Continuous orbit equivalence rigidity Xin Li Dynamical systems and - PowerPoint PPT Presentation

Continuous orbit equivalence rigidity Xin Li

Dynamical systems and operator algebras Dynamical systems ← − − − − − − − − − − − − → Operator algebras

Dynamical systems and operator algebras Dynamical systems ← − − − − − − − − − − − − → Operator algebras ◮ Q.: How to go from dynamical systems to operator algebras?

Dynamical systems and operator algebras Dynamical systems ← − − − − − − − − − − − − → Operator algebras ◮ Q.: How to go from dynamical systems to operator algebras? A.: Crossed product construction.

Dynamical systems and operator algebras Dynamical systems ← − − − − − − − − − − − − → Operator algebras ◮ Q.: How to go from dynamical systems to operator algebras? A.: Crossed product construction. ◮ Q.: Is there a way back???

Dynamical systems and operator algebras Dynamical systems ← − − − − − − − − − − − − → Operator algebras ◮ Q.: How to go from dynamical systems to operator algebras? A.: Crossed product construction. ◮ Q.: Is there a way back??? ◮ More precisely: Given G � X , H � Y , do we have C 0 ( X ) ⋊ r G ∼ = C 0 ( Y ) ⋊ r H ⇒ G � X ∼ H � Y ???

Continuous orbit equivalence

Continuous orbit equivalence G � X , H � Y : topological dynamical systems.

Continuous orbit equivalence G � X , H � Y : topological dynamical systems. Definition G � X and H � Y are conjugate if there exist a homeomorphism ∼ = ∼ = ϕ : X − → Y and an isomorphism ρ : G − → H with ϕ ( g . x ) = ρ ( g ) .ϕ ( x ) for all g ∈ G , x ∈ X .

Continuous orbit equivalence G � X , H � Y : topological dynamical systems. Definition G � X and H � Y are conjugate if there exist a homeomorphism ∼ = ∼ = ϕ : X − → Y and an isomorphism ρ : G − → H with ϕ ( g . x ) = ρ ( g ) .ϕ ( x ) for all g ∈ G , x ∈ X . Definition G � X and H � Y are continuously orbit equivalent if there ∼ = exists a homeomorphism ϕ : X − → Y together with continuous maps a : G × X → H and b : H × Y → G such that

Continuous orbit equivalence G � X , H � Y : topological dynamical systems. Definition G � X and H � Y are conjugate if there exist a homeomorphism ∼ = ∼ = ϕ : X − → Y and an isomorphism ρ : G − → H with ϕ ( g . x ) = ρ ( g ) .ϕ ( x ) for all g ∈ G , x ∈ X . Definition G � X and H � Y are continuously orbit equivalent if there ∼ = exists a homeomorphism ϕ : X − → Y together with continuous maps a : G × X → H and b : H × Y → G such that ϕ ( g . x ) = a ( g , x ) .ϕ ( x ) and ϕ − 1 ( h . y ) = b ( h , y ) .ϕ − 1 ( y ).

Topological dynamics and C*-algebras

Topological dynamics and C*-algebras Theorem G � X, H � Y : topologically free topological dynamical systems. G � X ∼ coe H � Y if and only if there is a C*-isomorphism ∼ = Φ : C 0 ( X ) ⋊ r G − → C 0 ( Y ) ⋊ r H with Φ( C 0 ( X )) = C 0 ( Y ) .

Topological dynamics and C*-algebras Theorem G � X, H � Y : topologically free topological dynamical systems. G � X ∼ coe H � Y if and only if there is a C*-isomorphism ∼ = Φ : C 0 ( X ) ⋊ r G − → C 0 ( Y ) ⋊ r H with Φ( C 0 ( X )) = C 0 ( Y ) . - top. free: for all e � = g ∈ G , { x ∈ X : g . x � = x } is dense in X .

Topological dynamics and C*-algebras Theorem G � X, H � Y : topologically free topological dynamical systems. G � X ∼ coe H � Y if and only if there is a C*-isomorphism ∼ = Φ : C 0 ( X ) ⋊ r G − → C 0 ( Y ) ⋊ r H with Φ( C 0 ( X )) = C 0 ( Y ) . - top. free: for all e � = g ∈ G , { x ∈ X : g . x � = x } is dense in X . ◮ Conjugacy ⇒ COE ⇔ Cartan-isom. ⇒ C*-isom.

Topological dynamics and C*-algebras Theorem G � X, H � Y : topologically free topological dynamical systems. G � X ∼ coe H � Y if and only if there is a C*-isomorphism ∼ = Φ : C 0 ( X ) ⋊ r G − → C 0 ( Y ) ⋊ r H with Φ( C 0 ( X )) = C 0 ( Y ) . - top. free: for all e � = g ∈ G , { x ∈ X : g . x � = x } is dense in X . ◮ Conjugacy ⇒ COE ⇔ Cartan-isom. ⇒ C*-isom. ◮ Can these arrows be reversed?

Continuous orbit equivalence rigidity Can we reverse the first arrow in ◮ Conjugacy ⇒ COE ⇔ Cartan-isom. ⇒ C*-isom.?

Continuous orbit equivalence rigidity Can we reverse the first arrow in ◮ Conjugacy ⇒ COE ⇔ Cartan-isom. ⇒ C*-isom.? ◮ Example (Boyle-Tomiyama 1998): For top. transitive topological dynamical systems of the form Z � X on compact spaces X , Conjugacy ⇐ COE.

Continuous orbit equivalence rigidity Can we reverse the first arrow in ◮ Conjugacy ⇒ COE ⇔ Cartan-isom. ⇒ C*-isom.? ◮ Example (Boyle-Tomiyama 1998): For top. transitive topological dynamical systems of the form Z � X on compact spaces X , Conjugacy ⇐ COE. ⇐ COE for certain Z n � X , ◮ Counterexamples: Conjugacy ✟ ✟

Continuous orbit equivalence rigidity Can we reverse the first arrow in ◮ Conjugacy ⇒ COE ⇔ Cartan-isom. ⇒ C*-isom.? ◮ Example (Boyle-Tomiyama 1998): For top. transitive topological dynamical systems of the form Z � X on compact spaces X , Conjugacy ⇐ COE. ⇐ COE for certain Z n � X , ◮ Counterexamples: Conjugacy ✟ ✟ and also for certain F n � X .

Continuous orbit equivalence and quasi-isometry

Continuous orbit equivalence and quasi-isometry Theorem G � X, H � Y : top. free systems on compact spaces X and Y . Assume G � X ∼ coe H � Y .

Continuous orbit equivalence and quasi-isometry Theorem G � X, H � Y : top. free systems on compact spaces X and Y . Assume G � X ∼ coe H � Y . If G is fin. gen., then so is H, and G and H are quasi-isometric.

Continuous orbit equivalence and quasi-isometry Theorem G � X, H � Y : top. free systems on compact spaces X and Y . Assume G � X ∼ coe H � Y . If G is fin. gen., then so is H, and G and H are quasi-isometric. ◮ A. Thom and R. Sauer have shown that for two groups G and H , there exist top. free systems G � X , H � Y on compact spaces X and Y with G � X ∼ coe H � Y if and only if G and H are bi-Lipschitz equivalent.

An abstract continuous orbit equivalence rigidity result

An abstract continuous orbit equivalence rigidity result Theorem G � X, H � Y : top. free systems.

An abstract continuous orbit equivalence rigidity result Theorem G � X, H � Y : top. free systems. Assume: ◮ X compact, C ( X , Z ) ∼ = Z · 1 ⊕ N as Z G-modules with pd Z G ( N ) < cd ( G ) − 1

An abstract continuous orbit equivalence rigidity result Theorem G � X, H � Y : top. free systems. Assume: ◮ X compact, C ( X , Z ) ∼ = Z · 1 ⊕ N as Z G-modules with pd Z G ( N ) < cd ( G ) − 1 ◮ G: duality group, H: solvable group.

An abstract continuous orbit equivalence rigidity result Theorem G � X, H � Y : top. free systems. Assume: ◮ X compact, C ( X , Z ) ∼ = Z · 1 ⊕ N as Z G-modules with pd Z G ( N ) < cd ( G ) − 1 ◮ G: duality group, H: solvable group. Then G � X ∼ coe H � Y ⇒ G � X ∼ conj H � Y .

A concrete continuous orbit equivalence rigidity result

A concrete continuous orbit equivalence rigidity result Theorem The following systems satisfy COER:

A concrete continuous orbit equivalence rigidity result Theorem The following systems satisfy COER: ◮ G � X G 0 , X 0 compact, | X 0 | > 1 , G: solvable duality group;

A concrete continuous orbit equivalence rigidity result Theorem The following systems satisfy COER: ◮ G � X G 0 , X 0 compact, | X 0 | > 1 , G: solvable duality group; ◮ top. free subshift of G � { 0 , . . . , N } G whose forbidden words avoid the letter 0 , G: solvable duality group;

A concrete continuous orbit equivalence rigidity result Theorem The following systems satisfy COER: ◮ G � X G 0 , X 0 compact, | X 0 | > 1 , G: solvable duality group; ◮ top. free subshift of G � { 0 , . . . , N } G whose forbidden words avoid the letter 0 , G: solvable duality group; ◮ chessboards Z 2 � X ( n ) with n ≥ 4 colours.

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