Quasi-uniform convergence in compact dynamical systems

Downarowicz, Tomasz; Iwanik, Anzelm

doi:10.4064/sm-89-1-11-25

Cited by 18 publications

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“…is the symmetric difference of the left-hand sides of (4)- (8) and f (∅) ∈ K, this proves our claim.…”

Section: ] Implies That There Is a Borel Function C Such That C(a) ∈ supporting

confidence: 68%

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Luzin gaps

Farah

2004

Trans. Amer. Math. Soc.

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Abstract. We isolate a class of F σδ ideals on N that includes all analytic P-ideals and all Fσ ideals, and introduce 'Luzin gaps' in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author's rigidity conjecture for quotients P(N)/I. We also prove that the ideals NWD(Q) and NULL(Q) have the Radon-Nikodým property, and (using OCA∞) a uniformization result for K-coherent families of continuous partial functions.One of the most fascinating facts about the Boolean algebra P(N)/ Fin was discovered by Hausdorff in 1908. In [20], he constructed two families A and B of sets of integers such that (a) A ∩ B is finite for all A ∈ A and all B ∈ B, (b) for every C ⊆ N either A \ C is infinite for some A ∈ A or B ∩ C is infinite for some B ∈ B, and (c) both and A and B have order-type equal to ω 1 with respect to the inclusion modulo finite. Families A and B satisfying (a) are orthogonal, those satisfying both (a) and (b) form a gap in P(N)/ Fin (or, they are not separated over Fin, the ideal of finite sets), and if they furthermore satisfy (c) they form an (ω 1 , ω 1 )-gap. If (a) and (b) hold and both A and B are linearly ordered by the inclusion modulo finite, a gap is linear. A gap (linear or not) is Hausdorff if both of its sides are σ-directed under the inclusion modulo finite.Another major advance in the study of gaps in P(N)/ Fin was made by Kunen ([29]), who used a condition originally isolated by Luzin in [32] to prove that for every (ω 1 , ω 1 )-gap there is a ccc poset that freezes it. (A gap is indestructible, or frozen, if it remains a gap in every ℵ 1 -preserving forcing extension.) Remarkably, a linear gap can be frozen by an ℵ 1 -preserving forcing if and only if it contains an (ω 1 , ω 1 )-gap. Kunen used freezing to prove that P(N)/ Fin need not be universal for linear orderings of size at most 2 ℵ0 even if Martin's Axiom, MA, is assumed. The technique of freezing gaps has played an important role in a variety of subjects, from automatic continuity in Banach algebras ([6]) to the study of automorphisms of P(N)/ Fin ([35]).

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“…is the symmetric difference of the left-hand sides of (4)- (8) and f (∅) ∈ K, this proves our claim.…”

Section: ] Implies That There Is a Borel Function C Such That C(a) ∈ supporting

confidence: 68%

“…The ideal Z W is sometimes called the Weyl ideal. Using results of [8] and [16], it can be proved that the quotient over Z W is not isomorphic to the quotient over NWD(Q), NULL(Q) or any analytic P-ideal. Proof.…”

Section: Lemma 24mentioning

confidence: 99%

Luzin gaps

Farah

2004

Trans. Amer. Math. Soc.

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“…Proposition 3.5 ( [26,40]). If x is WRAP then (X x , S) is uniquely ergodic and has zero topological entropy.…”

Section: Definition Of Rational Subshifts Examplesmentioning

confidence: 99%

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

Bergelson¹,

Kułaga-Przymus²,

Lemańczyk³

et al. 2018

Ergod. Th. Dynam. Sys.

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A set R ⊂ N is called rational if it is well-approximable by finite unions of arithmetic progressions, meaning that for every ǫ > 0 there exists a set B = r i=1 a i N+b i , where a 1 , . . . , a r , b 1 , . . . , b r ∈ N, such thatExamples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form Φ x := {n ∈ N :< x}, where x ∈ [0, 1] and ϕ is Euler's totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if R ⊂ N is a rational set with d(R) > 0, then the following are equivalent:(b) R is an averaging set of polynomial single recurrence.(c) R is an averaging set of polynomial multiple recurrence.As an application, we show that if R ⊂ N is rational and divisible, then for any set E ⊂ N with d(E) > 0 and any polynomials p i ∈ Q[t], i = 1, . . . , ℓ, which satisfy p i (Z) ⊂ Z and p i (0) = 0 for all i ∈ {1, . . . , ℓ}, there exists β > 0 such that the sethas positive lower density.Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if A is a finite alphabet, η ∈ A N is rationally almost periodic, S denotes the left-shift on A Z and X := {y ∈ A Z : each finite word appearing in y appears in η}, then η is a generic point for an S-invariant probability measure ν on X such that the measure preserving system (X, ν, S) is ergodic and has rational discrete spectrum.

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“…This term was introduced by Jacobs and Keane in , who proved that if

{false{x_{n} false}}_{n \in double-struckN}

tends to

x

D_{W} + ρ

and all points

x_{n}

are strictly transitive (that is there is only one Borel probability invariant measure on the closure of its orbit), then so is

x

. Further dynamical consequences of the quasi‐uniform convergence were studied by Downarowicz and Iwanik , Blanchard, Formenti and Kůrka , and Salo and Törmä .…”

Section: Introductionmentioning

confidence: 99%

Quasi-uniform convergence in dynamical systems generated by an amenable group action

Łącka

Straszak

2018

J. London Math. Soc.

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We study the Weyl pseudometric associated with an action of a countable amenable group on a compact metric space. We prove that the topological entropy and the number of minimal subsets of the closure of an orbit are both lower semicontinuous with respect to the Weyl pseudometric. Furthermore, the simplex of invariant measures supported on the orbit closure varies continuously. We apply the Weyl pseudometric to Toeplitz configurations for arbitrary amenable residually finite groups. We introduce the notion of a regular Toeplitz configuration and demonstrate that all regular Toeplitz configurations define minimal and uniquely ergodic systems. We prove that this family is path-connected in the Weyl pseudometric. This leads to a new proof of a theorem of Krieger, saying that Toeplitz configurations can have arbitrary finite entropy.is ergodic if for every G-invariant set A ⊂ X one has either μ(A) = 0 or μ(A) = 1. We denote by † We omit this assumption at the beginning of Section 7. ‡ We used the adjective to distinguish left Følner sequences from their right-and two-sided analogues. In a similar vain notions related to non-commutative groups have 'left' and 'right' variants. For brevity, we will use adjectives 'left/right' only in definitions and routinely omit them later, since the choice of 'left/right' is fixed throughout the paper.

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Quasi-uniform convergence in compact dynamical systems

Cited by 18 publications

References 0 publications

Luzin gaps

Luzin gaps

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

Quasi-uniform convergence in dynamical systems generated by an amenable group action

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