Model theory of measure-preserving group actions Todor Tsankov (joint with Tomás Ibarlucía) Université Paris Diderot – Paris 7 Model theory, Będlewo
We identify The basic setup of ergodic theory if . Thus becomes a (complete) metric space with metric also carries the the structure of a Boolean algebra and acts on by isometric isomorphisms. ▸ Γ – countable group; ▸ ( X , X , µ ) – a probability space; ▸ Γ ↷ X by measure-preserving transformations: µ ( γ A ) = µ ( A ) for all γ ∈ Γ, A ∈ X .
The basic setup of ergodic theory (complete) metric space with metric ▸ Γ – countable group; ▸ ( X , X , µ ) – a probability space; ▸ Γ ↷ X by measure-preserving transformations: µ ( γ A ) = µ ( A ) for all γ ∈ Γ, A ∈ X . We identify A 1 , A 2 ∈ X if µ ( A 1 △ A 2 ) = 0 . Thus X becomes a d ( A , B ) = µ ( A △ B ) . X also carries the the structure of a Boolean algebra and Γ acts on X by isometric isomorphisms.
subalgebra. In ergodic theory, this is known as a factor (and is Examples Examples (for map usually viewed dually, as a -equivariant, measure-preserving is a closed, -invariant Substructures: A substructure ). odometer (translation by on the dyadic integers ): the irrational rotation on the circle or the . is the quotient of the Haar measure on The measure on by is a closed subgroup, then group and is a homomorphism to a compact Compact actions: if . ▸ The Bernoulli shift Γ ↷ 2 Γ : ( γ ⋅ x )( γ ′ ) = x ( γ − 1 γ ′ ) . The measure on 2 Γ is ( p δ 0 + ( 1 − p ) δ 1 ) Γ for some p ∈ ( 0, 1 ) .
subalgebra. In ergodic theory, this is known as a factor (and is Substructures: A substructure Examples is a closed, -invariant usually viewed dually, as a -equivariant, measure-preserving map . ▸ The Bernoulli shift Γ ↷ 2 Γ : ( γ ⋅ x )( γ ′ ) = x ( γ − 1 γ ′ ) . The measure on 2 Γ is ( p δ 0 + ( 1 − p ) δ 1 ) Γ for some p ∈ ( 0, 1 ) . ▸ Compact actions: if ρ ∶ Γ → K is a homomorphism to a compact group and L ≤ K is a closed subgroup, then Γ ↷ K / L by γ ⋅ kL = ρ ( γ ) kL . The measure on K / L is the quotient of the Haar measure on K . Examples (for Γ = Z ): the irrational rotation on the circle or the odometer (translation by 1 on the dyadic integers Z 2 ).
Examples ▸ The Bernoulli shift Γ ↷ 2 Γ : ( γ ⋅ x )( γ ′ ) = x ( γ − 1 γ ′ ) . The measure on 2 Γ is ( p δ 0 + ( 1 − p ) δ 1 ) Γ for some p ∈ ( 0, 1 ) . ▸ Compact actions: if ρ ∶ Γ → K is a homomorphism to a compact group and L ≤ K is a closed subgroup, then Γ ↷ K / L by γ ⋅ kL = ρ ( γ ) kL . The measure on K / L is the quotient of the Haar measure on K . Examples (for Γ = Z ): the irrational rotation on the circle or the odometer (translation by 1 on the dyadic integers Z 2 ). ▸ Substructures: A substructure Y ⊆ X is a closed, Γ -invariant subalgebra. In ergodic theory, this is known as a factor (and is usually viewed dually, as a Γ -equivariant, measure-preserving map X → Y .
Continuous logic functions; as a complete set of connectives the constants, addition, and multiplication; ▸ Structures are complete metric spaces; ▸ The equality predicate is replaced by the metric d ( ⋅ , ⋅ ) ; ▸ Predicates are real-valued, bounded, uniformly continuous ▸ Connectives are continuous functions f ∶ R k → R . We can take ▸ Quantifiers are of the form inf y ϕ ( ¯ x , y ) and sup y ( ¯ x , y ) ; ▸ Uniform limits of formulas are again formulas; ▸ If ϕ ( ¯ x ) is an n -ary formula and M is a model, the interpretation of ϕ in M is a uniformly continuous, bounded function M n → R , where the modulus of continuity and the bound can determined syntactically from ϕ .
Measure theory in continuous logic Axioms: it is a boolean algebra, for example, is a probability measure on : is non-atomic: These axioms define a complete, -categorical, -stable theory. The language for ( X , µ ) : the language of Boolean algebras and a predicate µ for the measure (which defines the metric).
Measure theory in continuous logic These axioms define a complete, -categorical, -stable theory. Axioms: The language for ( X , µ ) : the language of Boolean algebras and a predicate µ for the measure (which defines the metric). ▸ it is a boolean algebra, for example, d ( A ∩ ( B ∪ C ) , ( A ∩ B ) ∪ ( A ∪ C )) = 0. sup A , B , C ▸ µ is a probability measure on X : µ ( A ∩ B ) + µ ( A ∪ B ) − µ ( A ) − µ ( B ) = 0. sup A , B ▸ µ is non-atomic: B ∣ µ ( A ∩ B ) − µ ( A )/ 2 ∣ = 0. sup inf A
Measure theory in continuous logic Axioms: The language for ( X , µ ) : the language of Boolean algebras and a predicate µ for the measure (which defines the metric). ▸ it is a boolean algebra, for example, d ( A ∩ ( B ∪ C ) , ( A ∩ B ) ∪ ( A ∪ C )) = 0. sup A , B , C ▸ µ is a probability measure on X : µ ( A ∩ B ) + µ ( A ∪ B ) − µ ( A ) − µ ( B ) = 0. sup A , B ▸ µ is non-atomic: B ∣ µ ( A ∩ B ) − µ ( A )/ 2 ∣ = 0. sup inf A These axioms define a complete, ω -categorical, ω -stable theory.
is called free if Ergodic theory in continuous logic actions of the resulting theory of free, measure-preserving We call , Axioms: for every for every The action To code the group action, one adds a function symbol for every on a non-atomic probability space. element γ ∈ Γ and the axioms: ▸ each γ is an automorphism of X ; ▸ sup A d (( γ 1 ⋯ γ n ) A , A ) = 0 for every γ 1 , . . . , γ n such that γ 1 ⋯ γ n = 1 Γ . Note that it is enough to add symbols for a generating set for Γ ; for example, if Γ = Z , one function symbol suffices.
Ergodic theory in continuous logic To code the group action, one adds a function symbol for every actions of the resulting theory of free, measure-preserving We call , Axioms: for every on a non-atomic probability space. element γ ∈ Γ and the axioms: ▸ each γ is an automorphism of X ; ▸ sup A d (( γ 1 ⋯ γ n ) A , A ) = 0 for every γ 1 , . . . , γ n such that γ 1 ⋯ γ n = 1 Γ . Note that it is enough to add symbols for a generating set for Γ ; for example, if Γ = Z , one function symbol suffices. The action Γ ↷ X is called free if µ ({ x ∈ X ∶ γ ⋅ x = x }) = 0. for every γ ≠ 1 Γ
Ergodic theory in continuous logic To code the group action, one adds a function symbol for every element γ ∈ Γ and the axioms: ▸ each γ is an automorphism of X ; ▸ sup A d (( γ 1 ⋯ γ n ) A , A ) = 0 for every γ 1 , . . . , γ n such that γ 1 ⋯ γ n = 1 Γ . Note that it is enough to add symbols for a generating set for Γ ; for example, if Γ = Z , one function symbol suffices. The action Γ ↷ X is called free if µ ({ x ∈ X ∶ γ ⋅ x = x }) = 0. for every γ ≠ 1 Γ Axioms: for every γ ≠ 1 Γ , B µ ( B ∖ A ) + µ ( B ∩ γ B ) + ∣ µ ( B ) − µ ( A )/ 3 ∣ = 0. sup inf A We call FR ( Γ ) the resulting theory of free, measure-preserving actions of Γ on a non-atomic probability space.
(and any group containing it) is not amenable. Theorem (ess. Ben Yaacov–Berenstein–Henson–Usvyatsov) Let be an amenable group. Then is a complete, stable theory that eliminates quantifiers. It is -categorical iff is finite. (They considered the case of but the proof extends, using the machinery of Ornstein–Weiss, to amenable group actions.) The case of amenable Γ A group Γ is called amenable if there exists a left-invariant, finitely additive, probability measure on Γ . Examples are finite groups, abelian (and more generally, solvable) groups. The free group F 2
(and any group containing it) is not amenable. Theorem (ess. Ben Yaacov–Berenstein–Henson–Usvyatsov) machinery of Ornstein–Weiss, to amenable group actions.) The case of amenable Γ A group Γ is called amenable if there exists a left-invariant, finitely additive, probability measure on Γ . Examples are finite groups, abelian (and more generally, solvable) groups. The free group F 2 Let Γ be an amenable group. Then FR ( Γ ) is a complete, stable theory that eliminates quantifiers. It is ω -categorical iff Γ is finite. (They considered the case of Γ = Z but the proof extends, using the
However, ergodicity is not, in general, an elementary property. This Ergodicity is always ergodic (if define it has a dense image. used to is ergodic iff the morphism compact action is infinite) and a The Bernoulli action components. measure-preserving action decomposes as an integral of ergodic The ergodic decomposition theorem states that every invariant sets: can be seen, for example, from the previous theorem. The action Γ ↷ X is called ergodic if there are no non-trivial ( ∀ γ ∈ Γ γ A = A ) � ⇒ µ ( A ) ∈ { 0, 1 } . ∀ A ∈ X
However, ergodicity is not, in general, an elementary property. This Ergodicity invariant sets: The ergodic decomposition theorem states that every measure-preserving action decomposes as an integral of ergodic components. define it has a dense image. can be seen, for example, from the previous theorem. The action Γ ↷ X is called ergodic if there are no non-trivial ( ∀ γ ∈ Γ γ A = A ) � ⇒ µ ( A ) ∈ { 0, 1 } . ∀ A ∈ X The Bernoulli action Γ ↷ 2 Γ is always ergodic (if Γ is infinite) and a compact action Γ ↷ K / L is ergodic iff the morphism Γ → K used to
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