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 (Fn)∞

n=0 of finite subsets of G. Let T = (Tg)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 (Bn)∞

n=0 of finite

measure in X and an increasing sequence of positive integers (ln)∞

n=1 such that for each n ≥ 0, the subsets TgBn, g ∈ Fln, are

mutually disjoint and for each subset B ⊂ X of finite measure, min

F⊂Fln

µ(B△

• g∈F

TgBn) → 0 then T is called of rank one along (Fn)∞

n=0.

Alexandre I. Danilenko 2/17

Rank one actions

Let G be a discrete infinite countable group. Fix a sequence of finite subsets (Fn)∞

n=0 of finite subsets of G. Let T = (Tg)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 (Bn)∞

n=0 of finite

measure in X and an increasing sequence of positive integers (ln)∞

n=1 such that for each n ≥ 0, the subsets TgBn, g ∈ Fln, are

mutually disjoint and for each subset B ⊂ X of finite measure, min

F⊂Fln

µ(B△

• g∈F

TgBn) → 0 then T is called of rank one along (Fn)∞

n=0.

Case G = Z: rank one, funny rank one.

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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]

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(C, F)-construction

Let (Fn)n≥0 and (Cn)n≥1 be two sequences of finite subsets in G such that for each n > 0, (I) 1 ∈ F0 ∩ Cn, #Cn > 1, (II) FnCn+1 ⊂ Fn+1, (III) Fnc ∩ Fnc′ = ∅ if c, c′ ∈ Cn+1 and c = c′.

Alexandre I. Danilenko 4/17

(C, F)-construction

Let (Fn)n≥0 and (Cn)n≥1 be two sequences of finite subsets in G such that for each n > 0, (I) 1 ∈ F0 ∩ Cn, #Cn > 1, (II) FnCn+1 ⊂ Fn+1, (III) Fnc ∩ Fnc′ = ∅ if c, c′ ∈ Cn+1 and c = c′. Then for each n ≥ 0, we can define a “tower” Xn as the union of disjoint “n-levels” [f ]n, f ∈ Fn, which are Cantor sets, by setting [f ]n := {f } × Cn+1 × Cn+2 × · · · . The n-levels are “congruent”, i.e. they differ only by the first

• coordinate. Partition [f ]n into clopen subsets, sublevels,

{f } × {c} × Cn+2 × · · · , c ∈ Cn+1, which we identify with [fc]n+1 via (f , c, cn+2, cn+3, . . . ) → (fc, cn+2, cn+3, . . . ) Thus we have a partition [f ]n =

c∈Cn+1[fc]n+1.

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(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 Xn.

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 Xn. 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 Xn. 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(X0) = 1. Moreover, µ([f ]n) = 1/#C1 · · · #Cn.

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(C, F)-actions on Polish spaces

For g ∈ G, we define Tg[f ]n := [gf ]n if f , gf ∈ Fn.

Alexandre I. Danilenko 6/17

(C, F)-actions on Polish spaces

For g ∈ G, we define Tg[f ]n := [gf ]n if f , gf ∈ Fn. Then there is a Gδ subset X ′ ⊂ X such that

• Tg extends to a homeomorphism of X ′ for each g ∈ G,
• T := (Tg)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,

lim

m→∞

#((gFnCn+1Cn+2 · · · Cm) ∩ Fm) #Fn#Cn+1 · · · #Cm = 1.

• if µ is finite then X ′ is µ-conull iff (Fn)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

(Fn)∞

n=1.

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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 gFnCn+1Cn+2Cm ⊂ Fm. 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

Theorem 1. Let T = (Tg)g∈G and T ′ = (T ′

g)g∈G be two

(C, F)-actions of G associated with some sequences (Cn, Fn−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 = l0 < l′

1 < l1 < l′ 2 < l2 < · · · and subsets An ⊂ F ′ ln, Bn ⊂ Fln,

such that AnBn = Cln−1+1 · · · Cln, BnAn+1 = C ′

l′

n+1 · · · C ′

l′

n+1,

F ′

l′

nBn ⊂ Fln, FlnAn+1 ⊂ F ′

ln+1,

(F ′

ln)−1F ′ ln ∩ BnB−1 n

= F −1

ln Fln ∩ An+1A−1 n+1 = {1}

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 (Cn, Fn−1)n≥1 and (C ′

n, F ′ n−1)n≥1 respectively and the two sequences satisfy (I)–(IV).

Suppose that Cn ∪ C ′

n ⊂ Z+. Then T and T ′ are topologically

isomorphic if and only if there is r > 0 and a subset R ⊂ F ′

r ∩ Z+

such that

i>0 Ci = R + i>r C ′ 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 (Cn, Fn−1)n≥1 and (C ′

n, F ′ n−1)n≥1 respectively and the two sequences satisfy (I)–(IV).

Suppose that Cn ∪ C ′

n ⊂ Z+. Then T and T ′ are topologically

isomorphic if and only if there is r > 0 and a subset R ⊂ F ′

r ∩ Z+

such that

i>0 Ci = R + i>r C ′ 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 = (Tg)g∈G of G on a locally compact Cantor space X, we let Ctop(T) := {θ ∈ Homeo(X) | θTg = Tgθ 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 = (Tg)g∈G of G on a locally compact Cantor space X, we let Ctop(T) := {θ ∈ Homeo(X) | θTg = Tgθ 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 (Cn, Fn−1)n≥1 satisfying (I)–(IV). If Cn ⊂ G+ for each n ≥ 1 then Ctop(T) = {Tg | g ∈ G}.

Alexandre I. Danilenko 10/17

Applications: topological centralizers and inverse actions

Definition 2. Given a topological action T = (Tg)g∈G of G on a locally compact Cantor space X, we let Ctop(T) := {θ ∈ Homeo(X) | θTg = Tgθ 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 (Cn, Fn−1)n≥1 satisfying (I)–(IV). If Cn ⊂ G+ for each n ≥ 1 then Ctop(T) = {Tg | g ∈ G}. Theorem 4. Let (G, G+) = (Z, Z+), Cn ⊂ G+ and Fn = (−αn, βn) ∩ Z for some integers αn, βn > 0 for each n ≥ 1. Then T is topologically isomorphic to T −1 if and only if Cn = −Cn + max Cn eventually.

Alexandre I. Danilenko 10/17

Standard definitions of rank-one transformations

A transformation T is associated with a sequence of integers rn > 1 and a sequence of maps σn : {1, . . . , rn} → Z+. Equivalently, let h0 := 1, hn+1 := rnhn + rn

i=1 σn(i). We set

Fn := {0, 1, . . . , hn − 1} and Cn+1 := {ihn + i

j=1 σn(j) | i = 0, . . . , rn − 1}. Then T is

associated with (Cn, Fn−1)∞

n=1.

Examples: Chacon 2-cuts map: rn = 2, σn(0) = 0, σn(1) = 1; Chacon 3-cuts map: rn = 3, σn(0) = 0, σn(1) = 1, σn(2) = 0. Definition 3. T is called adapted if σn(rn) = 0 for all n. Definition 4. We say that T has bounded parameters if there is R > 0 such that rn < R and max1≤i≤rn σn(i) < R.

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Rigidity and ergodicity of rank-one maps

Theorem 5. Let T be a rank-one map with bounded parameters. Then T is rigid if and only if for each N > 0, there are integers n, m such that m > n + N > n > N and the set Cn + · · · + Cm is an arithmetic progression. Theorem 6. Let T be a rank-one map. Then (i) If T d is ergodic then for each divisor p of d, there are infinitely many n > 0 such that some c ∈ Cn is not divisible by p. (ii) If the sequence (rn)∞

n=1 is bounded and for each divisor p of a

positive integer d, there are infinitely many n such that p does not divide some c ∈ Cn then T d is ergodic.

Alexandre I. Danilenko 12/17

Rigidity and ergodicity of rank-one maps

Theorem 5. Let T be a rank-one map with bounded parameters. Then T is rigid if and only if for each N > 0, there are integers n, m such that m > n + N > n > N and the set Cn + · · · + Cm is an arithmetic progression. Theorem 6. Let T be a rank-one map. Then (i) If T d is ergodic then for each divisor p of d, there are infinitely many n > 0 such that some c ∈ Cn is not divisible by p. (ii) If the sequence (rn)∞

n=1 is bounded and for each divisor p of a

positive integer d, there are infinitely many n such that p does not divide some c ∈ Cn then T d is ergodic. (iii) If the sequence (rn)∞

n=1 is bounded then T is totally ergodic if

and only if for each d > 1, there are infinitely many n > 0 such that some element c of Cn is not divisible by d.

Alexandre I. Danilenko 12/17

Rigidity and ergodicity of rank-one maps

Theorem 5. Let T be a rank-one map with bounded parameters. Then T is rigid if and only if for each N > 0, there are integers n, m such that m > n + N > n > N and the set Cn + · · · + Cm is an arithmetic progression. Theorem 6. Let T be a rank-one map. Then (i) If T d is ergodic then for each divisor p of d, there are infinitely many n > 0 such that some c ∈ Cn is not divisible by p. (ii) If the sequence (rn)∞

n=1 is bounded and for each divisor p of a

positive integer d, there are infinitely many n such that p does not divide some c ∈ Cn then T d is ergodic. (iii) If the sequence (rn)∞

n=1 is bounded then T is totally ergodic if

and only if for each d > 1, there are infinitely many n > 0 such that some element c of Cn is not divisible by d. Gao, Hill, Bounded rank-one transformations, J. d’Anal. Math. 129 (2016), 341–365.

Alexandre I. Danilenko 12/17

MSJ and disjointness

3-cuts Chacon has MSJ (del Junco, Rahe, Swanson, 1980) Theorem 7. (Ryzhikov) Let T be a rank-one transformation with bounded parameters. Suppose that T is not rigid and that T is totally ergodic. Then T has MSJ. (Hence T n and T m are Furstenberg disjoint if n > m > 0.)

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MSJ and disjointness

3-cuts Chacon has MSJ (del Junco, Rahe, Swanson, 1980) Theorem 7. (Ryzhikov) Let T be a rank-one transformation with bounded parameters. Suppose that T is not rigid and that T is totally ergodic. Then T has MSJ. (Hence T n and T m are Furstenberg disjoint if n > m > 0.) Ryzhikov, Minimal self-joinings, bounded constructions, and weak closure of ergodic actions, preprint, arXiv:1212.2602v2. Gao, Hill, Disjointness between bounded rank-one transformations, arXiv:1601.04119v1

Alexandre I. Danilenko 13/17

Disjointness

Theorem 8. Let T and T ′ be two rank-one transformations with bounded parameters and hn = h′

n (and hence rn = r′ n). Let either

T or T ′ be non-rigid. (i) Then T and T ′ are isomorphic if and only if sn = s′

n eventually.

(ii) If sn = s′

n for infinitely many n and for each n > 0 and either

Tn or T ′

n is ergodic then T and T ′ are Furstenberg disjoint.

Alexandre I. Danilenko 14/17

Disjointness

Theorem 8. Let T and T ′ be two rank-one transformations with bounded parameters and hn = h′

n (and hence rn = r′ n). Let either

T or T ′ be non-rigid. (i) Then T and T ′ are isomorphic if and only if sn = s′

n eventually.

(ii) If sn = s′

n for infinitely many n and for each n > 0 and either

Tn or T ′

n is ergodic then T and T ′ are Furstenberg disjoint.

Theorem 9. Let T be a rank-one transformation with bounded

• parameters. Let T be non-rigid.

(i) Then T and T −1 are isomorphic if and only if −Cn + max Cn = Cn eventually. (ii) If T is totally ergodic and T is not isomorphic to T −1 then T and T −1 are Furstenberg disjoint. Gao and Hill proved Theorems 8, 9 under assumption that T and T ′ are adapted.

Alexandre I. Danilenko 14/17

Light mixing and bounded rank-one constructions

A transformation S of a probability space (Y , ν) is called lightly mixing if for all A, B ⊂ Y of positive measure, lim inf

n→∞ µ(T nA ∩ B) > 0

Chacon 2-cuts is lightly mixing (Friedman, King). Theorem 10. Let T be an adapted rank-one transformation ∼ (rn, σn)n≥1. If there is K > 0 such that supn≥1 max1≤i≤rn σn(i) ≤ K then there is a polynomial P(Z) = a0 + a1Z + · · · + aKZ K with real nonnegative coefficients such that k

i=0 ai = 1 and P(T) belongs to the WCP(T). (in the

weak operator topology). It follows that T is not lightly mixing. Hence the Chacon 2-cuts map is not isomorphic to an adapted rank-one transformation with bounded parameters.

Alexandre I. Danilenko 15/17

Trichotomy theorem

El Abdalaoui, Lema´ nczyk, de la Rue, On spectral disjointness of powers for rank-one transformations and Mobius orthogonality, J.

• Funct. Anal., 2014

Let T be a rank-one transformation with bounded parameters. Then either... or... (i) T is weakly mixing; (ii) the group of eigenvalues ΛT of T is finite and nontrivial; (iii) T is an odometer.

Alexandre I. Danilenko 16/17

Quadchotomy theorem

The following refines the trichotomy theorem. Theorem 11. Let T be a rank-one transformation with bounded parameters (rn, σn)n≥1 and let K := supn≥1 max1≤j≤rn σn(j). Then

• ne of the following four properties holds:

(i) T has MSJ. In particular, T is weakly mixing and C(T) = {T n | n ∈ Z}. (ii) T is non-rigid, the group ΛT ⊂ T of eigenvalues of T is nontrivial but finite and the order of each λ ∈ ΛT does not exceed

• K. For each ρ ∈ Je

2(T) which is neither a graph-joining nor µ × µ,

there is λ ∈ ΛT \ {1} and 0 < n ≤ K such that λn = 1 and n−1 n−1

i=0 ρ ◦ (I × T i) = µ × µ.

(iii) T is rigid, ΛT is finite and the order of each λ ∈ ΛT does not exceed K. (iv) T is an odometer of bounded type.

Alexandre I. Danilenko 17/17