On a problem of Ellis and Pestov’s Conjecture a 1 Andy Zucker 2 Dana Bartoˇ sov´ 1 Universidade de S˜ ao Paulo 2 Carnegie Mellon University Toposym 2016 Praha July 27, 2016 The first author is supported by the grant FAPESP 2013/14458-9. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Flow Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Flow � X – a continuous action G × X Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Flow � X – a continuous action G × X ↑ ↑ topological compact group Hausdorff space Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Flow � X – a continuous action G × X ↑ ↑ topological compact group Hausdorff space g ( hx ) = ( gh ) x ex = x Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Flow � X – a continuous action G × X ↑ ↑ topological compact group Hausdorff space g ( hx ) = ( gh ) x ex = x � X – a homeomorphism ( g, · ) : X Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Flow � X – a continuous action G × X ↑ ↑ topological compact group Hausdorff space g ( hx ) = ( gh ) x ex = x � X – a homeomorphism ( g, · ) : X � Homeo(X) – a continuous homomorphism G Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Flow � X – a continuous action G × X ↑ ↑ topological compact group Hausdorff space g ( hx ) = ( gh ) x ex = x � X – a homeomorphism ( g, · ) : X � Homeo(X) – a continuous homomorphism G We call X a G -flow. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Homomorphism X, Y – G -flows Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Homomorphism X, Y – G -flows � Y is a G -homomorphism if A continuous map φ : X φ ( gx ) = gφ ( x ) . Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
� � � Homomorphism X, Y – G -flows � Y is a G -homomorphism if A continuous map φ : X φ ( gx ) = gφ ( x ) . � X G × X G × X X id × φ id × φ G × Y G × Y Y Y Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
� � � Homomorphism X, Y – G -flows � Y is a G -homomorphism if A continuous map φ : X φ ( gx ) = gφ ( x ) . � X G × X G × X X id × φ id × φ G × Y G × Y Y Y Y is a factor of X if there is a surjective homomorphism � Y. X Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Ellis semigroup � X – G-flow π : G × X Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Ellis semigroup � X – G-flow π : G × X � X X , π : G where X X is considered with the product topology. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Ellis semigroup � X – G-flow π : G × X � X X , π : G where X X is considered with the product topology. E ( X ) = π ( G ) is the Ellis semigroup of X . Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Ellis semigroup � X – G-flow π : G × X � X X , π : G where X X is considered with the product topology. E ( X ) = π ( G ) is the Ellis semigroup of X . E ( X ) is a sort of “semigroup compactification” of G Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Ellis semigroup � X – G-flow π : G × X � X X , π : G where X X is considered with the product topology. E ( X ) = π ( G ) is the Ellis semigroup of X . E ( X ) is a sort of “semigroup compactification” of G π extends to a continuous semigroup action � X E ( X ) × X Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Ellis semigroup � X – G-flow π : G × X � X X , π : G where X X is considered with the product topology. E ( X ) = π ( G ) is the Ellis semigroup of X . E ( X ) is a sort of “semigroup compactification” of G π extends to a continuous semigroup action � X E ( X ) × X Gx = E ( X ) x for every x ∈ X. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Ellis semigroup � X – G-flow π : G × X � X X , π : G where X X is considered with the product topology. E ( X ) = π ( G ) is the Ellis semigroup of X . E ( X ) is a sort of “semigroup compactification” of G π extends to a continuous semigroup action � X E ( X ) × X Gx = E ( X ) x for every x ∈ X. ( x, y ) ∈ X 2 is proximal iff there is p ∈ E ( X ) such that px = py. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
(Universal) minimal flow Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
(Universal) minimal flow X is minimal if it has no proper closed G -invariant subset. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
(Universal) minimal flow X is minimal if it has no proper closed G -invariant subset. M ( G ) is the universal minimal G -flow if every minimal G -flow is its quotient. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
� � � (Universal) minimal flow X is minimal if it has no proper closed G -invariant subset. M ( G ) is the universal minimal G -flow if every minimal G -flow is its quotient. � M ( G ) G × M ( G ) G × M ( G ) M ( G ) id × q id × q G × M G × M M M Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
� � � (Universal) minimal flow X is minimal if it has no proper closed G -invariant subset. M ( G ) is the universal minimal G -flow if every minimal G -flow is its quotient. � M ( G ) G × M ( G ) G × M ( G ) M ( G ) id × q id × q G × M G × M M M Theorem (Ellis) The universal minimal flow exists and it is unique up to isomorphism. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Greatest ambit Ambit is a pointed G -flow ( X, x 0 ) such that Gx 0 is dense in X. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Greatest ambit Ambit is a pointed G -flow ( X, x 0 ) such that Gx 0 is dense in X. A homomorphism of G -ambits ( X, x 0 ) and ( Y, y 0 ) is a � Y such that φ ( x 0 ) = y 0 . homomorphism φ : X Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Greatest ambit Ambit is a pointed G -flow ( X, x 0 ) such that Gx 0 is dense in X. A homomorphism of G -ambits ( X, x 0 ) and ( Y, y 0 ) is a � Y such that φ ( x 0 ) = y 0 . homomorphism φ : X The greatest ambit ( S ( G ) , e ) is an ambit that has every ambit as its quotient. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Greatest ambit Ambit is a pointed G -flow ( X, x 0 ) such that Gx 0 is dense in X. A homomorphism of G -ambits ( X, x 0 ) and ( Y, y 0 ) is a � Y such that φ ( x 0 ) = y 0 . homomorphism φ : X The greatest ambit ( S ( G ) , e ) is an ambit that has every ambit as its quotient. � ( βG, e ) is the greatest ambit of G. G - discrete Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Greatest ambit Ambit is a pointed G -flow ( X, x 0 ) such that Gx 0 is dense in X. A homomorphism of G -ambits ( X, x 0 ) and ( Y, y 0 ) is a � Y such that φ ( x 0 ) = y 0 . homomorphism φ : X The greatest ambit ( S ( G ) , e ) is an ambit that has every ambit as its quotient. � ( βG, e ) is the greatest ambit of G. G - discrete S ( G ) is the minimal compactification of G to which all uniformly continuous maps into [0 , 1] extend. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Greatest ambit Ambit is a pointed G -flow ( X, x 0 ) such that Gx 0 is dense in X. A homomorphism of G -ambits ( X, x 0 ) and ( Y, y 0 ) is a � Y such that φ ( x 0 ) = y 0 . homomorphism φ : X The greatest ambit ( S ( G ) , e ) is an ambit that has every ambit as its quotient. � ( βG, e ) is the greatest ambit of G. G - discrete S ( G ) is the minimal compactification of G to which all uniformly continuous maps into [0 , 1] extend. S ( G ) - compact right-topological semigroup. Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Problem of Ellis � E ( M ( G ) , e ) an Is the ambit homomorphism ( S ( G ) , e ) isomorphism? Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Problem of Ellis � E ( M ( G ) , e ) an Is the ambit homomorphism ( S ( G ) , e ) isomorphism? GLASNER NO for G = Z . Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Problem of Ellis � E ( M ( G ) , e ) an Is the ambit homomorphism ( S ( G ) , e ) isomorphism? GLASNER NO for G = Z . PESTOV NO for Homeo + ( S 1 ) . Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
Problem of Ellis � E ( M ( G ) , e ) an Is the ambit homomorphism ( S ( G ) , e ) isomorphism? GLASNER NO for G = Z . PESTOV NO for Homeo + ( S 1 ) . NO for extremely amenable groups ≡ groups with trivial universal minimal flows (e.g., Aut( Q , < ) , U ( l 2 ), Iso( U , d)). Dana Bartoˇ sov´ a On a problem of Ellis and Pestov’s Conjecture
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