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Topological dynamics of stable groups Ludomir Newelski Instytut - PowerPoint PPT Presentation

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Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet Wroc lawski July 2017 Newelski Topological dynamics of stable groups Set-up G is a stable group in countable language L T = Th ( G ) G G is a

  1. Functional representation Sometimes X ∼ = E ( X ) Let A ⊆ P ( G ) be a G -algebra of sets (i.e. closed under left translation in G ). Then S ( A ) is a G -flow. For p ∈ S ( A ) we define d p : A → P ( G ) by: d p ( U ) = { g ∈ G : g − 1 U ∈ p } Definition A is d -closed if A is closed under d p for every p ∈ S ( A ). Example A = Def ( G ) is d -closed, because every p ∈ S ( G ) is definable. Newelski Topological dynamics of stable groups

  2. Functional representation Sometimes X ∼ = E ( X ) Let A ⊆ P ( G ) be a G -algebra of sets (i.e. closed under left translation in G ). Then S ( A ) is a G -flow. For p ∈ S ( A ) we define d p : A → P ( G ) by: d p ( U ) = { g ∈ G : g − 1 U ∈ p } Definition A is d -closed if A is closed under d p for every p ∈ S ( A ). Example A = Def ( G ) is d -closed, because every p ∈ S ( G ) is definable. Newelski Topological dynamics of stable groups

  3. Functional representation Sometimes X ∼ = E ( X ) Let A ⊆ P ( G ) be a G -algebra of sets (i.e. closed under left translation in G ). Then S ( A ) is a G -flow. For p ∈ S ( A ) we define d p : A → P ( G ) by: d p ( U ) = { g ∈ G : g − 1 U ∈ p } Definition A is d -closed if A is closed under d p for every p ∈ S ( A ). Example A = Def ( G ) is d -closed, because every p ∈ S ( G ) is definable. Newelski Topological dynamics of stable groups

  4. Functional representation Sometimes X ∼ = E ( X ) Let A ⊆ P ( G ) be a G -algebra of sets (i.e. closed under left translation in G ). Then S ( A ) is a G -flow. For p ∈ S ( A ) we define d p : A → P ( G ) by: d p ( U ) = { g ∈ G : g − 1 U ∈ p } Definition A is d -closed if A is closed under d p for every p ∈ S ( A ). Example A = Def ( G ) is d -closed, because every p ∈ S ( G ) is definable. Newelski Topological dynamics of stable groups

  5. Functional representation Sometimes X ∼ = E ( X ) Let A ⊆ P ( G ) be a G -algebra of sets (i.e. closed under left translation in G ). Then S ( A ) is a G -flow. For p ∈ S ( A ) we define d p : A → P ( G ) by: d p ( U ) = { g ∈ G : g − 1 U ∈ p } Definition A is d -closed if A is closed under d p for every p ∈ S ( A ). Example A = Def ( G ) is d -closed, because every p ∈ S ( G ) is definable. Newelski Topological dynamics of stable groups

  6. Functional representation Sometimes X ∼ = E ( X ) Let A ⊆ P ( G ) be a G -algebra of sets (i.e. closed under left translation in G ). Then S ( A ) is a G -flow. For p ∈ S ( A ) we define d p : A → P ( G ) by: d p ( U ) = { g ∈ G : g − 1 U ∈ p } Definition A is d -closed if A is closed under d p for every p ∈ S ( A ). Example A = Def ( G ) is d -closed, because every p ∈ S ( G ) is definable. Newelski Topological dynamics of stable groups

  7. Functional representation Assume A is d -closed. For p ∈ S ( A ), d p ∈ End ( A ) := { G -endomorphisms of A} . Let d : S ( A ) → End ( A ) map p to d p . Then d is a bijection. d induces ∗ on S ( A ) so that ∼ = d : ( S ( A ) , ∗ ) → ( End ( A ) , ◦ ) Theorem 1 = 1 ( S ( A ) , ∗ ) ∼ = 2 ( End ( A ) , ◦ ) ( E ( S ( A )) , ◦ ) ∼ Proof 1. For p ∈ S ( A ) let l p ( q ) = p ∗ q . Then l p ∈ E ( S ( A )) and p �→ l p gives ∼ = 1 . 2. This is d . Newelski Topological dynamics of stable groups

  8. Functional representation Assume A is d -closed. For p ∈ S ( A ), d p ∈ End ( A ) := { G -endomorphisms of A} . Let d : S ( A ) → End ( A ) map p to d p . Then d is a bijection. d induces ∗ on S ( A ) so that ∼ = d : ( S ( A ) , ∗ ) → ( End ( A ) , ◦ ) Theorem 1 = 1 ( S ( A ) , ∗ ) ∼ = 2 ( End ( A ) , ◦ ) ( E ( S ( A )) , ◦ ) ∼ Proof 1. For p ∈ S ( A ) let l p ( q ) = p ∗ q . Then l p ∈ E ( S ( A )) and p �→ l p gives ∼ = 1 . 2. This is d . Newelski Topological dynamics of stable groups

  9. Functional representation Assume A is d -closed. For p ∈ S ( A ), d p ∈ End ( A ) := { G -endomorphisms of A} . Let d : S ( A ) → End ( A ) map p to d p . Then d is a bijection. d induces ∗ on S ( A ) so that ∼ = d : ( S ( A ) , ∗ ) → ( End ( A ) , ◦ ) Theorem 1 = 1 ( S ( A ) , ∗ ) ∼ = 2 ( End ( A ) , ◦ ) ( E ( S ( A )) , ◦ ) ∼ Proof 1. For p ∈ S ( A ) let l p ( q ) = p ∗ q . Then l p ∈ E ( S ( A )) and p �→ l p gives ∼ = 1 . 2. This is d . Newelski Topological dynamics of stable groups

  10. Functional representation Assume A is d -closed. For p ∈ S ( A ), d p ∈ End ( A ) := { G -endomorphisms of A} . Let d : S ( A ) → End ( A ) map p to d p . Then d is a bijection. d induces ∗ on S ( A ) so that ∼ = d : ( S ( A ) , ∗ ) → ( End ( A ) , ◦ ) Theorem 1 = 1 ( S ( A ) , ∗ ) ∼ = 2 ( End ( A ) , ◦ ) ( E ( S ( A )) , ◦ ) ∼ Proof 1. For p ∈ S ( A ) let l p ( q ) = p ∗ q . Then l p ∈ E ( S ( A )) and p �→ l p gives ∼ = 1 . 2. This is d . Newelski Topological dynamics of stable groups

  11. Functional representation Assume A is d -closed. For p ∈ S ( A ), d p ∈ End ( A ) := { G -endomorphisms of A} . Let d : S ( A ) → End ( A ) map p to d p . Then d is a bijection. d induces ∗ on S ( A ) so that ∼ = d : ( S ( A ) , ∗ ) → ( End ( A ) , ◦ ) Theorem 1 = 1 ( S ( A ) , ∗ ) ∼ = 2 ( End ( A ) , ◦ ) ( E ( S ( A )) , ◦ ) ∼ Proof 1. For p ∈ S ( A ) let l p ( q ) = p ∗ q . Then l p ∈ E ( S ( A )) and p �→ l p gives ∼ = 1 . 2. This is d . Newelski Topological dynamics of stable groups

  12. Functional representation Assume A is d -closed. For p ∈ S ( A ), d p ∈ End ( A ) := { G -endomorphisms of A} . Let d : S ( A ) → End ( A ) map p to d p . Then d is a bijection. d induces ∗ on S ( A ) so that ∼ = d : ( S ( A ) , ∗ ) → ( End ( A ) , ◦ ) Theorem 1 = 1 ( S ( A ) , ∗ ) ∼ = 2 ( End ( A ) , ◦ ) ( E ( S ( A )) , ◦ ) ∼ Proof 1. For p ∈ S ( A ) let l p ( q ) = p ∗ q . Then l p ∈ E ( S ( A )) and p �→ l p gives ∼ = 1 . 2. This is d . Newelski Topological dynamics of stable groups

  13. Functional representation Assume A is d -closed. For p ∈ S ( A ), d p ∈ End ( A ) := { G -endomorphisms of A} . Let d : S ( A ) → End ( A ) map p to d p . Then d is a bijection. d induces ∗ on S ( A ) so that ∼ = d : ( S ( A ) , ∗ ) → ( End ( A ) , ◦ ) Theorem 1 = 1 ( S ( A ) , ∗ ) ∼ = 2 ( End ( A ) , ◦ ) ( E ( S ( A )) , ◦ ) ∼ Proof 1. For p ∈ S ( A ) let l p ( q ) = p ∗ q . Then l p ∈ E ( S ( A )) and p �→ l p gives ∼ = 1 . 2. This is d . Newelski Topological dynamics of stable groups

  14. Functional representation Assume A is d -closed. For p ∈ S ( A ), d p ∈ End ( A ) := { G -endomorphisms of A} . Let d : S ( A ) → End ( A ) map p to d p . Then d is a bijection. d induces ∗ on S ( A ) so that ∼ = d : ( S ( A ) , ∗ ) → ( End ( A ) , ◦ ) Theorem 1 = 1 ( S ( A ) , ∗ ) ∼ = 2 ( End ( A ) , ◦ ) ( E ( S ( A )) , ◦ ) ∼ Proof 1. For p ∈ S ( A ) let l p ( q ) = p ∗ q . Then l p ∈ E ( S ( A )) and p �→ l p gives ∼ = 1 . 2. This is d . Newelski Topological dynamics of stable groups

  15. Functional representation Example If A = Def ( G ) then A is d -closed and ∗ on S G ( M ) = S ( A ) from Theorem 1 is just the free multiplication of G -types. Newelski Topological dynamics of stable groups

  16. Functional representation Example If A = Def ( G ) then A is d -closed and ∗ on S G ( M ) = S ( A ) from Theorem 1 is just the free multiplication of G -types. Newelski Topological dynamics of stable groups

  17. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  18. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  19. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  20. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  21. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  22. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  23. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  24. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  25. ( S ( G ) , ∗ ) in the definable realm Definition 1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsets of G is closed under left and right translation in G , and also under taking inverse. 2. Let Inv = { ∆ ⊆ fin L : ∆ is invariant } . Fact Inv is cofinal in [ L ] <ω . Let ∆ ∈ Inv . Notation Def ∆ ( G ) = { relatively ∆-definable subsets of G } S ∆ ( G ) = S ( Def ∆ ( G )) , the space of complete ∆-types over G . Newelski Topological dynamics of stable groups

  26. ( S ( G ) , ∗ ) in the definable realm 1 Def G , ∆ ( M ) is a d -closed G -algebra of sets. (this relies on the full definability lemma in local stability theory) 2 ( S G , ∆ ( M ) , ∗ ) ∼ = ( E ( S G , ∆ ( M )) , ◦ ) ∼ = ( End ( Def G , ∆ ( M )) , ◦ ) (this is by Theorem 1) 3 � Def G ( M ) = Def G , ∆ ( M ) ∆ ∈ Inv 4 � S G , ∆ ( M ) , ∆ ∈ Inv � is an inverse system of G - flows and semi-groups (the connecting functions are restrictions) 5 S G ( M ) = invlim ∆ ∈ Inv S G , ∆ ( M ) (as G -flows and semigroups) Newelski Topological dynamics of stable groups

  27. ( S ( G ) , ∗ ) in the definable realm 1 Def G , ∆ ( M ) is a d -closed G -algebra of sets. (this relies on the full definability lemma in local stability theory) 2 ( S G , ∆ ( M ) , ∗ ) ∼ = ( E ( S G , ∆ ( M )) , ◦ ) ∼ = ( End ( Def G , ∆ ( M )) , ◦ ) (this is by Theorem 1) 3 � Def G ( M ) = Def G , ∆ ( M ) ∆ ∈ Inv 4 � S G , ∆ ( M ) , ∆ ∈ Inv � is an inverse system of G - flows and semi-groups (the connecting functions are restrictions) 5 S G ( M ) = invlim ∆ ∈ Inv S G , ∆ ( M ) (as G -flows and semigroups) Newelski Topological dynamics of stable groups

  28. ( S ( G ) , ∗ ) in the definable realm 1 Def G , ∆ ( M ) is a d -closed G -algebra of sets. (this relies on the full definability lemma in local stability theory) 2 ( S G , ∆ ( M ) , ∗ ) ∼ = ( E ( S G , ∆ ( M )) , ◦ ) ∼ = ( End ( Def G , ∆ ( M )) , ◦ ) (this is by Theorem 1) 3 � Def G ( M ) = Def G , ∆ ( M ) ∆ ∈ Inv 4 � S G , ∆ ( M ) , ∆ ∈ Inv � is an inverse system of G - flows and semi-groups (the connecting functions are restrictions) 5 S G ( M ) = invlim ∆ ∈ Inv S G , ∆ ( M ) (as G -flows and semigroups) Newelski Topological dynamics of stable groups

  29. ( S ( G ) , ∗ ) in the definable realm 1 Def G , ∆ ( M ) is a d -closed G -algebra of sets. (this relies on the full definability lemma in local stability theory) 2 ( S G , ∆ ( M ) , ∗ ) ∼ = ( E ( S G , ∆ ( M )) , ◦ ) ∼ = ( End ( Def G , ∆ ( M )) , ◦ ) (this is by Theorem 1) 3 � Def G ( M ) = Def G , ∆ ( M ) ∆ ∈ Inv 4 � S G , ∆ ( M ) , ∆ ∈ Inv � is an inverse system of G - flows and semi-groups (the connecting functions are restrictions) 5 S G ( M ) = invlim ∆ ∈ Inv S G , ∆ ( M ) (as G -flows and semigroups) Newelski Topological dynamics of stable groups

  30. ( S ( G ) , ∗ ) in the definable realm 1 Def G , ∆ ( M ) is a d -closed G -algebra of sets. (this relies on the full definability lemma in local stability theory) 2 ( S G , ∆ ( M ) , ∗ ) ∼ = ( E ( S G , ∆ ( M )) , ◦ ) ∼ = ( End ( Def G , ∆ ( M )) , ◦ ) (this is by Theorem 1) 3 � Def G ( M ) = Def G , ∆ ( M ) ∆ ∈ Inv 4 � S G , ∆ ( M ) , ∆ ∈ Inv � is an inverse system of G - flows and semi-groups (the connecting functions are restrictions) 5 S G ( M ) = invlim ∆ ∈ Inv S G , ∆ ( M ) (as G -flows and semigroups) Newelski Topological dynamics of stable groups

  31. ( S ( G ) , ∗ ) in the definable realm 1 Def G , ∆ ( M ) is a d -closed G -algebra of sets. (this relies on the full definability lemma in local stability theory) 2 ( S G , ∆ ( M ) , ∗ ) ∼ = ( E ( S G , ∆ ( M )) , ◦ ) ∼ = ( End ( Def G , ∆ ( M )) , ◦ ) (this is by Theorem 1) 3 � Def G ( M ) = Def G , ∆ ( M ) ∆ ∈ Inv 4 � S G , ∆ ( M ) , ∆ ∈ Inv � is an inverse system of G - flows and semi-groups (the connecting functions are restrictions) 5 S G ( M ) = invlim ∆ ∈ Inv S G , ∆ ( M ) (as G -flows and semigroups) Newelski Topological dynamics of stable groups

  32. ( S ( G ) , ∗ ) in the definable realm 1 Def G , ∆ ( M ) is a d -closed G -algebra of sets. (this relies on the full definability lemma in local stability theory) 2 ( S G , ∆ ( M ) , ∗ ) ∼ = ( E ( S G , ∆ ( M )) , ◦ ) ∼ = ( End ( Def G , ∆ ( M )) , ◦ ) (this is by Theorem 1) 3 � Def G ( M ) = Def G , ∆ ( M ) ∆ ∈ Inv 4 � S G , ∆ ( M ) , ∆ ∈ Inv � is an inverse system of G - flows and semi-groups (the connecting functions are restrictions) 5 S G ( M ) = invlim ∆ ∈ Inv S G , ∆ ( M ) (as G -flows and semigroups) Newelski Topological dynamics of stable groups

  33. ( S ( G ) , ∗ ) in the definable realm 1 Def G , ∆ ( M ) is a d -closed G -algebra of sets. (this relies on the full definability lemma in local stability theory) 2 ( S G , ∆ ( M ) , ∗ ) ∼ = ( E ( S G , ∆ ( M )) , ◦ ) ∼ = ( End ( Def G , ∆ ( M )) , ◦ ) (this is by Theorem 1) 3 � Def G ( M ) = Def G , ∆ ( M ) ∆ ∈ Inv 4 � S G , ∆ ( M ) , ∆ ∈ Inv � is an inverse system of G - flows and semi-groups (the connecting functions are restrictions) 5 S G ( M ) = invlim ∆ ∈ Inv S G , ∆ ( M ) (as G -flows and semigroups) Newelski Topological dynamics of stable groups

  34. Types as functions S ( G ) ∋ p � d p : Def ( G ) → Def ( G ) S ∆ ( G ) ∋ p � d p : Def ∆ ( G ) → Def ∆ ( G ) d p � Ker ( d p ) , Im ( d p ) Ker ( d p ) = { U ∈ Def ∆ ( G ) : [ U ] ∩ cl ( Gp ) = ∅} Idea The larger the type p ∈ S G ( M ) , p ∈ S G , ∆ ( M ) The smaller the flow cl ( Gp ). The larger the kernel Ker ( d p ). The smaller the image Im ( d p ). The larger the (local) Morley rank of p . Newelski Topological dynamics of stable groups

  35. Types as functions S ( G ) ∋ p � d p : Def ( G ) → Def ( G ) S ∆ ( G ) ∋ p � d p : Def ∆ ( G ) → Def ∆ ( G ) d p � Ker ( d p ) , Im ( d p ) Ker ( d p ) = { U ∈ Def ∆ ( G ) : [ U ] ∩ cl ( Gp ) = ∅} Idea The larger the type p ∈ S G ( M ) , p ∈ S G , ∆ ( M ) The smaller the flow cl ( Gp ). The larger the kernel Ker ( d p ). The smaller the image Im ( d p ). The larger the (local) Morley rank of p . Newelski Topological dynamics of stable groups

  36. Types as functions S ( G ) ∋ p � d p : Def ( G ) → Def ( G ) S ∆ ( G ) ∋ p � d p : Def ∆ ( G ) → Def ∆ ( G ) d p � Ker ( d p ) , Im ( d p ) Ker ( d p ) = { U ∈ Def ∆ ( G ) : [ U ] ∩ cl ( Gp ) = ∅} Idea The larger the type p ∈ S G ( M ) , p ∈ S G , ∆ ( M ) The smaller the flow cl ( Gp ). The larger the kernel Ker ( d p ). The smaller the image Im ( d p ). The larger the (local) Morley rank of p . Newelski Topological dynamics of stable groups

  37. Types as functions S ( G ) ∋ p � d p : Def ( G ) → Def ( G ) S ∆ ( G ) ∋ p � d p : Def ∆ ( G ) → Def ∆ ( G ) d p � Ker ( d p ) , Im ( d p ) Ker ( d p ) = { U ∈ Def ∆ ( G ) : [ U ] ∩ cl ( Gp ) = ∅} Idea The larger the type p ∈ S G ( M ) , p ∈ S G , ∆ ( M ) The smaller the flow cl ( Gp ). The larger the kernel Ker ( d p ). The smaller the image Im ( d p ). The larger the (local) Morley rank of p . Newelski Topological dynamics of stable groups

  38. Types as functions S ( G ) ∋ p � d p : Def ( G ) → Def ( G ) S ∆ ( G ) ∋ p � d p : Def ∆ ( G ) → Def ∆ ( G ) d p � Ker ( d p ) , Im ( d p ) Ker ( d p ) = { U ∈ Def ∆ ( G ) : [ U ] ∩ cl ( Gp ) = ∅} Idea The larger the type p ∈ S G ( M ) , p ∈ S G , ∆ ( M ) The smaller the flow cl ( Gp ). The larger the kernel Ker ( d p ). The smaller the image Im ( d p ). The larger the (local) Morley rank of p . Newelski Topological dynamics of stable groups

  39. Types as functions S ( G ) ∋ p � d p : Def ( G ) → Def ( G ) S ∆ ( G ) ∋ p � d p : Def ∆ ( G ) → Def ∆ ( G ) d p � Ker ( d p ) , Im ( d p ) Ker ( d p ) = { U ∈ Def ∆ ( G ) : [ U ] ∩ cl ( Gp ) = ∅} Idea The larger the type p ∈ S G ( M ) , p ∈ S G , ∆ ( M ) The smaller the flow cl ( Gp ). The larger the kernel Ker ( d p ). The smaller the image Im ( d p ). The larger the (local) Morley rank of p . Newelski Topological dynamics of stable groups

  40. Types as functions S ( G ) ∋ p � d p : Def ( G ) → Def ( G ) S ∆ ( G ) ∋ p � d p : Def ∆ ( G ) → Def ∆ ( G ) d p � Ker ( d p ) , Im ( d p ) Ker ( d p ) = { U ∈ Def ∆ ( G ) : [ U ] ∩ cl ( Gp ) = ∅} Idea The larger the type p ∈ S G ( M ) , p ∈ S G , ∆ ( M ) The smaller the flow cl ( Gp ). The larger the kernel Ker ( d p ). The smaller the image Im ( d p ). The larger the (local) Morley rank of p . Newelski Topological dynamics of stable groups

  41. Types as functions S ( G ) ∋ p � d p : Def ( G ) → Def ( G ) S ∆ ( G ) ∋ p � d p : Def ∆ ( G ) → Def ∆ ( G ) d p � Ker ( d p ) , Im ( d p ) Ker ( d p ) = { U ∈ Def ∆ ( G ) : [ U ] ∩ cl ( Gp ) = ∅} Idea The larger the type p ∈ S G ( M ) , p ∈ S G , ∆ ( M ) The smaller the flow cl ( Gp ). The larger the kernel Ker ( d p ). The smaller the image Im ( d p ). The larger the (local) Morley rank of p . Newelski Topological dynamics of stable groups

  42. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  43. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  44. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  45. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  46. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  47. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  48. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  49. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  50. Types as functions Ker ( d p ) , Im ( d p ): measures of the size of p . Let p ∈ S ( G ) (or p ∈ S ∆ ( G )...) Let p ∗ n = p ∗ · · · ∗ p .So d p ∗ n = d p ◦ · · · ◦ d p . � �� � � �� � n n Let R ( p ) = � RM ∆ ( p ) : ∆ ∈ Inv � . Lemma 1. R ( p ∗ n ) grow (coordinatewise), Ker ( d p ∗ n ) grow and Im ( d p ∗ n ) shrink with n = 1 , 2 , 3 , . . . . 2. The growth/shrinking of these three sequences is strictly correlated. There are similar connections between RM ∆ , Ker and Im in S ∆ ( G ). In particular, if p ∈ S ∆ ( G ), U ∈ Def ∆ ( G ) and RM ∆ ( U ) < RM ∆ ( p ), then U ∈ Ker ( d p ) Newelski Topological dynamics of stable groups

  51. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  52. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  53. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  54. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  55. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  56. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  57. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  58. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  59. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  60. Subalgebras of Def ∆ ( G ) Assume A ⊆ Def ∆ ( G ) is a G -subalgebra. A is scattered, has finite CB -rank, MR ∆ -rank. A is atomic. For g ∈ G let U g ∈ A be the atom containing g . It exists: there is some atom U ∈ A , then U = U h for any h ∈ U . then 1 ∈ h − 1 U h = U 1 g ∈ gU 1 = U g U 1 < G U g = gU 1 is the left coset of U 1 containing g . Atoms almost determine A Newelski Topological dynamics of stable groups

  61. A := Im ( d p ) explained Let p ∈ S ∆ ( G ) and A = Im ( d p ). What is U 1 ? For V ∈ Def ∆ ( G ): 1 ∈ d p ( V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB ( V ) = CB ( p ) and Mlt ( V ) = Mlt ( p ). Here CB and Mlt is meant in Def ∆ ( G ). Then U 1 = d p ( V ) U g = gU 1 for all g ∈ G In fact, U 1 = Stab ( p ) = { g ∈ G : gp = p } . Newelski Topological dynamics of stable groups

  62. A := Im ( d p ) explained Let p ∈ S ∆ ( G ) and A = Im ( d p ). What is U 1 ? For V ∈ Def ∆ ( G ): 1 ∈ d p ( V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB ( V ) = CB ( p ) and Mlt ( V ) = Mlt ( p ). Here CB and Mlt is meant in Def ∆ ( G ). Then U 1 = d p ( V ) U g = gU 1 for all g ∈ G In fact, U 1 = Stab ( p ) = { g ∈ G : gp = p } . Newelski Topological dynamics of stable groups

  63. A := Im ( d p ) explained Let p ∈ S ∆ ( G ) and A = Im ( d p ). What is U 1 ? For V ∈ Def ∆ ( G ): 1 ∈ d p ( V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB ( V ) = CB ( p ) and Mlt ( V ) = Mlt ( p ). Here CB and Mlt is meant in Def ∆ ( G ). Then U 1 = d p ( V ) U g = gU 1 for all g ∈ G In fact, U 1 = Stab ( p ) = { g ∈ G : gp = p } . Newelski Topological dynamics of stable groups

  64. A := Im ( d p ) explained Let p ∈ S ∆ ( G ) and A = Im ( d p ). What is U 1 ? For V ∈ Def ∆ ( G ): 1 ∈ d p ( V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB ( V ) = CB ( p ) and Mlt ( V ) = Mlt ( p ). Here CB and Mlt is meant in Def ∆ ( G ). Then U 1 = d p ( V ) U g = gU 1 for all g ∈ G In fact, U 1 = Stab ( p ) = { g ∈ G : gp = p } . Newelski Topological dynamics of stable groups

  65. A := Im ( d p ) explained Let p ∈ S ∆ ( G ) and A = Im ( d p ). What is U 1 ? For V ∈ Def ∆ ( G ): 1 ∈ d p ( V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB ( V ) = CB ( p ) and Mlt ( V ) = Mlt ( p ). Here CB and Mlt is meant in Def ∆ ( G ). Then U 1 = d p ( V ) U g = gU 1 for all g ∈ G In fact, U 1 = Stab ( p ) = { g ∈ G : gp = p } . Newelski Topological dynamics of stable groups

  66. A := Im ( d p ) explained Let p ∈ S ∆ ( G ) and A = Im ( d p ). What is U 1 ? For V ∈ Def ∆ ( G ): 1 ∈ d p ( V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB ( V ) = CB ( p ) and Mlt ( V ) = Mlt ( p ). Here CB and Mlt is meant in Def ∆ ( G ). Then U 1 = d p ( V ) U g = gU 1 for all g ∈ G In fact, U 1 = Stab ( p ) = { g ∈ G : gp = p } . Newelski Topological dynamics of stable groups

  67. A := Im ( d p ) explained Let p ∈ S ∆ ( G ) and A = Im ( d p ). What is U 1 ? For V ∈ Def ∆ ( G ): 1 ∈ d p ( V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB ( V ) = CB ( p ) and Mlt ( V ) = Mlt ( p ). Here CB and Mlt is meant in Def ∆ ( G ). Then U 1 = d p ( V ) U g = gU 1 for all g ∈ G In fact, U 1 = Stab ( p ) = { g ∈ G : gp = p } . Newelski Topological dynamics of stable groups

  68. A := Im ( d p ) explained Let p ∈ S ∆ ( G ) and A = Im ( d p ). What is U 1 ? For V ∈ Def ∆ ( G ): 1 ∈ d p ( V ) ⇐ ⇒ V ∈ p Choose V ∈ p with CB ( V ) = CB ( p ) and Mlt ( V ) = Mlt ( p ). Here CB and Mlt is meant in Def ∆ ( G ). Then U 1 = d p ( V ) U g = gU 1 for all g ∈ G In fact, U 1 = Stab ( p ) = { g ∈ G : gp = p } . Newelski Topological dynamics of stable groups

  69. Test case: the 2-step theorem Assume p ∈ S ( G ). Recall that G ∗ ≻ G is a monster model. p ( G ∗ ) generates a subgroup � p ( G ∗ ) � < G ∗ , invariant under Aut ( G ∗ / G ). Let � p � be the minimal type-definable subgroup of G ∗ containing � p ( G ∗ ) � . Let Cl ∗ ( p ) = cl ( { p n : n ∈ N + } ), where p n = p ∗ · · · ∗ p � �� � n So p n ( G ∗ ) ⊆ � p � . Theorem 2 The generic types of � p � are precisely the types in Cl ∗ ( p ) with maximal ranks RM ∆ , ∆ ⊆ L finite, invariant. Newelski Topological dynamics of stable groups

  70. Test case: the 2-step theorem Assume p ∈ S ( G ). Recall that G ∗ ≻ G is a monster model. p ( G ∗ ) generates a subgroup � p ( G ∗ ) � < G ∗ , invariant under Aut ( G ∗ / G ). Let � p � be the minimal type-definable subgroup of G ∗ containing � p ( G ∗ ) � . Let Cl ∗ ( p ) = cl ( { p n : n ∈ N + } ), where p n = p ∗ · · · ∗ p � �� � n So p n ( G ∗ ) ⊆ � p � . Theorem 2 The generic types of � p � are precisely the types in Cl ∗ ( p ) with maximal ranks RM ∆ , ∆ ⊆ L finite, invariant. Newelski Topological dynamics of stable groups

  71. Test case: the 2-step theorem Assume p ∈ S ( G ). Recall that G ∗ ≻ G is a monster model. p ( G ∗ ) generates a subgroup � p ( G ∗ ) � < G ∗ , invariant under Aut ( G ∗ / G ). Let � p � be the minimal type-definable subgroup of G ∗ containing � p ( G ∗ ) � . Let Cl ∗ ( p ) = cl ( { p n : n ∈ N + } ), where p n = p ∗ · · · ∗ p � �� � n So p n ( G ∗ ) ⊆ � p � . Theorem 2 The generic types of � p � are precisely the types in Cl ∗ ( p ) with maximal ranks RM ∆ , ∆ ⊆ L finite, invariant. Newelski Topological dynamics of stable groups

  72. Test case: the 2-step theorem Assume p ∈ S ( G ). Recall that G ∗ ≻ G is a monster model. p ( G ∗ ) generates a subgroup � p ( G ∗ ) � < G ∗ , invariant under Aut ( G ∗ / G ). Let � p � be the minimal type-definable subgroup of G ∗ containing � p ( G ∗ ) � . Let Cl ∗ ( p ) = cl ( { p n : n ∈ N + } ), where p n = p ∗ · · · ∗ p � �� � n So p n ( G ∗ ) ⊆ � p � . Theorem 2 The generic types of � p � are precisely the types in Cl ∗ ( p ) with maximal ranks RM ∆ , ∆ ⊆ L finite, invariant. Newelski Topological dynamics of stable groups

  73. Test case: the 2-step theorem Assume p ∈ S ( G ). Recall that G ∗ ≻ G is a monster model. p ( G ∗ ) generates a subgroup � p ( G ∗ ) � < G ∗ , invariant under Aut ( G ∗ / G ). Let � p � be the minimal type-definable subgroup of G ∗ containing � p ( G ∗ ) � . Let Cl ∗ ( p ) = cl ( { p n : n ∈ N + } ), where p n = p ∗ · · · ∗ p � �� � n So p n ( G ∗ ) ⊆ � p � . Theorem 2 The generic types of � p � are precisely the types in Cl ∗ ( p ) with maximal ranks RM ∆ , ∆ ⊆ L finite, invariant. Newelski Topological dynamics of stable groups

  74. Test case: the 2-step theorem Assume p ∈ S ( G ). Recall that G ∗ ≻ G is a monster model. p ( G ∗ ) generates a subgroup � p ( G ∗ ) � < G ∗ , invariant under Aut ( G ∗ / G ). Let � p � be the minimal type-definable subgroup of G ∗ containing � p ( G ∗ ) � . Let Cl ∗ ( p ) = cl ( { p n : n ∈ N + } ), where p n = p ∗ · · · ∗ p � �� � n So p n ( G ∗ ) ⊆ � p � . Theorem 2 The generic types of � p � are precisely the types in Cl ∗ ( p ) with maximal ranks RM ∆ , ∆ ⊆ L finite, invariant. Newelski Topological dynamics of stable groups

  75. The 2-step theorem RM ∆ ( p 0 ∗ p 1 ) ≥ RM ∆ ( p i ) RM ∆ ( p n ) , n < ω , is non-decreasing. If q = lim i p n i , then RM ∆ ( q ) ≥ RM ∆ ( p n i ). If q ∈ Cl ∗ ( p ) is a generic type of � p � , then q is an accumulation point of the set { p n : n > 0 } . Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of { p n : n > 0 } and r is an accumulation point of { q n : n > 0 } . Then r is a generic type of � p � . Newelski Topological dynamics of stable groups

  76. The 2-step theorem RM ∆ ( p 0 ∗ p 1 ) ≥ RM ∆ ( p i ) RM ∆ ( p n ) , n < ω , is non-decreasing. If q = lim i p n i , then RM ∆ ( q ) ≥ RM ∆ ( p n i ). If q ∈ Cl ∗ ( p ) is a generic type of � p � , then q is an accumulation point of the set { p n : n > 0 } . Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of { p n : n > 0 } and r is an accumulation point of { q n : n > 0 } . Then r is a generic type of � p � . Newelski Topological dynamics of stable groups

  77. The 2-step theorem RM ∆ ( p 0 ∗ p 1 ) ≥ RM ∆ ( p i ) RM ∆ ( p n ) , n < ω , is non-decreasing. If q = lim i p n i , then RM ∆ ( q ) ≥ RM ∆ ( p n i ). If q ∈ Cl ∗ ( p ) is a generic type of � p � , then q is an accumulation point of the set { p n : n > 0 } . Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of { p n : n > 0 } and r is an accumulation point of { q n : n > 0 } . Then r is a generic type of � p � . Newelski Topological dynamics of stable groups

  78. The 2-step theorem RM ∆ ( p 0 ∗ p 1 ) ≥ RM ∆ ( p i ) RM ∆ ( p n ) , n < ω , is non-decreasing. If q = lim i p n i , then RM ∆ ( q ) ≥ RM ∆ ( p n i ). If q ∈ Cl ∗ ( p ) is a generic type of � p � , then q is an accumulation point of the set { p n : n > 0 } . Is the converse true? Not necessarily. The 2-step theorem Assume q is an accumulation point of { p n : n > 0 } and r is an accumulation point of { q n : n > 0 } . Then r is a generic type of � p � . Newelski Topological dynamics of stable groups

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