[PPT] - Applications of Ramsey theory in topological dynamics a 1 Aleksandra PowerPoint Presentation

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Applications of Ramsey theory in topological dynamics

Dana Bartoˇ sov´ a 1 Aleksandra Kwiatkowska 2 Jordi Lopez Abad 3 Brice R. Mbombo 4

1,4University of S˜

ao Paulo

2UCLA 3ICMAT Madrid and University of S˜

ao Paulo

Forcing and its Applications April 1

The first author was supported by the grants FAPESP 2013/14458-9 and FAPESP 2014/12405-8.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Outline

(KPT) Topological dynamics and Ramsey theory

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Outline

(KPT) Topological dynamics and Ramsey theory (G) Gurarij space

group of linear isometries approximate Ramsey propety for finite dimensional normed spaces

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Outline

(KPT) Topological dynamics and Ramsey theory (G) Gurarij space

group of linear isometries approximate Ramsey propety for finite dimensional normed spaces

(S) Poulsen simplex

new characterization group of affine homeomorphisms approximate Ramsey propety for finite dimensional simplexes

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Outline

(KPT) Topological dynamics and Ramsey theory (G) Gurarij space

group of linear isometries approximate Ramsey propety for finite dimensional normed spaces

(S) Poulsen simplex

new characterization group of affine homeomorphisms approximate Ramsey propety for finite dimensional simplexes

(L) Lelek fan

group of homeomorphism exact Ramsey propety for sequences in FINk

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Topological dynamics

G-flow G × X

X - a continuous action

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Topological dynamics

G-flow G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Topological dynamics

G-flow G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space ex = x g(hx) = (gh)x

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Topological dynamics

G-flow G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space ex = x g(hx) = (gh)x X is a minimal G-flow ← → X has no proper closed invariant subset.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Topological dynamics

G-flow G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space ex = x g(hx) = (gh)x X is a minimal G-flow ← → X has no proper closed invariant subset. The universal minimal flow M(G) is a minimal flow which has every other minimal flow as its factor.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Topological dynamics

G-flow G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space ex = x g(hx) = (gh)x X is a minimal G-flow ← → X has no proper closed invariant subset. The universal minimal flow M(G) is a minimal flow which has every other minimal flow as its factor. G is extremely amenable ← → its universal minimal flow is a singleton (← → every G-flow has a fixed point).

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Structural Ramsey property

Theorem (Ramsey) For every k ≤ m and r ≥ 2, there exists n such that for every colouring of k-element subsets of n with r-many colours there is a subset X of n of size m such that all k-element subsets of X have the same colour.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Structural Ramsey property

Theorem (Ramsey) For every k ≤ m and r ≥ 2, there exists n such that for every colouring of k-element subsets of n with r-many colours there is a subset X of n of size m such that all k-element subsets of X have the same colour. A class K of finite structures satisfies the Ramsey property if for every A ≤ B ∈ K

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Structural Ramsey property

Theorem (Ramsey) For every k ≤ m and r ≥ 2, there exists n such that for every colouring of k-element subsets of n with r-many colours there is a subset X of n of size m such that all k-element subsets of X have the same colour. A class K of finite structures satisfies the Ramsey property if for every A ≤ B ∈ K and r ≥ 2 a natural number

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Structural Ramsey property

Theorem (Ramsey) For every k ≤ m and r ≥ 2, there exists n such that for every colouring of k-element subsets of n with r-many colours there is a subset X of n of size m such that all k-element subsets of X have the same colour. A class K of finite structures satisfies the Ramsey property if for every A ≤ B ∈ K and r ≥ 2 a natural number there exists C ∈ K such that

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Structural Ramsey property

Theorem (Ramsey) For every k ≤ m and r ≥ 2, there exists n such that for every colouring of k-element subsets of n with r-many colours there is a subset X of n of size m such that all k-element subsets of X have the same colour. A class K of finite structures satisfies the Ramsey property if for every A ≤ B ∈ K and r ≥ 2 a natural number there exists C ∈ K such that for every colouring of copies of A in C by r colours, there is a copy B′ of B in C, such that all copies of A in B′ have the same colour.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Ramsey classes and extremely amenable groups

Ramsey classes finite linear orders (Ramsey) finite linearly ordered graphs (Neˇ setˇ ril and R¨

dl)

finite linearly ordered metric spaces (Neˇ setˇ ril) finite Boolean algebras (Graham and Rothschild)

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Ramsey classes and extremely amenable groups

Ramsey classes finite linear orders (Ramsey) finite linearly ordered graphs (Neˇ setˇ ril and R¨

dl)

finite linearly ordered metric spaces (Neˇ setˇ ril) finite Boolean algebras (Graham and Rothschild) Extremely amenable groups Aut(Q, <) (Pestov) Aut(OR) – OR the random ordered graph (Kechris, Pestov & Todorˇ cevi´ c) Iso(U, d) (Pestov) Homeo(C, C) – (C, C) the Cantor space with a generic maximal chain of closed subsets (KPT; Glasner & Weiss)

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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What allows us to use the Ramsey property?

A - a first order structures

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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What allows us to use the Ramsey property?

A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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What allows us to use the Ramsey property?

A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A. G = Aut(A) with topology of pointwise convergence

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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What allows us to use the Ramsey property?

A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A. G = Aut(A) with topology of pointwise convergence A - a finitely-generated substructure of A

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What allows us to use the Ramsey property?

A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can 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}

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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What allows us to use the Ramsey property?

A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can 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.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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What allows us to use the Ramsey property?

A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can 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-generated substructures of A satisfy the Ramsey property and are rigid.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Universal minimal flows

G = Aut(A) – A ultrahomogeneous

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Universal minimal flows

G = Aut(A) – A ultrahomogeneous G∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of A

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Universal minimal flows

G = Aut(A) – A ultrahomogeneous G∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of A Finite substructures of A∗ satisfy the Ramsey property and are rigid.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Universal minimal flows

G = Aut(A) – A ultrahomogeneous G∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of A Finite substructures of A∗ satisfy the Ramsey property and are rigid. OFTEN M(G) ∼ = G/G∗

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Universal minimal flows

G = Aut(A) – A ultrahomogeneous G∗ = Aut(A∗) – A∗ ultrahomogeneous expansion of A Finite substructures of A∗ satisfy the Ramsey property and are rigid. 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

f

closed subsets of C Glasner and Weiss

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Other settings

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Other settings

ultrahomogeneous Ramsey property

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Other settings

ultrahomogeneous approximately ultrahomogeneous Ramsey property approximate Ramsey property

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Other settings

ultrahomogeneous approximately ultrahomogeneous projectively ultrahomogeneous Ramsey property approximate Ramsey property dual Ramsey property

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Other settings

ultrahomogeneous approximately ultrahomogeneous projectively ultrahomogeneous approximately projectively ultrahomogeneous Ramsey property approximate Ramsey property dual Ramsey property approximate dual Ramsey property

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Other settings

ultrahomogeneous approximately ultrahomogeneous projectively ultrahomogeneous approximately projectively ultrahomogeneous Ramsey property approximate Ramsey property dual Ramsey property approximate dual Ramsey property Structure... ...homogeneous w.r.t. N, R embeddings Gurarij space linear isometric embeddings Lelek fan epimorphisms Poulsen simplex affine epimorphisms

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Gurarij space G

(1) separable Banach space

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Gurarij space G

(1) separable Banach space (2) contains isometric copy of every finite dimensional Banach space

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Gurarij space G

(1) separable Banach space (2) contains isometric copy of every finite dimensional Banach space (3) for every E finite dimensional, i : E ֒ → G isometric embedding and ε > 0 there is a linear isometry f : G

G

i − f ↾ E < ε

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Gurarij space G

(1) separable Banach space (2) contains isometric copy of every finite dimensional Banach space (3) for every E finite dimensional, i : E ֒ → G isometric embedding and ε > 0 there is a linear isometry f : G

G

i − f ↾ E < ε LUSKY Conditions (1),(2),(3) uniquely define G up to a linear isometry.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Gurarij space G

(1) separable Banach space (2) contains isometric copy of every finite dimensional Banach space (3) for every E finite dimensional, i : E ֒ → G isometric embedding and ε > 0 there is a linear isometry f : G

G

i − f ↾ E < ε LUSKY Conditions (1),(2),(3) uniquely define G up to a linear isometry. KUBI´ S-SOLECKI; HENSON Simple proof - metric Fra¨ ıss´ e theory.

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Linear isometries

Isol(G) + point-wise convergence topology = Polish group

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Linear isometries

Isol(G) + point-wise convergence topology = Polish group BASIS

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Linear isometries

Isol(G) + point-wise convergence topology = Polish group BASIS E - finite dimensional subspace of G

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Linear isometries

Isol(G) + point-wise convergence topology = Polish group BASIS E - finite dimensional subspace of G ε > 0

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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}

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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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 YAACOV Isol(G) is a universal Polish group.

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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 YAACOV Isol(G) is a universal Polish group. Katˇ etov construction

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Approximate Ramsey property for finite-dimensional normed spaces

E, F - finite dimensional spaces θ ≥ 1 Embθ(E, F) = {T : E

F : T embedding & T

T−1

≤ θ}

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Approximate Ramsey property for finite-dimensional normed spaces

E, F - finite dimensional spaces θ ≥ 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) = ∅

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 Bartoˇ sov´ a Ramsey theory in topological dynamics

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Pestov’s characterization of extreme amenability

G - topological group

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Pestov’s characterization of extreme amenability

G - topological group f : G

R is finitely oscillation stable if

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Pestov’s characterization of extreme amenability

G - topological group f : G

R is finitely oscillation stable if

∀X ⊂ G finite and ε > 0

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Pestov’s characterization of extreme amenability

G - topological group f : G

R is finitely oscillation stable if

∀X ⊂ G finite and ε > 0 ∃g ∈ G such that osc(f ↾ gX) < ε.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Pestov’s characterization of extreme amenability

G - topological group f : G

R is finitely oscillation stable if

∀X ⊂ G finite and ε > 0 ∃g ∈ G such that osc(f ↾ gX) < ε. Theorem (Pestov) TFAE G is extremely amenable, every f : G

R bounded left-uniformly continuous is

finite oscillation stable.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Pestov’s characterization of extreme amenability

G - topological group f : G

R is finitely oscillation stable if

∀X ⊂ G finite and ε > 0 ∃g ∈ G such that osc(f ↾ gX) < ε. Theorem (Pestov) TFAE G is extremely amenable, every f : G

R bounded left-uniformly continuous is

finite oscillation stable. Theorem (B-LA-M) Isol(G) is extremely amenable.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Urysohn space U

Theorem Finite metric spaces satisfy the approximate Ramsey property.

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Urysohn space U

Theorem Finite metric spaces satisfy the approximate Ramsey property. Corollary (Pestov) Iso(U) is extremely amenable.

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Poulsen simplex P

(1) metrizable

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Poulsen simplex P

(1) metrizable (2) contains every metrizable simplex as its face

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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 finite dimension, there is an affine autohomeomorphism of P mapping E onto F

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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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 finite dimension, there is an affine autohomeomorphism of P mapping E onto F LINDENSTRAUSS-OLSEN-STERNFELD Properties (1),(2) and (3) uniquely determine P up to an affine homeomorphism.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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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 finite dimension, there is an affine autohomeomorphism of P mapping E onto F LINDENSTRAUSS-OLSEN-STERNFELD Properties (1),(2) and (3) uniquely determine P up to an affine homeomorphism. POULSEN The set of extreme points of P is dense in P.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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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 finite dimension, there is an affine autohomeomorphism of P mapping E onto F LINDENSTRAUSS-OLSEN-STERNFELD Properties (1),(2) and (3) uniquely determine P up to an affine homeomorphism. POULSEN The set of extreme points of P is dense in P. FACT T : {0, 1}Z

{0, 1}Z the shift ⇒ T-invariant probability

measures form P

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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A projective characterization of P

Sn := positive part of the unit ball of ln

1 – finite-dimensional

simplex with n + 1 extreme points

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A projective characterization of P

Sn := positive part of the unit ball of ln

1 – finite-dimensional

simplex with n + 1 extreme points Epi(Sn, Sm) := continuous affine surjections Sn

Sm

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A projective characterization of P

Sn := positive part of the unit ball of ln

1 – finite-dimensional

simplex with n + 1 extreme points Epi(Sn, Sm) := continuous affine surjections Sn

Sm

AH(P) := group of affine homeomorphisms of P + compact-open topology

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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A projective characterization of P

Sn := positive part of the unit ball of ln

1 – finite-dimensional

simplex 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) with

d(φ1, φ2 ◦ f) < ε

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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A projective characterization of P

Sn := positive part of the unit ball of ln

1 – finite-dimensional

simplex 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) with

d(φ1, φ2 ◦ f) < ε Theorem (B-LA-M) (U) + (APU) characterize P among non-trivial metrizable simplexes up to affine homeomorphism.

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Approximate Ramsey property for P

Epi0(Sn, Sm) - continuous affine surjections preserving 0

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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 that

for 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

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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 that

for 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 P AHp(P) = {f ∈ AH(P) : f(p) = p}

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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 that

for 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 P AHp(P) = {f ∈ AH(P) : f(p) = p} Theorem (B-LA-M) AHp(P) is extremely amenable.

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Universal minimal flow of AH(P)

Theorem (B-LA-M) M(AH(P)) ∼ =

AH(P)/AHp(P) ∼

= P

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Hilbert cube Q = [−1, 1]N

PROBLEM What is the universal minimal flow of Homeo(Q)?

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Hilbert cube Q = [−1, 1]N

PROBLEM What is the universal minimal flow of Homeo(Q)? Q is homeomorphic to P.

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Hilbert cube Q = [−1, 1]N

PROBLEM What 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 minimal flow being the natural action on Q.

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Hilbert cube Q = [−1, 1]N

PROBLEM What 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 minimal flow being the natural action on Q. Q with its natural convex structure.

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Hilbert cube Q = [−1, 1]N

PROBLEM What 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 minimal flow 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∞.

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Hilbert cube Q = [−1, 1]N

PROBLEM What 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 minimal flow 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 Bartoˇ sov´ a Ramsey theory in topological dynamics

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Lelek fan L

= unique non-trivial subcontinuum of the Cantor fan with a dense set of endpoints (Bula-Oversteegen, Charatonik) continuum = connected compact metric Hausdorff space

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Lelek fan L

= unique non-trivial subcontinuum of the Cantor fan with a dense set of endpoints (Bula-Oversteegen, Charatonik) continuum = connected compact metric Hausdorff space fan.jpg

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Pre-Lelek fan

(L, RL

s ) - compact, 0-dim, RL s ⊂ L2 closed with one or two

element equivalence classes

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Pre-Lelek fan

(L, RL

s ) - compact, 0-dim, RL s ⊂ L2 closed with one or two

element equivalence classes L/RL

s ∼

= L

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Pre-Lelek fan

(L, RL

s ) - compact, 0-dim, RL s ⊂ L2 closed with one or two

element equivalence classes L/RL

s ∼

= L F = {finite fans} + surjective homomorphisms

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Pre-Lelek fan

(L, RL

s ) - compact, 0-dim, RL s ⊂ L2 closed with one or two

element equivalence classes L/RL

s ∼

= L F = {finite fans} + surjective homomorphisms (U) T ∈ F ∃φ : (L, RL)

T - continuous surjective

homomorphism (R) X finite, f : L

X continuous ∃T ∈ F, φ : L T and

g : T

X such that f = g ◦ φ

(PU) T ∈ F, φ1, φ2 : L

T ∃g : L L automorphism with

φ1 = φ2 ◦ g

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Aut

Homeo

Aut(L, RL

s ) and Homeo(L) + the compact-open topology

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Aut

Homeo

Aut(L, RL

s ) and Homeo(L) + the compact-open topology

π : L

L/RL

s ∼

= L

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Aut

Homeo

Aut(L, RL

s ) and Homeo(L) + the compact-open topology

π : L

L/RL

s ∼

= L induces a continuous embedding Aut(L, RL

s ) ֒

→ Homeo(L)

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Aut

Homeo

Aut(L, RL

s ) and Homeo(L) + the compact-open topology

π : L

L/RL

s ∼

= L induces a continuous embedding Aut(L, RL

s ) ֒

→ Homeo(L) with a dense image

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

SLIDE 91

Aut

Homeo

Aut(L, RL

s ) and Homeo(L) + the compact-open topology

π : L

L/RL

s ∼

= L induces a continuous embedding Aut(L, RL

s ) ֒

→ Homeo(L) with a dense image h → h∗ π ◦ h = h∗ ◦ π.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

SLIDE 92

Ramsey property for L

F< - finite fans with a linear order extending the natural order {C

A} := all epimorphisms from C onto A

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

SLIDE 93

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.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

SLIDE 94

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 every colouring c : {C

A} {1, 2, . . . , r}

there exists f : C

B such that {B A} ◦ f is

monochromatic.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

SLIDE 95

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 every colouring c : {C

A} {1, 2, . . . , r}

there exists f : C

B such that {B A} ◦ f is

monochromatic. Theorem (B-K) Let L< be the limit of F<. Then Aut(L<) is extremely amenable.

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Universal minimal flow of Homeo(L)

Theorem (B-K) M(Aut(L)) ∼ =

Aut(L)/Aut(L<)

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

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Universal minimal flow of Homeo(L)

Theorem (B-K) M(Aut(L)) ∼ =

Aut(L)/Aut(L<)

M(Homeo(L)) ∼ =

Homeo(L)/Homeo(L<)

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

SLIDE 98

A question

Is there a non-trivial simplex with extremely amenable group of affine homeomorphisms?

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics

SLIDE 99

THANK YOU HAPPY FOOLS’ DAY!

Dana Bartoˇ sov´ a Ramsey theory in topological dynamics