ∈ p } . The family { A : A ⊂ N } forms a basis for the open sets of β N . 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 = A 1 ∪ . . . ∪ A r , then some A i ∈ p . (In other words, an ultrafilter is a maximal filter.) The space of ultrafilters on N is denoted by β N (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 ∈ β N : A ∈ p } . The family { A : A ⊂ N } forms a basis for the open sets of β N . Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters
Ultrafilters Extension of addition on N to β N : Given p , q ∈ β N 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 ( β N , +) a compact left-continuous semitopological semigroup, meaning that for each p ∈ β N the function λ p ( q ) = p + q is continuous. Idempotents in β N : By Ellis’ lemma, any compact left-continuous semitopological semigroup has an idempotent. Note that if p ∈ β N is an idempotent ( p ∈ E ( β N ) ) 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 β N : Given p , q ∈ β N 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 ( β N , +) a compact left-continuous semitopological semigroup, meaning that for each p ∈ β N the function λ p ( q ) = p + q is continuous. Idempotents in β N : By Ellis’ lemma, any compact left-continuous semitopological semigroup has an idempotent. Note that if p ∈ β N is an idempotent ( p ∈ E ( β N ) ) 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 β N : Given p , q ∈ β N 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 ( β N , +) a compact left-continuous semitopological semigroup, meaning that for each p ∈ β N the function λ p ( q ) = p + q is continuous. Idempotents in β N : By Ellis’ lemma, any compact left-continuous semitopological semigroup has an idempotent. Note that if p ∈ β N is an idempotent ( p ∈ E ( β N ) ) 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 β N : Given p , q ∈ β N 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 ( β N , +) a compact left-continuous semitopological semigroup, meaning that for each p ∈ β N the function λ p ( q ) = p + q is continuous. Idempotents in β N : By Ellis’ lemma, any compact left-continuous semitopological semigroup has an idempotent. Note that if p ∈ β N is an idempotent ( p ∈ E ( β N ) ) 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 ( β N ) . Since � k − 1 i = 0 ( k N + i ) = N is a disjoint union, only for one 0 � i < k we have k N + i ∈ p . However p + p = p , so B := { n ∈ N : ( k N + i ) − n ∈ p } ∈ p . It follows that ( k N + i ) ∩ B � = ∅ . Take n ∈ ( k N + i ) ∩ B . Then for some r ∈ N , n = kr + i and also ( k N + i ) − n ∈ p . It follows immediately that k N ∈ p . If p ∈ E ( β N ) then k N ∈ 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 ( β N ) . Since � k − 1 i = 0 ( k N + i ) = N is a disjoint union, only for one 0 � i < k we have k N + i ∈ p . However p + p = p , so B := { n ∈ N : ( k N + i ) − n ∈ p } ∈ p . It follows that ( k N + i ) ∩ B � = ∅ . Take n ∈ ( k N + i ) ∩ B . Then for some r ∈ N , n = kr + i and also ( k N + i ) − n ∈ p . It follows immediately that k N ∈ p . If p ∈ E ( β N ) then k N ∈ 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 ( β N ) . Since � k − 1 i = 0 ( k N + i ) = N is a disjoint union, only for one 0 � i < k we have k N + i ∈ p . However p + p = p , so B := { n ∈ N : ( k N + i ) − n ∈ p } ∈ p . It follows that ( k N + i ) ∩ B � = ∅ . Take n ∈ ( k N + i ) ∩ B . Then for some r ∈ N , n = kr + i and also ( k N + i ) − n ∈ p . It follows immediately that k N ∈ p . If p ∈ E ( β N ) then k N ∈ 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 ( β N ) . Since � k − 1 i = 0 ( k N + i ) = N is a disjoint union, only for one 0 � i < k we have k N + i ∈ p . However p + p = p , so B := { n ∈ N : ( k N + i ) − n ∈ p } ∈ p . It follows that ( k N + i ) ∩ B � = ∅ . Take n ∈ ( k N + i ) ∩ B . Then for some r ∈ N , n = kr + i and also ( k N + i ) − n ∈ p . It follows immediately that k N ∈ p . If p ∈ E ( β N ) then k N ∈ p for each k ∈ N . Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters
IP-sets Let p ∈ E ( β N ) . By Hindman’s theorem, each A ∈ p must contain a set of the form FS (( n i ) i � 1 ) := { n i 1 + n i 2 + . . . + n i k : i 1 < i 2 < . . . < i k , k � 1 } . In ergodic theory and topological dynamics the sets of the form FS (( n i ) i � 1 ) are called IP- sets . On the other hand, given a sequence ( n i ) i � 1 we can find p ∈ E ( β N ) for which FS (( n i ) i � 1 ) ∈ p . Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters
IP-sets Let p ∈ E ( β N ) . By Hindman’s theorem, each A ∈ p must contain a set of the form FS (( n i ) i � 1 ) := { n i 1 + n i 2 + . . . + n i k : i 1 < i 2 < . . . < i k , k � 1 } . In ergodic theory and topological dynamics the sets of the form FS (( n i ) i � 1 ) are called IP- sets . On the other hand, given a sequence ( n i ) i � 1 we can find p ∈ E ( β N ) for which FS (( n i ) i � 1 ) ∈ p . Mariusz Lemańczyk Polynomial actions of unitary operators and idempotent ultrafilters
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