Stone duality, more duality, and dynamics in Will Brian May 22, - PowerPoint PPT Presentation

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality, more duality, and dynamics in βω Will Brian May 22, 2014 Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Table of contents 1 A quick look at βω 2 Filters and families 3 Ultrafilters on families 4 A Filter Dichotomy Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality for βω βω is the space of all ultrafilters on ω . Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality for βω βω is the space of all ultrafilters on ω . βω is the set of all ultrafilters containing If A ⊆ ω , then ˆ A = A A . Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality for βω βω is the space of all ultrafilters on ω . βω is the set of all ultrafilters containing If A ⊆ ω , then ˆ A = A A . F = � � � If F is any filter on ω , then ˆ ˆ A : A ∈ F is a closed subset of βω . Conversely, if C is a closed subset of βω then there is a unique filter F such that ˆ F = C , namely � � A ⊆ ω : C ⊆ ˆ F = A Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality for βω βω is the space of all ultrafilters on ω . βω is the set of all ultrafilters containing If A ⊆ ω , then ˆ A = A A . F = � � � If F is any filter on ω , then ˆ ˆ A : A ∈ F is a closed subset of βω . Conversely, if C is a closed subset of βω then there is a unique filter F such that ˆ F = C , namely � � A ⊆ ω : C ⊆ ˆ F = A The above correspondence is called Stone duality . It represents a special case of a famous theorem proved by Marshal Stone in 1936. Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a dynamical system By a dynamical system we mean a compact space X together with a map f : X → X . Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a dynamical system By a dynamical system we mean a compact space X together with a map f : X → X . Define the shift map σ : βω → βω by σ ( p ) = ↑{ A + 1: A ∈ p } , where A + 1 = { n + 1: n ∈ A } and ↑B is the set of all supersets of elements of B . Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a dynamical system By a dynamical system we mean a compact space X together with a map f : X → X . Define the shift map σ : βω → βω by σ ( p ) = ↑{ A + 1: A ∈ p } , where A + 1 = { n + 1: n ∈ A } and ↑B is the set of all supersets of elements of B . This map is continuous, and almost a surjection. σ is the unique continuous extension to βω of the map on ω given by n �→ n + 1. Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup Recall that every continuous function f : ω → βω extends to a continuous function β f : βω → βω : ω βω f β f βω Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup Recall that every continuous function f : ω → βω extends to a continuous function β f : βω → βω : ω βω f β f βω In particular, we can extend the function n �→ n + m to βω and thus define p + m for any m ∈ ω and p ∈ βω . We can then extend the function n �→ p + n to βω and thus define p + q for any p , q ∈ βω . Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup Equivalently, we can write p + q = { A ⊆ ω : { n : ( A − n ) ∈ p } ∈ q } . The function q �→ p + q is continuous for every p , but the function p �→ p + q is continuous only when q is principal. Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup Equivalently, we can write p + q = { A ⊆ ω : { n : ( A − n ) ∈ p } ∈ q } . The function q �→ p + q is continuous for every p , but the function p �→ p + q is continuous only when q is principal. This makes βω into a left-topological semigroup. Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup Equivalently, we can write p + q = { A ⊆ ω : { n : ( A − n ) ∈ p } ∈ q } . The function q �→ p + q is continuous for every p , but the function p �→ p + q is continuous only when q is principal. This makes βω into a left-topological semigroup. If we consider that an ultrafilter p is a (non- σ -additive) measure in which every set has measure 0 or 1, then the ultrafilter p + q simply represents the convolution of the measures p and q . Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy dynamics and algebra Theorem (Bergelson) The subsystems of ( β N , σ ) are the closed right ideals of ( β N , +) , and the minimal subsystems are the minimal right ideals. Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy dynamics and algebra Theorem (Bergelson) The subsystems of ( β N , σ ) are the closed right ideals of ( β N , +) , and the minimal subsystems are the minimal right ideals. Proof. If R is a right ideal then R + β N ⊆ R and, in particular, if p ∈ R then p + 1 = σ ( p ) ∈ R . Thus every right ideal is σ -invariant, and if closed is a subsystem of ( β N , σ ). Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy dynamics and algebra Theorem (Bergelson) The subsystems of ( β N , σ ) are the closed right ideals of ( β N , +) , and the minimal subsystems are the minimal right ideals. Proof. If R is a right ideal then R + β N ⊆ R and, in particular, if p ∈ R then p + 1 = σ ( p ) ∈ R . Thus every right ideal is σ -invariant, and if closed is a subsystem of ( β N , σ ). Conversely, if X is closed and σ -invariant, then p + β N = p + N = p + N ⊆ X = X for every p ∈ X . Thus every subsystem of ( β N , σ ) is a right ideal. Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy dynamics and algebra Theorem (Bergelson) The subsystems of ( β N , σ ) are the closed right ideals of ( β N , +) , and the minimal subsystems are the minimal right ideals. Proof. If R is a right ideal then R + β N ⊆ R and, in particular, if p ∈ R then p + 1 = σ ( p ) ∈ R . Thus every right ideal is σ -invariant, and if closed is a subsystem of ( β N , σ ). Conversely, if X is closed and σ -invariant, then p + β N = p + N = p + N ⊆ X = X for every p ∈ X . Thus every subsystem of ( β N , σ ) is a right ideal. For the second assertion, we need only prove that every minimal right ideal is closed. If R is a minimal right ideal then p + β N = R . Since x �→ p + x is a continuous function, R is compact. Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A familiar definition A filter F on ω is a set of subsets of ω satisfying: 1 Nontriviality: ∅ / ∈ F and ω ∈ F . 2 Upwards heredity: if A ∈ F and A ⊆ B , then B ∈ F . 3 Finite intersection property: if A , B ∈ F then A ∩ B ∈ F . Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A familiar definition A filter F on ω is a set of subsets of ω satisfying: 1 Nontriviality: ∅ / ∈ F and ω ∈ F . 2 Upwards heredity: if A ∈ F and A ⊆ B , then B ∈ F . 3 Finite intersection property: if A , B ∈ F then A ∩ B ∈ F . If we omit (2) then we get the definition of a filter base . Will Brian Stone duality, more duality, and dynamics in βω

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A familiar definition A filter F on ω is a set of subsets of ω satisfying: 1 Nontriviality: ∅ / ∈ F and ω ∈ F . 2 Upwards heredity: if A ∈ F and A ⊆ B , then B ∈ F . 3 Finite intersection property: if A , B ∈ F then A ∩ B ∈ F . If we omit (2) then we get the definition of a filter base . If we omit (3) then we get the definition of a Furstenberg family , or simply a family . Will Brian Stone duality, more duality, and dynamics in βω