Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in...
Transcript of Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in...
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Applications of Ramsey theory in topologicaldynamics
Dana Bartosova 1 Aleksandra Kwiatkowska 2
Jordi Lopez Abad 3 Brice R. Mbombo 4
1,4University of Sao Paulo
2UCLA
3ICMAT Madrid and University of Sao Paulo
Forcing and its ApplicationsApril 1
The first author was supported by the grants FAPESP 2013/14458-9
and FAPESP 2014/12405-8.
Dana Bartosova Ramsey theory in topological dynamics
![Page 2: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/2.jpg)
Outline
(KPT) Topological dynamics and Ramsey theory
(G) Gurarij space
group of linear isometriesapproximate Ramsey propety for finite dimensional normedspaces
(S) Poulsen simplex
new characterizationgroup of affine homeomorphismsapproximate Ramsey propety for finite dimensionalsimplexes
(L) Lelek fan
group of homeomorphismexact Ramsey propety for sequences in FINk
Dana Bartosova Ramsey theory in topological dynamics
![Page 3: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/3.jpg)
Outline
(KPT) Topological dynamics and Ramsey theory
(G) Gurarij space
group of linear isometriesapproximate Ramsey propety for finite dimensional normedspaces
(S) Poulsen simplex
new characterizationgroup of affine homeomorphismsapproximate Ramsey propety for finite dimensionalsimplexes
(L) Lelek fan
group of homeomorphismexact Ramsey propety for sequences in FINk
Dana Bartosova Ramsey theory in topological dynamics
![Page 4: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/4.jpg)
Outline
(KPT) Topological dynamics and Ramsey theory
(G) Gurarij space
group of linear isometriesapproximate Ramsey propety for finite dimensional normedspaces
(S) Poulsen simplex
new characterizationgroup of affine homeomorphismsapproximate Ramsey propety for finite dimensionalsimplexes
(L) Lelek fan
group of homeomorphismexact Ramsey propety for sequences in FINk
Dana Bartosova Ramsey theory in topological dynamics
![Page 5: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/5.jpg)
Outline
(KPT) Topological dynamics and Ramsey theory
(G) Gurarij space
group of linear isometriesapproximate Ramsey propety for finite dimensional normedspaces
(S) Poulsen simplex
new characterizationgroup of affine homeomorphismsapproximate Ramsey propety for finite dimensionalsimplexes
(L) Lelek fan
group of homeomorphismexact Ramsey propety for sequences in FINk
Dana Bartosova Ramsey theory in topological dynamics
![Page 6: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/6.jpg)
Topological dynamics
G-flowG × X // X - a continuous action
↑ ↑topological compactgroup Hausdorff space
ex = x
g(hx) = (gh)x
X is a minimal G-flow ←→ X has no proper closed invariantsubset.The universal minimal flow M(G) is a minimal flow which hasevery other minimal flow as its factor.G is extremely amenable ←→ its universal minimal flow is asingleton (←→ every G-flow has a fixed point).
Dana Bartosova Ramsey theory in topological dynamics
![Page 7: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/7.jpg)
Topological dynamics
G-flowG × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space
ex = x
g(hx) = (gh)x
X is a minimal G-flow ←→ X has no proper closed invariantsubset.The universal minimal flow M(G) is a minimal flow which hasevery other minimal flow as its factor.G is extremely amenable ←→ its universal minimal flow is asingleton (←→ every G-flow has a fixed point).
Dana Bartosova Ramsey theory in topological dynamics
![Page 8: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/8.jpg)
Topological dynamics
G-flowG × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space
ex = x
g(hx) = (gh)x
X is a minimal G-flow ←→ X has no proper closed invariantsubset.The universal minimal flow M(G) is a minimal flow which hasevery other minimal flow as its factor.G is extremely amenable ←→ its universal minimal flow is asingleton (←→ every G-flow has a fixed point).
Dana Bartosova Ramsey theory in topological dynamics
![Page 9: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/9.jpg)
Topological dynamics
G-flowG × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space
ex = x
g(hx) = (gh)x
X is a minimal G-flow ←→ X has no proper closed invariantsubset.
The universal minimal flow M(G) is a minimal flow which hasevery other minimal flow as its factor.G is extremely amenable ←→ its universal minimal flow is asingleton (←→ every G-flow has a fixed point).
Dana Bartosova Ramsey theory in topological dynamics
![Page 10: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/10.jpg)
Topological dynamics
G-flowG × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space
ex = x
g(hx) = (gh)x
X is a minimal G-flow ←→ X has no proper closed invariantsubset.The universal minimal flow M(G) is a minimal flow which hasevery other minimal flow as its factor.
G is extremely amenable ←→ its universal minimal flow is asingleton (←→ every G-flow has a fixed point).
Dana Bartosova Ramsey theory in topological dynamics
![Page 11: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/11.jpg)
Topological dynamics
G-flowG × X // X - a continuous action↑ ↑topological compactgroup Hausdorff space
ex = x
g(hx) = (gh)x
X is a minimal G-flow ←→ X has no proper closed invariantsubset.The universal minimal flow M(G) is a minimal flow which hasevery other minimal flow as its factor.G is extremely amenable ←→ its universal minimal flow is asingleton (←→ every G-flow has a fixed point).
Dana Bartosova Ramsey theory in topological dynamics
![Page 12: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/12.jpg)
Structural Ramsey property
Theorem (Ramsey)
For every k ≤ m and r ≥ 2, there exists n such that for everycolouring of k-element subsets of n with r-many colours there isa subset X of n of size m such that all k-element subsets of Xhave the same colour.
A class K of finite structures satisfies the Ramsey property iffor every A ≤ B ∈ K and r ≥ 2 a natural number there existsC ∈ K such thatfor every colouring of copies of A in C by r colours, there is acopy B′ of B in C, such that all copies of A in B′ have the samecolour.
Dana Bartosova Ramsey theory in topological dynamics
![Page 13: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/13.jpg)
Structural Ramsey property
Theorem (Ramsey)
For every k ≤ m and r ≥ 2, there exists n such that for everycolouring of k-element subsets of n with r-many colours there isa subset X of n of size m such that all k-element subsets of Xhave the same colour.
A class K of finite structures satisfies the Ramsey property iffor every A ≤ B ∈ K
and r ≥ 2 a natural number there existsC ∈ K such thatfor every colouring of copies of A in C by r colours, there is acopy B′ of B in C, such that all copies of A in B′ have the samecolour.
Dana Bartosova Ramsey theory in topological dynamics
![Page 14: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/14.jpg)
Structural Ramsey property
Theorem (Ramsey)
For every k ≤ m and r ≥ 2, there exists n such that for everycolouring of k-element subsets of n with r-many colours there isa subset X of n of size m such that all k-element subsets of Xhave the same colour.
A class K of finite structures satisfies the Ramsey property iffor every A ≤ B ∈ K and r ≥ 2 a natural number
there existsC ∈ K such thatfor every colouring of copies of A in C by r colours, there is acopy B′ of B in C, such that all copies of A in B′ have the samecolour.
Dana Bartosova Ramsey theory in topological dynamics
![Page 15: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/15.jpg)
Structural Ramsey property
Theorem (Ramsey)
For every k ≤ m and r ≥ 2, there exists n such that for everycolouring of k-element subsets of n with r-many colours there isa subset X of n of size m such that all k-element subsets of Xhave the same colour.
A class K of finite structures satisfies the Ramsey property iffor every A ≤ B ∈ K and r ≥ 2 a natural number there existsC ∈ K such that
for every colouring of copies of A in C by r colours, there is acopy B′ of B in C, such that all copies of A in B′ have the samecolour.
Dana Bartosova Ramsey theory in topological dynamics
![Page 16: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/16.jpg)
Structural Ramsey property
Theorem (Ramsey)
For every k ≤ m and r ≥ 2, there exists n such that for everycolouring of k-element subsets of n with r-many colours there isa subset X of n of size m such that all k-element subsets of Xhave the same colour.
A class K of finite structures satisfies the Ramsey property iffor every A ≤ B ∈ K and r ≥ 2 a natural number there existsC ∈ K such thatfor every colouring of copies of A in C by r colours, there is acopy B′ of B in C, such that all copies of A in B′ have the samecolour.
Dana Bartosova Ramsey theory in topological dynamics
![Page 17: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/17.jpg)
Ramsey classes and extremely amenable groups
Ramsey classes
finite linear orders (Ramsey)
finite linearly ordered graphs (Nesetril and Rodl)
finite linearly ordered metric spaces (Nesetril)
finite Boolean algebras (Graham and Rothschild)
Extremely amenable groups
Aut(Q, <) (Pestov)
Aut(OR) – OR the random ordered graph (Kechris,Pestov & Todorcevic)
Iso(U, d) (Pestov)
Homeo(C, C) – (C, C) the Cantor space with a genericmaximal chain of closed subsets (KPT; Glasner & Weiss)
Dana Bartosova Ramsey theory in topological dynamics
![Page 18: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/18.jpg)
Ramsey classes and extremely amenable groups
Ramsey classes
finite linear orders (Ramsey)
finite linearly ordered graphs (Nesetril and Rodl)
finite linearly ordered metric spaces (Nesetril)
finite Boolean algebras (Graham and Rothschild)
Extremely amenable groups
Aut(Q, <) (Pestov)
Aut(OR) – OR the random ordered graph (Kechris,Pestov & Todorcevic)
Iso(U, d) (Pestov)
Homeo(C, C) – (C, C) the Cantor space with a genericmaximal chain of closed subsets (KPT; Glasner & Weiss)
Dana Bartosova Ramsey theory in topological dynamics
![Page 19: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/19.jpg)
What allows us to use the Ramsey property?
A - a first order structures
A is ultrahomogeneous ←→ every partial finite isomorphismcan be extended to an automorphism of A.G = Aut(A) with topology of pointwise convergence
A - a finitely-generated substructure of A
GA = {g ∈ G : ga = a ∀a ∈ A}
form a basis of neighbourhoods of the identity.
Theorem (KPT; NvT)
Aut(A) is extremely amenable ←→ finitely-generatedsubstructures of A satisfy the Ramsey property and are rigid.
Dana Bartosova Ramsey theory in topological dynamics
![Page 20: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/20.jpg)
What allows us to use the Ramsey property?
A - a first order structuresA is ultrahomogeneous ←→ every partial finite isomorphismcan be extended to an automorphism of A.
G = Aut(A) with topology of pointwise convergence
A - a finitely-generated substructure of A
GA = {g ∈ G : ga = a ∀a ∈ A}
form a basis of neighbourhoods of the identity.
Theorem (KPT; NvT)
Aut(A) is extremely amenable ←→ finitely-generatedsubstructures of A satisfy the Ramsey property and are rigid.
Dana Bartosova Ramsey theory in topological dynamics
![Page 21: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/21.jpg)
What allows us to use the Ramsey property?
A - a first order structuresA is ultrahomogeneous ←→ every partial finite isomorphismcan be extended to an automorphism of A.G = Aut(A) with topology of pointwise convergence
A - a finitely-generated substructure of A
GA = {g ∈ G : ga = a ∀a ∈ A}
form a basis of neighbourhoods of the identity.
Theorem (KPT; NvT)
Aut(A) is extremely amenable ←→ finitely-generatedsubstructures of A satisfy the Ramsey property and are rigid.
Dana Bartosova Ramsey theory in topological dynamics
![Page 22: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/22.jpg)
What allows us to use the Ramsey property?
A - a first order structuresA is ultrahomogeneous ←→ every partial finite isomorphismcan be extended to an automorphism of A.G = Aut(A) with topology of pointwise convergence
A - a finitely-generated substructure of A
GA = {g ∈ G : ga = a ∀a ∈ A}
form a basis of neighbourhoods of the identity.
Theorem (KPT; NvT)
Aut(A) is extremely amenable ←→ finitely-generatedsubstructures of A satisfy the Ramsey property and are rigid.
Dana Bartosova Ramsey theory in topological dynamics
![Page 23: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/23.jpg)
What allows us to use the Ramsey property?
A - a first order structuresA is ultrahomogeneous ←→ every partial finite isomorphismcan be extended to an automorphism of A.G = Aut(A) with topology of pointwise convergence
A - a finitely-generated substructure of A
GA = {g ∈ G : ga = a ∀a ∈ A}
form a basis of neighbourhoods of the identity.
Theorem (KPT; NvT)
Aut(A) is extremely amenable ←→ finitely-generatedsubstructures of A satisfy the Ramsey property and are rigid.
Dana Bartosova Ramsey theory in topological dynamics
![Page 24: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/24.jpg)
What allows us to use the Ramsey property?
A - a first order structuresA is ultrahomogeneous ←→ every partial finite isomorphismcan be extended to an automorphism of A.G = Aut(A) with topology of pointwise convergence
A - a finitely-generated substructure of A
GA = {g ∈ G : ga = a ∀a ∈ A}
form a basis of neighbourhoods of the identity.
Theorem (KPT; NvT)
Aut(A) is extremely amenable ←→ finitely-generatedsubstructures of A satisfy the Ramsey property and are rigid.
Dana Bartosova Ramsey theory in topological dynamics
![Page 25: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/25.jpg)
What allows us to use the Ramsey property?
A - a first order structuresA is ultrahomogeneous ←→ every partial finite isomorphismcan be extended to an automorphism of A.G = Aut(A) with topology of pointwise convergence
A - a finitely-generated substructure of A
GA = {g ∈ G : ga = a ∀a ∈ A}
form a basis of neighbourhoods of the identity.
Theorem (KPT; NvT)
Aut(A) is extremely amenable ←→ finitely-generatedsubstructures of A satisfy the Ramsey property and are rigid.
Dana Bartosova Ramsey theory in topological dynamics
![Page 26: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/26.jpg)
Universal minimal flows
G = Aut(A) – A ultrahomogeneous
G∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of AFinite substructures of A∗ satisfy the Ramsey property and arerigid.
OFTEN M(G) ∼= G/G∗
Structure A M(Aut(A)) authors
N linear orders on N Glasner and Weiss
random graph R linear orders on R KPT
Cantor space C maximal chains ofclosed subsets of C
Glasner and Weiss
Dana Bartosova Ramsey theory in topological dynamics
![Page 27: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/27.jpg)
Universal minimal flows
G = Aut(A) – A ultrahomogeneousG∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of A
Finite substructures of A∗ satisfy the Ramsey property and arerigid.
OFTEN M(G) ∼= G/G∗
Structure A M(Aut(A)) authors
N linear orders on N Glasner and Weiss
random graph R linear orders on R KPT
Cantor space C maximal chains ofclosed subsets of C
Glasner and Weiss
Dana Bartosova Ramsey theory in topological dynamics
![Page 28: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/28.jpg)
Universal minimal flows
G = Aut(A) – A ultrahomogeneousG∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of AFinite substructures of A∗ satisfy the Ramsey property and arerigid.
OFTEN M(G) ∼= G/G∗
Structure A M(Aut(A)) authors
N linear orders on N Glasner and Weiss
random graph R linear orders on R KPT
Cantor space C maximal chains ofclosed subsets of C
Glasner and Weiss
Dana Bartosova Ramsey theory in topological dynamics
![Page 29: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/29.jpg)
Universal minimal flows
G = Aut(A) – A ultrahomogeneousG∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of AFinite substructures of A∗ satisfy the Ramsey property and arerigid.
OFTEN M(G) ∼= G/G∗
Structure A M(Aut(A)) authors
N linear orders on N Glasner and Weiss
random graph R linear orders on R KPT
Cantor space C maximal chains ofclosed subsets of C
Glasner and Weiss
Dana Bartosova Ramsey theory in topological dynamics
![Page 30: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/30.jpg)
Universal minimal flows
G = Aut(A) – A ultrahomogeneousG∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of AFinite substructures of A∗ satisfy the Ramsey property and arerigid.
OFTEN M(G) ∼= G/G∗
Structure A M(Aut(A)) authors
N linear orders on N Glasner and Weiss
random graph R linear orders on R KPT
Cantor space C maximal chains ofclosed subsets of C
Glasner and Weiss
Dana Bartosova Ramsey theory in topological dynamics
![Page 31: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/31.jpg)
Other settings
ultrahomogeneous
approximately ultrahomogeneous
projectively ultrahomogeneous
approximately projectivelyultrahomogeneous
Ramsey property
approximate Ramsey property
dual Ramsey property
approximate dual Ramseyproperty
Structure... ...homogeneous w.r.t.
N, R embeddings
Gurarij space linear isometric embeddings
Lelek fan epimorphisms
Poulsen simplex affine epimorphisms
Dana Bartosova Ramsey theory in topological dynamics
![Page 32: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/32.jpg)
Other settings
ultrahomogeneous
approximately ultrahomogeneous
projectively ultrahomogeneous
approximately projectivelyultrahomogeneous
Ramsey property
approximate Ramsey property
dual Ramsey property
approximate dual Ramseyproperty
Structure... ...homogeneous w.r.t.
N, R embeddings
Gurarij space linear isometric embeddings
Lelek fan epimorphisms
Poulsen simplex affine epimorphisms
Dana Bartosova Ramsey theory in topological dynamics
![Page 33: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/33.jpg)
Other settings
ultrahomogeneous
approximately ultrahomogeneous
projectively ultrahomogeneous
approximately projectivelyultrahomogeneous
Ramsey property
approximate Ramsey property
dual Ramsey property
approximate dual Ramseyproperty
Structure... ...homogeneous w.r.t.
N, R embeddings
Gurarij space linear isometric embeddings
Lelek fan epimorphisms
Poulsen simplex affine epimorphisms
Dana Bartosova Ramsey theory in topological dynamics
![Page 34: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/34.jpg)
Other settings
ultrahomogeneous
approximately ultrahomogeneous
projectively ultrahomogeneous
approximately projectivelyultrahomogeneous
Ramsey property
approximate Ramsey property
dual Ramsey property
approximate dual Ramseyproperty
Structure... ...homogeneous w.r.t.
N, R embeddings
Gurarij space linear isometric embeddings
Lelek fan epimorphisms
Poulsen simplex affine epimorphisms
Dana Bartosova Ramsey theory in topological dynamics
![Page 35: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/35.jpg)
Other settings
ultrahomogeneous
approximately ultrahomogeneous
projectively ultrahomogeneous
approximately projectivelyultrahomogeneous
Ramsey property
approximate Ramsey property
dual Ramsey property
approximate dual Ramseyproperty
Structure... ...homogeneous w.r.t.
N, R embeddings
Gurarij space linear isometric embeddings
Lelek fan epimorphisms
Poulsen simplex affine epimorphisms
Dana Bartosova Ramsey theory in topological dynamics
![Page 36: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/36.jpg)
Other settings
ultrahomogeneous
approximately ultrahomogeneous
projectively ultrahomogeneous
approximately projectivelyultrahomogeneous
Ramsey property
approximate Ramsey property
dual Ramsey property
approximate dual Ramseyproperty
Structure... ...homogeneous w.r.t.
N, R embeddings
Gurarij space linear isometric embeddings
Lelek fan epimorphisms
Poulsen simplex affine epimorphisms
Dana Bartosova Ramsey theory in topological dynamics
![Page 37: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/37.jpg)
Gurarij space G
(1) separable Banach space
(2) contains isometric copy of every finite dimensional Banachspace
(3) for every E finite dimensional, i : E ↪→ G isometricembedding and ε > 0 there is a linear isometry f : G //G
‖i− f � E‖ < ε
LUSKYConditions (1),(2),(3) uniquely define G up to a linearisometry.
KUBIS-SOLECKI; HENSONSimple proof - metric Fraısse theory.
Dana Bartosova Ramsey theory in topological dynamics
![Page 38: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/38.jpg)
Gurarij space G
(1) separable Banach space
(2) contains isometric copy of every finite dimensional Banachspace
(3) for every E finite dimensional, i : E ↪→ G isometricembedding and ε > 0 there is a linear isometry f : G //G
‖i− f � E‖ < ε
LUSKYConditions (1),(2),(3) uniquely define G up to a linearisometry.
KUBIS-SOLECKI; HENSONSimple proof - metric Fraısse theory.
Dana Bartosova Ramsey theory in topological dynamics
![Page 39: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/39.jpg)
Gurarij space G
(1) separable Banach space
(2) contains isometric copy of every finite dimensional Banachspace
(3) for every E finite dimensional, i : E ↪→ G isometricembedding and ε > 0 there is a linear isometry f : G //G
‖i− f � E‖ < ε
LUSKYConditions (1),(2),(3) uniquely define G up to a linearisometry.
KUBIS-SOLECKI; HENSONSimple proof - metric Fraısse theory.
Dana Bartosova Ramsey theory in topological dynamics
![Page 40: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/40.jpg)
Gurarij space G
(1) separable Banach space
(2) contains isometric copy of every finite dimensional Banachspace
(3) for every E finite dimensional, i : E ↪→ G isometricembedding and ε > 0 there is a linear isometry f : G //G
‖i− f � E‖ < ε
LUSKYConditions (1),(2),(3) uniquely define G up to a linearisometry.
KUBIS-SOLECKI; HENSONSimple proof - metric Fraısse theory.
Dana Bartosova Ramsey theory in topological dynamics
![Page 41: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/41.jpg)
Gurarij space G
(1) separable Banach space
(2) contains isometric copy of every finite dimensional Banachspace
(3) for every E finite dimensional, i : E ↪→ G isometricembedding and ε > 0 there is a linear isometry f : G //G
‖i− f � E‖ < ε
LUSKYConditions (1),(2),(3) uniquely define G up to a linearisometry.
KUBIS-SOLECKI; HENSONSimple proof - metric Fraısse theory.
Dana Bartosova Ramsey theory in topological dynamics
![Page 42: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/42.jpg)
Linear isometries
Isol(G) + point-wise convergence topology = Polish group
BASIS
E - finite dimensional subspace of Gε > 0
Vε(E) = {g ∈ Iso(G) : ‖g � E − id � E‖ < ε}
BEN YAACOVIsol(G) is a universal Polish group.
Katetov construction
Dana Bartosova Ramsey theory in topological dynamics
![Page 43: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/43.jpg)
Linear isometries
Isol(G) + point-wise convergence topology = Polish group
BASIS
E - finite dimensional subspace of Gε > 0
Vε(E) = {g ∈ Iso(G) : ‖g � E − id � E‖ < ε}
BEN YAACOVIsol(G) is a universal Polish group.
Katetov construction
Dana Bartosova Ramsey theory in topological dynamics
![Page 44: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/44.jpg)
Linear isometries
Isol(G) + point-wise convergence topology = Polish group
BASIS
E - finite dimensional subspace of G
ε > 0
Vε(E) = {g ∈ Iso(G) : ‖g � E − id � E‖ < ε}
BEN YAACOVIsol(G) is a universal Polish group.
Katetov construction
Dana Bartosova Ramsey theory in topological dynamics
![Page 45: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/45.jpg)
Linear isometries
Isol(G) + point-wise convergence topology = Polish group
BASIS
E - finite dimensional subspace of Gε > 0
Vε(E) = {g ∈ Iso(G) : ‖g � E − id � E‖ < ε}
BEN YAACOVIsol(G) is a universal Polish group.
Katetov construction
Dana Bartosova Ramsey theory in topological dynamics
![Page 46: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/46.jpg)
Linear isometries
Isol(G) + point-wise convergence topology = Polish group
BASIS
E - finite dimensional subspace of Gε > 0
Vε(E) = {g ∈ Iso(G) : ‖g � E − id � E‖ < ε}
BEN YAACOVIsol(G) is a universal Polish group.
Katetov construction
Dana Bartosova Ramsey theory in topological dynamics
![Page 47: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/47.jpg)
Linear isometries
Isol(G) + point-wise convergence topology = Polish group
BASIS
E - finite dimensional subspace of Gε > 0
Vε(E) = {g ∈ Iso(G) : ‖g � E − id � E‖ < ε}
BEN YAACOVIsol(G) is a universal Polish group.
Katetov construction
Dana Bartosova Ramsey theory in topological dynamics
![Page 48: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/48.jpg)
Linear isometries
Isol(G) + point-wise convergence topology = Polish group
BASIS
E - finite dimensional subspace of Gε > 0
Vε(E) = {g ∈ Iso(G) : ‖g � E − id � E‖ < ε}
BEN YAACOVIsol(G) is a universal Polish group.
Katetov construction
Dana Bartosova Ramsey theory in topological dynamics
![Page 49: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/49.jpg)
Approximate Ramsey property for finite-dimensionalnormed spaces
E,F - finite dimensional spacesθ ≥ 1
Embθ(E,F ) = {T : E // F : T embedding & ‖T‖∥∥T−1
∥∥ ≤ θ}
Theorem (B-LA-M)
r - number of colours, ε > 0 // ∃H f.d. with Emb(F,H) 6= ∅such that for every
c : Embθ(E,H) // {0, 1, . . . , r − 1}
∃i ∈ Embθ(F,H) and α < r such that
i ◦ Embθ(E,F ) ⊂ (c−1(α))θ−1+ε
Dana Bartosova Ramsey theory in topological dynamics
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Approximate Ramsey property for finite-dimensionalnormed spaces
E,F - finite dimensional spacesθ ≥ 1
Embθ(E,F ) = {T : E // F : T embedding & ‖T‖∥∥T−1
∥∥ ≤ θ}Theorem (B-LA-M)
r - number of colours, ε > 0 // ∃H f.d. with Emb(F,H) 6= ∅such that for every
c : Embθ(E,H) // {0, 1, . . . , r − 1}
∃i ∈ Embθ(F,H) and α < r such that
i ◦ Embθ(E,F ) ⊂ (c−1(α))θ−1+ε
Dana Bartosova Ramsey theory in topological dynamics
![Page 51: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/51.jpg)
Pestov’s characterization of extreme amenability
G - topological group
f : G // R is finitely oscillation stable if ∀X ⊂ G finite andε > 0 ∃g ∈ G such that osc(f � gX) < ε.
Theorem (Pestov)
TFAE
G is extremely amenable,
every f : G // R bounded left-uniformly continuous isfinite oscillation stable.
Theorem (B-LA-M)
Isol(G) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 52: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/52.jpg)
Pestov’s characterization of extreme amenability
G - topological groupf : G // R is finitely oscillation stable if
∀X ⊂ G finite andε > 0 ∃g ∈ G such that osc(f � gX) < ε.
Theorem (Pestov)
TFAE
G is extremely amenable,
every f : G // R bounded left-uniformly continuous isfinite oscillation stable.
Theorem (B-LA-M)
Isol(G) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 53: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/53.jpg)
Pestov’s characterization of extreme amenability
G - topological groupf : G // R is finitely oscillation stable if ∀X ⊂ G finite andε > 0
∃g ∈ G such that osc(f � gX) < ε.
Theorem (Pestov)
TFAE
G is extremely amenable,
every f : G // R bounded left-uniformly continuous isfinite oscillation stable.
Theorem (B-LA-M)
Isol(G) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 54: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/54.jpg)
Pestov’s characterization of extreme amenability
G - topological groupf : G // R is finitely oscillation stable if ∀X ⊂ G finite andε > 0 ∃g ∈ G such that osc(f � gX) < ε.
Theorem (Pestov)
TFAE
G is extremely amenable,
every f : G // R bounded left-uniformly continuous isfinite oscillation stable.
Theorem (B-LA-M)
Isol(G) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 55: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/55.jpg)
Pestov’s characterization of extreme amenability
G - topological groupf : G // R is finitely oscillation stable if ∀X ⊂ G finite andε > 0 ∃g ∈ G such that osc(f � gX) < ε.
Theorem (Pestov)
TFAE
G is extremely amenable,
every f : G // R bounded left-uniformly continuous isfinite oscillation stable.
Theorem (B-LA-M)
Isol(G) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 56: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/56.jpg)
Pestov’s characterization of extreme amenability
G - topological groupf : G // R is finitely oscillation stable if ∀X ⊂ G finite andε > 0 ∃g ∈ G such that osc(f � gX) < ε.
Theorem (Pestov)
TFAE
G is extremely amenable,
every f : G // R bounded left-uniformly continuous isfinite oscillation stable.
Theorem (B-LA-M)
Isol(G) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 57: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/57.jpg)
Urysohn space U
Theorem
Finite metric spaces satisfy the approximate Ramsey property.
Corollary (Pestov)
Iso(U) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 58: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/58.jpg)
Urysohn space U
Theorem
Finite metric spaces satisfy the approximate Ramsey property.
Corollary (Pestov)
Iso(U) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 59: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/59.jpg)
Poulsen simplex P
(1) metrizable
(2) contains every metrizable simplex as its face
(3) for every two faces E,F of P with the same finitedimension, there is an affine autohomeomorphism of Pmapping E onto F
LINDENSTRAUSS-OLSEN-STERNFELDProperties (1),(2) and (3) uniquely determine P up to an affinehomeomorphism.
POULSENThe set of extreme points of P is dense in P.
FACTT : {0, 1}Z // {0, 1}Z the shift ⇒ T -invariant probabilitymeasures form P
Dana Bartosova Ramsey theory in topological dynamics
![Page 60: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/60.jpg)
Poulsen simplex P
(1) metrizable
(2) contains every metrizable simplex as its face
(3) for every two faces E,F of P with the same finitedimension, there is an affine autohomeomorphism of Pmapping E onto F
LINDENSTRAUSS-OLSEN-STERNFELDProperties (1),(2) and (3) uniquely determine P up to an affinehomeomorphism.
POULSENThe set of extreme points of P is dense in P.
FACTT : {0, 1}Z // {0, 1}Z the shift ⇒ T -invariant probabilitymeasures form P
Dana Bartosova Ramsey theory in topological dynamics
![Page 61: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/61.jpg)
Poulsen simplex P
(1) metrizable
(2) contains every metrizable simplex as its face
(3) for every two faces E,F of P with the same finitedimension, there is an affine autohomeomorphism of Pmapping E onto F
LINDENSTRAUSS-OLSEN-STERNFELDProperties (1),(2) and (3) uniquely determine P up to an affinehomeomorphism.
POULSENThe set of extreme points of P is dense in P.
FACTT : {0, 1}Z // {0, 1}Z the shift ⇒ T -invariant probabilitymeasures form P
Dana Bartosova Ramsey theory in topological dynamics
![Page 62: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/62.jpg)
Poulsen simplex P
(1) metrizable
(2) contains every metrizable simplex as its face
(3) for every two faces E,F of P with the same finitedimension, there is an affine autohomeomorphism of Pmapping E onto F
LINDENSTRAUSS-OLSEN-STERNFELDProperties (1),(2) and (3) uniquely determine P up to an affinehomeomorphism.
POULSENThe set of extreme points of P is dense in P.
FACTT : {0, 1}Z // {0, 1}Z the shift ⇒ T -invariant probabilitymeasures form P
Dana Bartosova Ramsey theory in topological dynamics
![Page 63: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/63.jpg)
Poulsen simplex P
(1) metrizable
(2) contains every metrizable simplex as its face
(3) for every two faces E,F of P with the same finitedimension, there is an affine autohomeomorphism of Pmapping E onto F
LINDENSTRAUSS-OLSEN-STERNFELDProperties (1),(2) and (3) uniquely determine P up to an affinehomeomorphism.
POULSENThe set of extreme points of P is dense in P.
FACTT : {0, 1}Z // {0, 1}Z the shift ⇒ T -invariant probabilitymeasures form P
Dana Bartosova Ramsey theory in topological dynamics
![Page 64: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/64.jpg)
Poulsen simplex P
(1) metrizable
(2) contains every metrizable simplex as its face
(3) for every two faces E,F of P with the same finitedimension, there is an affine autohomeomorphism of Pmapping E onto F
LINDENSTRAUSS-OLSEN-STERNFELDProperties (1),(2) and (3) uniquely determine P up to an affinehomeomorphism.
POULSENThe set of extreme points of P is dense in P.
FACTT : {0, 1}Z // {0, 1}Z the shift ⇒ T -invariant probabilitymeasures form P
Dana Bartosova Ramsey theory in topological dynamics
![Page 65: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/65.jpg)
A projective characterization of P
Sn := positive part of the unit ball of ln1 – finite-dimensionalsimplex with n+ 1 extreme points
Epi(Sn, Sm) := continuous affine surjections Sn // Sm
AH(P ) := group of affine homeomorphisms of P +compact-open topology
(U) ∀n ∃φ : P // Sn – continuous affine surjection
(APU) ∀ε > 0 ∀n ∀φ1, φ2 : P // Sn ∃f ∈ AH(P ) withd(φ1, φ2 ◦ f) < ε
Theorem (B-LA-M)
(U) + (APU) characterize P among non-trivial metrizablesimplexes up to affine homeomorphism.
Dana Bartosova Ramsey theory in topological dynamics
![Page 66: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/66.jpg)
A projective characterization of P
Sn := positive part of the unit ball of ln1 – finite-dimensionalsimplex with n+ 1 extreme points
Epi(Sn, Sm) := continuous affine surjections Sn // Sm
AH(P ) := group of affine homeomorphisms of P +compact-open topology
(U) ∀n ∃φ : P // Sn – continuous affine surjection
(APU) ∀ε > 0 ∀n ∀φ1, φ2 : P // Sn ∃f ∈ AH(P ) withd(φ1, φ2 ◦ f) < ε
Theorem (B-LA-M)
(U) + (APU) characterize P among non-trivial metrizablesimplexes up to affine homeomorphism.
Dana Bartosova Ramsey theory in topological dynamics
![Page 67: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/67.jpg)
A projective characterization of P
Sn := positive part of the unit ball of ln1 – finite-dimensionalsimplex with n+ 1 extreme points
Epi(Sn, Sm) := continuous affine surjections Sn // Sm
AH(P ) := group of affine homeomorphisms of P +compact-open topology
(U) ∀n ∃φ : P // Sn – continuous affine surjection
(APU) ∀ε > 0 ∀n ∀φ1, φ2 : P // Sn ∃f ∈ AH(P ) withd(φ1, φ2 ◦ f) < ε
Theorem (B-LA-M)
(U) + (APU) characterize P among non-trivial metrizablesimplexes up to affine homeomorphism.
Dana Bartosova Ramsey theory in topological dynamics
![Page 68: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/68.jpg)
A projective characterization of P
Sn := positive part of the unit ball of ln1 – finite-dimensionalsimplex with n+ 1 extreme points
Epi(Sn, Sm) := continuous affine surjections Sn // Sm
AH(P ) := group of affine homeomorphisms of P +compact-open topology
(U) ∀n ∃φ : P // Sn – continuous affine surjection
(APU) ∀ε > 0 ∀n ∀φ1, φ2 : P // Sn ∃f ∈ AH(P ) withd(φ1, φ2 ◦ f) < ε
Theorem (B-LA-M)
(U) + (APU) characterize P among non-trivial metrizablesimplexes up to affine homeomorphism.
Dana Bartosova Ramsey theory in topological dynamics
![Page 69: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/69.jpg)
A projective characterization of P
Sn := positive part of the unit ball of ln1 – finite-dimensionalsimplex with n+ 1 extreme points
Epi(Sn, Sm) := continuous affine surjections Sn // Sm
AH(P ) := group of affine homeomorphisms of P +compact-open topology
(U) ∀n ∃φ : P // Sn – continuous affine surjection
(APU) ∀ε > 0 ∀n ∀φ1, φ2 : P // Sn ∃f ∈ AH(P ) withd(φ1, φ2 ◦ f) < ε
Theorem (B-LA-M)
(U) + (APU) characterize P among non-trivial metrizablesimplexes up to affine homeomorphism.
Dana Bartosova Ramsey theory in topological dynamics
![Page 70: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/70.jpg)
Approximate Ramsey property for P
Epi0(Sn, Sm) - continuous affine surjections preserving 0
Theorem (B-LA-M)
d ≤ m and r natural numbers and ε > 0 given // ∃n such thatfor every colouring
c : Epi0(Sn, Sd) // {0, 1, . . . , r}
there is π ∈ Epi0(Sn, Sm) and α < r such that
Epi0(Sm, Sd) ◦ π ⊂ (c−1(α))ε
p - extreme point of PAHp(P ) = {f ∈ AH(P ) : f(p) = p}
Theorem (B-LA-M)
AHp(P ) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 71: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/71.jpg)
Approximate Ramsey property for P
Epi0(Sn, Sm) - continuous affine surjections preserving 0
Theorem (B-LA-M)
d ≤ m and r natural numbers and ε > 0 given // ∃n such thatfor every colouring
c : Epi0(Sn, Sd) // {0, 1, . . . , r}
there is π ∈ Epi0(Sn, Sm) and α < r such that
Epi0(Sm, Sd) ◦ π ⊂ (c−1(α))ε
p - extreme point of PAHp(P ) = {f ∈ AH(P ) : f(p) = p}
Theorem (B-LA-M)
AHp(P ) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 72: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/72.jpg)
Approximate Ramsey property for P
Epi0(Sn, Sm) - continuous affine surjections preserving 0
Theorem (B-LA-M)
d ≤ m and r natural numbers and ε > 0 given // ∃n such thatfor every colouring
c : Epi0(Sn, Sd) // {0, 1, . . . , r}
there is π ∈ Epi0(Sn, Sm) and α < r such that
Epi0(Sm, Sd) ◦ π ⊂ (c−1(α))ε
p - extreme point of PAHp(P ) = {f ∈ AH(P ) : f(p) = p}
Theorem (B-LA-M)
AHp(P ) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 73: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/73.jpg)
Approximate Ramsey property for P
Epi0(Sn, Sm) - continuous affine surjections preserving 0
Theorem (B-LA-M)
d ≤ m and r natural numbers and ε > 0 given // ∃n such thatfor every colouring
c : Epi0(Sn, Sd) // {0, 1, . . . , r}
there is π ∈ Epi0(Sn, Sm) and α < r such that
Epi0(Sm, Sd) ◦ π ⊂ (c−1(α))ε
p - extreme point of PAHp(P ) = {f ∈ AH(P ) : f(p) = p}
Theorem (B-LA-M)
AHp(P ) is extremely amenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 74: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/74.jpg)
Universal minimal flow of AH(P )
Theorem (B-LA-M)
M(AH(P )) ∼= AH(P )/AHp(P ) ∼= P
Dana Bartosova Ramsey theory in topological dynamics
![Page 75: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/75.jpg)
Hilbert cube Q = [−1, 1]N
PROBLEMWhat is the universal minimal flow of Homeo(Q)?
Q is homeomorphic to P.
Theorem (B-LA-M)
Homeo(Q) admits a closed subgroup with the universal minimalflow being the natural action on Q.
Q with its natural convex structure.
Theorem (B-LA-M)
Aut(Q) is topologically isomorphic to {−1, 1}N × S∞.
Theorem (B-LA-M)
M(Aut(Q)) = {−1, 1}N × LO(N).
Dana Bartosova Ramsey theory in topological dynamics
![Page 76: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/76.jpg)
Hilbert cube Q = [−1, 1]N
PROBLEMWhat is the universal minimal flow of Homeo(Q)?
Q is homeomorphic to P.
Theorem (B-LA-M)
Homeo(Q) admits a closed subgroup with the universal minimalflow being the natural action on Q.
Q with its natural convex structure.
Theorem (B-LA-M)
Aut(Q) is topologically isomorphic to {−1, 1}N × S∞.
Theorem (B-LA-M)
M(Aut(Q)) = {−1, 1}N × LO(N).
Dana Bartosova Ramsey theory in topological dynamics
![Page 77: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/77.jpg)
Hilbert cube Q = [−1, 1]N
PROBLEMWhat is the universal minimal flow of Homeo(Q)?
Q is homeomorphic to P.
Theorem (B-LA-M)
Homeo(Q) admits a closed subgroup with the universal minimalflow being the natural action on Q.
Q with its natural convex structure.
Theorem (B-LA-M)
Aut(Q) is topologically isomorphic to {−1, 1}N × S∞.
Theorem (B-LA-M)
M(Aut(Q)) = {−1, 1}N × LO(N).
Dana Bartosova Ramsey theory in topological dynamics
![Page 78: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/78.jpg)
Hilbert cube Q = [−1, 1]N
PROBLEMWhat is the universal minimal flow of Homeo(Q)?
Q is homeomorphic to P.
Theorem (B-LA-M)
Homeo(Q) admits a closed subgroup with the universal minimalflow being the natural action on Q.
Q with its natural convex structure.
Theorem (B-LA-M)
Aut(Q) is topologically isomorphic to {−1, 1}N × S∞.
Theorem (B-LA-M)
M(Aut(Q)) = {−1, 1}N × LO(N).
Dana Bartosova Ramsey theory in topological dynamics
![Page 79: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/79.jpg)
Hilbert cube Q = [−1, 1]N
PROBLEMWhat is the universal minimal flow of Homeo(Q)?
Q is homeomorphic to P.
Theorem (B-LA-M)
Homeo(Q) admits a closed subgroup with the universal minimalflow being the natural action on Q.
Q with its natural convex structure.
Theorem (B-LA-M)
Aut(Q) is topologically isomorphic to {−1, 1}N × S∞.
Theorem (B-LA-M)
M(Aut(Q)) = {−1, 1}N × LO(N).
Dana Bartosova Ramsey theory in topological dynamics
![Page 80: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/80.jpg)
Hilbert cube Q = [−1, 1]N
PROBLEMWhat is the universal minimal flow of Homeo(Q)?
Q is homeomorphic to P.
Theorem (B-LA-M)
Homeo(Q) admits a closed subgroup with the universal minimalflow being the natural action on Q.
Q with its natural convex structure.
Theorem (B-LA-M)
Aut(Q) is topologically isomorphic to {−1, 1}N × S∞.
Theorem (B-LA-M)
M(Aut(Q)) = {−1, 1}N × LO(N).
Dana Bartosova Ramsey theory in topological dynamics
![Page 81: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/81.jpg)
Lelek fan L
= unique non-trivial subcontinuum of the Cantor fan with adense set of endpoints (Bula-Oversteegen, Charatonik)
continuum = connected compact metric Hausdorff space
fan.jpg
Dana Bartosova Ramsey theory in topological dynamics
![Page 82: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/82.jpg)
Lelek fan L
= unique non-trivial subcontinuum of the Cantor fan with adense set of endpoints (Bula-Oversteegen, Charatonik)
continuum = connected compact metric Hausdorff space
fan.jpg
Dana Bartosova Ramsey theory in topological dynamics
![Page 83: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/83.jpg)
Pre-Lelek fan
(L, RLs ) - compact, 0-dim, RL
s ⊂ L2 closed with one or twoelement equivalence classes
L/RLs∼= L
F = {finite fans} + surjective homomorphisms
(U) T ∈ F ∃φ : (L, RL) // T - continuous surjectivehomomorphism
(R) X finite, f : L //X continuous ∃T ∈ F , φ : L // T andg : T //X such that f = g ◦ φ
(PU) T ∈ F , φ1, φ2 : L // T ∃g : L // L automorphism withφ1 = φ2 ◦ g
Dana Bartosova Ramsey theory in topological dynamics
![Page 84: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/84.jpg)
Pre-Lelek fan
(L, RLs ) - compact, 0-dim, RL
s ⊂ L2 closed with one or twoelement equivalence classes
L/RLs∼= L
F = {finite fans} + surjective homomorphisms
(U) T ∈ F ∃φ : (L, RL) // T - continuous surjectivehomomorphism
(R) X finite, f : L //X continuous ∃T ∈ F , φ : L // T andg : T //X such that f = g ◦ φ
(PU) T ∈ F , φ1, φ2 : L // T ∃g : L // L automorphism withφ1 = φ2 ◦ g
Dana Bartosova Ramsey theory in topological dynamics
![Page 85: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/85.jpg)
Pre-Lelek fan
(L, RLs ) - compact, 0-dim, RL
s ⊂ L2 closed with one or twoelement equivalence classes
L/RLs∼= L
F = {finite fans} + surjective homomorphisms
(U) T ∈ F ∃φ : (L, RL) // T - continuous surjectivehomomorphism
(R) X finite, f : L //X continuous ∃T ∈ F , φ : L // T andg : T //X such that f = g ◦ φ
(PU) T ∈ F , φ1, φ2 : L // T ∃g : L // L automorphism withφ1 = φ2 ◦ g
Dana Bartosova Ramsey theory in topological dynamics
![Page 86: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/86.jpg)
Pre-Lelek fan
(L, RLs ) - compact, 0-dim, RL
s ⊂ L2 closed with one or twoelement equivalence classes
L/RLs∼= L
F = {finite fans} + surjective homomorphisms
(U) T ∈ F ∃φ : (L, RL) // T - continuous surjectivehomomorphism
(R) X finite, f : L //X continuous ∃T ∈ F , φ : L // T andg : T //X such that f = g ◦ φ
(PU) T ∈ F , φ1, φ2 : L // T ∃g : L // L automorphism withφ1 = φ2 ◦ g
Dana Bartosova Ramsey theory in topological dynamics
![Page 87: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/87.jpg)
Aut // Homeo
Aut(L, RLs ) and Homeo(L) + the compact-open topology
π : L // L/RLs∼= L
induces a continuous embedding Aut(L, RLs ) ↪→ Homeo(L)
with a dense image
h 7→ h∗
π ◦ h = h∗ ◦ π.
Dana Bartosova Ramsey theory in topological dynamics
![Page 88: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/88.jpg)
Aut // Homeo
Aut(L, RLs ) and Homeo(L) + the compact-open topology
π : L // L/RLs∼= L
induces a continuous embedding Aut(L, RLs ) ↪→ Homeo(L)
with a dense image
h 7→ h∗
π ◦ h = h∗ ◦ π.
Dana Bartosova Ramsey theory in topological dynamics
![Page 89: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/89.jpg)
Aut // Homeo
Aut(L, RLs ) and Homeo(L) + the compact-open topology
π : L // L/RLs∼= L
induces a continuous embedding Aut(L, RLs ) ↪→ Homeo(L)
with a dense image
h 7→ h∗
π ◦ h = h∗ ◦ π.
Dana Bartosova Ramsey theory in topological dynamics
![Page 90: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/90.jpg)
Aut // Homeo
Aut(L, RLs ) and Homeo(L) + the compact-open topology
π : L // L/RLs∼= L
induces a continuous embedding Aut(L, RLs ) ↪→ Homeo(L)
with a dense image
h 7→ h∗
π ◦ h = h∗ ◦ π.
Dana Bartosova Ramsey theory in topological dynamics
![Page 91: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/91.jpg)
Aut // Homeo
Aut(L, RLs ) and Homeo(L) + the compact-open topology
π : L // L/RLs∼= L
induces a continuous embedding Aut(L, RLs ) ↪→ Homeo(L)
with a dense image
h 7→ h∗
π ◦ h = h∗ ◦ π.
Dana Bartosova Ramsey theory in topological dynamics
![Page 92: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/92.jpg)
Ramsey property for L
F< - finite fans with a linear order extending the natural order{C //A} := all epimorphisms from C onto A
Theorem
F< satisfies the Ramsey property.
For every A,B ∈ F< there exists C ∈ F< such that for everycolouring
c : {C //A} // {1, 2, . . . , r}
there exists f : C //B such that {B //A} ◦ f ismonochromatic.
Theorem (B-K)
Let L< be the limit of F<. Then Aut(L<) is extremelyamenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 93: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/93.jpg)
Ramsey property for L
F< - finite fans with a linear order extending the natural order{C //A} := all epimorphisms from C onto A
Theorem
F< satisfies the Ramsey property.
For every A,B ∈ F< there exists C ∈ F< such that for everycolouring
c : {C //A} // {1, 2, . . . , r}
there exists f : C //B such that {B //A} ◦ f ismonochromatic.
Theorem (B-K)
Let L< be the limit of F<. Then Aut(L<) is extremelyamenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 94: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/94.jpg)
Ramsey property for L
F< - finite fans with a linear order extending the natural order{C //A} := all epimorphisms from C onto A
Theorem
F< satisfies the Ramsey property.
For every A,B ∈ F< there exists C ∈ F< such that for everycolouring
c : {C //A} // {1, 2, . . . , r}
there exists f : C //B such that {B //A} ◦ f ismonochromatic.
Theorem (B-K)
Let L< be the limit of F<. Then Aut(L<) is extremelyamenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 95: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/95.jpg)
Ramsey property for L
F< - finite fans with a linear order extending the natural order{C //A} := all epimorphisms from C onto A
Theorem
F< satisfies the Ramsey property.
For every A,B ∈ F< there exists C ∈ F< such that for everycolouring
c : {C //A} // {1, 2, . . . , r}
there exists f : C //B such that {B //A} ◦ f ismonochromatic.
Theorem (B-K)
Let L< be the limit of F<. Then Aut(L<) is extremelyamenable.
Dana Bartosova Ramsey theory in topological dynamics
![Page 96: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/96.jpg)
Universal minimal flow of Homeo(L)
Theorem (B-K)
M(Aut(L)) ∼= Aut(L)/Aut(L<)
M(Homeo(L)) ∼= Homeo(L)/Homeo(L<)
Dana Bartosova Ramsey theory in topological dynamics
![Page 97: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/97.jpg)
Universal minimal flow of Homeo(L)
Theorem (B-K)
M(Aut(L)) ∼= Aut(L)/Aut(L<)
M(Homeo(L)) ∼= Homeo(L)/Homeo(L<)
Dana Bartosova Ramsey theory in topological dynamics
![Page 98: Applications of Ramsey theory in topological dynamics€¦ · Applications of Ramsey theory in topological dynamics Dana Barto sov a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3](https://reader033.fdocuments.us/reader033/viewer/2022042217/5ec221f989961924c01c2ff9/html5/thumbnails/98.jpg)
A question
Is there a non-trivial simplex with extremely amenable group ofaffine homeomorphisms?
Dana Bartosova Ramsey theory in topological dynamics
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THANK YOU
HAPPY FOOLS’ DAY!
Dana Bartosova Ramsey theory in topological dynamics