Rank-one actions and constructions with bounded parameters - PowerPoint PPT Presentation

Rank-one actions and constructions with bounded parameters Alexandre I. Danilenko Insitute for Low Temperature Physics and Engineering, NAS of Ukraine June 10, 2018 Alexandre I. Danilenko 1/17

Rank one actions Let G be a discrete infinite countable group. Fix a sequence of finite subsets ( F n ) ∞ n =0 of finite subsets of G . Let T = ( T g ) g ∈ G be a measure preserving action of G on a standard σ -finite measure space ( X , B , µ ). Definition 1. If there is a sequence of subsets ( B n ) ∞ n =0 of finite measure in X and an increasing sequence of positive integers ( l n ) ∞ n =1 such that for each n ≥ 0, the subsets T g B n , g ∈ F l n , are mutually disjoint and for each subset B ⊂ X of finite measure, � min µ ( B △ T g B n ) → 0 F ⊂ F ln g ∈ F then T is called of rank one along ( F n ) ∞ n =0 . Alexandre I. Danilenko 2/17

Rank one actions Let G be a discrete infinite countable group. Fix a sequence of finite subsets ( F n ) ∞ n =0 of finite subsets of G . Let T = ( T g ) g ∈ G be a measure preserving action of G on a standard σ -finite measure space ( X , B , µ ). Definition 1. If there is a sequence of subsets ( B n ) ∞ n =0 of finite measure in X and an increasing sequence of positive integers ( l n ) ∞ n =1 such that for each n ≥ 0, the subsets T g B n , g ∈ F l n , are mutually disjoint and for each subset B ⊂ X of finite measure, � min µ ( B △ T g B n ) → 0 F ⊂ F ln g ∈ F then T is called of rank one along ( F n ) ∞ n =0 . Case G = Z : rank one, funny rank one. Alexandre I. Danilenko 2/17

Producing rank-one actions It appeared (measure-theoretical version) in [del Junco, 1998] for G amenable, Alexandre I. Danilenko 3/17

Producing rank-one actions It appeared (measure-theoretical version) in [del Junco, 1998] for G amenable, (topological version) in [D, 2001] for G Abelian, Alexandre I. Danilenko 3/17

Producing rank-one actions It appeared (measure-theoretical version) in [del Junco, 1998] for G amenable, (topological version) in [D, 2001] for G Abelian, (the most general version, including top. and measure-theoretical, G is arbitary) in [D, 2015] Alexandre I. Danilenko 3/17

( C , F )-construction Let ( F n ) n ≥ 0 and ( C n ) n ≥ 1 be two sequences of finite subsets in G such that for each n > 0, (I) 1 ∈ F 0 ∩ C n , # C n > 1, (II) F n C n +1 ⊂ F n +1 , (III) F n c ∩ F n c ′ = ∅ if c , c ′ ∈ C n +1 and c � = c ′ . Alexandre I. Danilenko 4/17

( C , F )-construction Let ( F n ) n ≥ 0 and ( C n ) n ≥ 1 be two sequences of finite subsets in G such that for each n > 0, (I) 1 ∈ F 0 ∩ C n , # C n > 1, (II) F n C n +1 ⊂ F n +1 , (III) F n c ∩ F n c ′ = ∅ if c , c ′ ∈ C n +1 and c � = c ′ . Then for each n ≥ 0, we can define a “tower” X n as the union of disjoint “ n -levels” [ f ] n , f ∈ F n , which are Cantor sets, by setting [ f ] n := { f } × C n +1 × C n +2 × · · · . The n -levels are “congruent”, i.e. they differ only by the first coordinate. Partition [ f ] n into clopen subsets, sublevels, { f } × { c } × C n +2 × · · · , c ∈ C n +1 , which we identify with [ fc ] n +1 via ( f , c , c n +2 , c n +3 , . . . ) �→ ( fc , c n +2 , c n +3 , . . . ) Thus we have a partition [ f ] n = � c ∈ C n +1 [ fc ] n +1 . Alexandre I. Danilenko 4/17

( C , F )-construction: the tail equivalence relation Let X be the union of all n -levels, n ∈ N . It is a locally compact Cantor space. Define an equivalence relation on X : two points are tail equivalent if they belong eventually (in n ) to the same level of X n . Alexandre I. Danilenko 5/17

( C , F )-construction: the tail equivalence relation Let X be the union of all n -levels, n ∈ N . It is a locally compact Cantor space. Define an equivalence relation on X : two points are tail equivalent if they belong eventually (in n ) to the same level of X n . The tail equivalence relation is minimal. Alexandre I. Danilenko 5/17

( C , F )-construction: the tail equivalence relation Let X be the union of all n -levels, n ∈ N . It is a locally compact Cantor space. Define an equivalence relation on X : two points are tail equivalent if they belong eventually (in n ) to the same level of X n . The tail equivalence relation is minimal. The tail equivalence relation is Radon uniquely ergodic, i.e. there is a unique Radon (hence σ -finite) measure µ on X such that µ ( X 0 ) = 1. Moreover, µ ([ f ] n ) = 1 / # C 1 · · · # C n . Alexandre I. Danilenko 5/17

( C , F )-actions on Polish spaces For g ∈ G , we define T g [ f ] n := [ gf ] n if f , gf ∈ F n . Alexandre I. Danilenko 6/17

( C , F )-actions on Polish spaces For g ∈ G , we define T g [ f ] n := [ gf ] n if f , gf ∈ F n . Then there is a G δ subset X ′ ⊂ X such that - T g extends to a homeomorphism of X ′ for each g ∈ G , - T := ( T g ) g ∈ G is a continuous action of G on X ′ , - T is free and minimal on X ′ , - the T -orbit equivalence relation is the tail equivalence relation restricted to X ′ . - X ′ is either µ -null or µ -conull, - X ′ is µ -conull if and only if for each g ∈ G and n > 0, #(( gF n C n +1 C n +2 · · · C m ) ∩ F m ) lim = 1 . # F n # C n +1 · · · # C m m →∞ - if µ is finite then X ′ is µ -conull iff ( F n ) n ≥ 0 is a Følner sequence and hence G is amenable, - if X ′ is µ -conull then T is hyperfinite and of rank one along ( F n ) ∞ n =1 . Alexandre I. Danilenko 6/17

Topological ( C , F )-actions on locally compact Cantor spaces X ′ = X if and only if (IV) For each g ∈ G and n > 0, there is m > n such that gF n C n +1 C n +2 C m ⊂ F m . In this case T is minimal, amenable, Radon uniquely ergodic ( T is defined on a locally compact Cantor space), µ is the invariant measure. Alexandre I. Danilenko 7/17

Classification of topological ( C , F )-actions Let T = ( T g ) g ∈ G and T ′ = ( T ′ g ) g ∈ G be two Theorem 1. ( C , F )-actions of G associated with some sequences ( C n , F n − 1 ) n ≥ 1 and ( C ′ n , F ′ n − 1 ) n ≥ 1 respectively and the two sequences satisfy (I)–(IV). Then T and T ′ are topologically isomorphic if and only if there is an increasing sequence of integers 0 = l 0 < l ′ 1 < l 1 < l ′ 2 < l 2 < · · · and subsets A n ⊂ F ′ l n , B n ⊂ F l n , such that A n B n = C l n − 1 +1 · · · C l n , B n A n +1 = C ′ n +1 · · · C ′ n +1 , l ′ l ′ F ′ n B n ⊂ F l n , F l n A n +1 ⊂ F ′ l n +1 , l ′ l n ) − 1 F ′ l n ∩ B n B − 1 = F − 1 l n F l n ∩ A n +1 A − 1 ( F ′ n +1 = { 1 } n Alexandre I. Danilenko 8/17

Classification of topological ( C , F )-actions of Z Theorem 2. Let G = Z . Let T and T ′ be two ( C , F )-actions of G associated with some sequences ( C n , F n − 1 ) n ≥ 1 and ( C ′ n , F ′ n − 1 ) n ≥ 1 respectively and the two sequences satisfy (I)–(IV). n ⊂ Z + . Then T and T ′ are topologically Suppose that C n ∪ C ′ isomorphic if and only if there is r > 0 and a subset R ⊂ F ′ r ∩ Z + i > r C ′ such that � i > 0 C i = R + � i . Alexandre I. Danilenko 9/17

Classification of topological ( C , F )-actions of Z Theorem 2. Let G = Z . Let T and T ′ be two ( C , F )-actions of G associated with some sequences ( C n , F n − 1 ) n ≥ 1 and ( C ′ n , F ′ n − 1 ) n ≥ 1 respectively and the two sequences satisfy (I)–(IV). n ⊂ Z + . Then T and T ′ are topologically Suppose that C n ∪ C ′ isomorphic if and only if there is r > 0 and a subset R ⊂ F ′ r ∩ Z + i > r C ′ such that � i > 0 C i = R + � i . Gao and Hill, Topological isomorphism for rank-1 systems J. d’Anal. Math., 128 (2016), 1-49. Alexandre I. Danilenko 9/17

Applications: topological centralizers and inverse actions Definition 2. Given a topological action T = ( T g ) g ∈ G of G on a locally compact Cantor space X , we let C top ( T ) := { θ ∈ Homeo( X ) | θ T g = T g θ for each g ∈ G } and call this set the topological centralizer of T . Alexandre I. Danilenko 10/17

Applications: topological centralizers and inverse actions Definition 2. Given a topological action T = ( T g ) g ∈ G of G on a locally compact Cantor space X , we let C top ( T ) := { θ ∈ Homeo( X ) | θ T g = T g θ for each g ∈ G } and call this set the topological centralizer of T . Theorem 3. Let ( G , G + ) be a linearly ordered discrete countable Abelian group. Let T be a ( C , F )-action of G associated with a sequence ( C n , F n − 1 ) n ≥ 1 satisfying (I)–(IV). If C n ⊂ G + for each n ≥ 1 then C top ( T ) = { T g | g ∈ G } . Alexandre I. Danilenko 10/17

Applications: topological centralizers and inverse actions Definition 2. Given a topological action T = ( T g ) g ∈ G of G on a locally compact Cantor space X , we let C top ( T ) := { θ ∈ Homeo( X ) | θ T g = T g θ for each g ∈ G } and call this set the topological centralizer of T . Theorem 3. Let ( G , G + ) be a linearly ordered discrete countable Abelian group. Let T be a ( C , F )-action of G associated with a sequence ( C n , F n − 1 ) n ≥ 1 satisfying (I)–(IV). If C n ⊂ G + for each n ≥ 1 then C top ( T ) = { T g | g ∈ G } . Theorem 4. Let ( G , G + ) = ( Z , Z + ), C n ⊂ G + and F n = ( − α n , β n ) ∩ Z for some integers α n , β n > 0 for each n ≥ 1. Then T is topologically isomorphic to T − 1 if and only if C n = − C n + max C n eventually. Alexandre I. Danilenko 10/17