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Continuous orbit equivalence rigidity Xin Li Dynamical systems and - - PowerPoint PPT Presentation
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
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Dynamical systems and operator algebras
Dynamical systems ← − − − − − − − − − − − − → Operator algebras
◮ Q.: How to go from dynamical systems to operator algebras?
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Dynamical systems and operator algebras
Dynamical systems ← − − − − − − − − − − − − → Operator algebras
◮ Q.: How to go from dynamical systems to operator algebras?
A.: Crossed product construction.
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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???
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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
C0(X) ⋊r G ∼ = C0(Y ) ⋊r H ⇒ G X ∼ H Y ???
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Continuous orbit equivalence
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Continuous orbit equivalence
G X, H Y : topological dynamical systems.
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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.
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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
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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).
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Topological dynamics and C*-algebras
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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 : C0(X) ⋊r G
∼ =
− → C0(Y ) ⋊r H with Φ(C0(X)) = C0(Y ).
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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 : C0(X) ⋊r G
∼ =
− → C0(Y ) ⋊r H with Φ(C0(X)) = C0(Y ).
- top. free: for all e = g ∈ G, {x ∈ X: g.x = x} is dense in X.
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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 : C0(X) ⋊r G
∼ =
− → C0(Y ) ⋊r H with Φ(C0(X)) = C0(Y ).
- top. free: for all e = g ∈ G, {x ∈ X: g.x = x} is dense in X.
◮ Conjugacy ⇒ COE ⇔ Cartan-isom. ⇒ C*-isom.
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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 : C0(X) ⋊r G
∼ =
− → C0(Y ) ⋊r H with Φ(C0(X)) = C0(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?
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Continuous orbit equivalence rigidity
Can we reverse the first arrow in
◮ Conjugacy ⇒ COE ⇔ Cartan-isom. ⇒ C*-isom.?
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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.
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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.
◮ Counterexamples: Conjugacy ✟
✟
⇐ COE for certain Zn X,
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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.
◮ Counterexamples: Conjugacy ✟
✟
⇐ COE for certain Zn X, and also for certain Fn X.
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Continuous orbit equivalence and quasi-isometry
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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 .
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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.
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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.
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An abstract continuous orbit equivalence rigidity result
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An abstract continuous orbit equivalence rigidity result
Theorem
G X, H Y : top. free systems.
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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 ZG-modules with pdZG(N) < cd(G) − 1
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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 ZG-modules with pdZG(N) < cd(G) − 1
◮ G: duality group, H: solvable group.
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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 ZG-modules with pdZG(N) < cd(G) − 1
◮ G: duality group, H: solvable group.
Then G X ∼coe H Y ⇒ G X ∼conj H Y .
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A concrete continuous orbit equivalence rigidity result
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A concrete continuous orbit equivalence rigidity result
Theorem
The following systems satisfy COER:
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A concrete continuous orbit equivalence rigidity result
Theorem
The following systems satisfy COER:
◮ G X G 0 , X0 compact, |X0| > 1, G: solvable duality group;
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A concrete continuous orbit equivalence rigidity result
Theorem
The following systems satisfy COER:
◮ G X G 0 , X0 compact, |X0| > 1, G: solvable duality group; ◮ top. free subshift of G {0, . . . , N}G whose forbidden words
avoid the letter 0, G: solvable duality group;
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A concrete continuous orbit equivalence rigidity result
Theorem
The following systems satisfy COER:
◮ G X G 0 , X0 compact, |X0| > 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 Z2 X (n) with n ≥ 4 colours.
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