Refined scales of weak-mixing dynamical systems: typical behaviour

Carvalho, Silas L.; Oliveira, César R. de

doi:10.1017/etds.2019.40

Cited by 4 publications

(1 citation statement)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If T is partially weakly f -mixing, then there exists µ f of U T which is uniformly √ f -continuous [17]. Carvalho and De Oliveira [6] proved some properties of lim sup N N α C N and lim inf N N α C N for 0 < α < 1.…”

Section: Introductionmentioning

confidence: 99%

Size of exceptional sets in weakly mixing systems

Park¹,

Park²

2023

Preprint

View full text Add to dashboard Cite

For any weakly mixing system (X, B, µ, T ) and A, B ∈ B, there is a density zero set JA,B such that µ(A ∩ T −n B) converges to µ(A)µ(B) for n / ∈ J. In this paper, we study bounds on the size of this exceptional set J. First, we show that, given the rate of weak mixing, we can find an upper bound on the size of J. We use this property to show the existence of an exceptional set J of an interval exchange system with eitherdepending on whether or not the transformation is of rotation class. We also explicitly construct an exceptional set J for the Chacon transformation and give upper and lower bounds for its size. More precisely, we show that there is a constant C > 0 such that for any increasing function h : R+ → R+ diverging to infinity, there is an exceptional set n) . For the lower bound, we prove that for any t > 0, there exists Lebesgue measurable sets A, B ⊆ [0, 1] and C, N > 0 such that |JA,B ∩ [0, n]| ≥ C(log n) t for every n > N .

show abstract

Section: Introductionmentioning

confidence: 99%