Dynamics of spacing shifts

Banks, John; Nguyen, Thi Trang; Oprocha, Piotr; Stanley, Brett; Trotta, Belinda

doi:10.3934/dcds.2013.33.4207

Cited by 18 publications

(42 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is clearly open, and it is dense by (2). Then by Baire category theorem, Y is a dense G δ subset of X .…”

Section: Proofmentioning

confidence: 95%

“…. , x) is a transitive point of (X r , f (a) ); (2) -transitive if it is -transitive with respect to (1, 2, . . .…”

Section: Proofmentioning

confidence: 99%

“…which implies that [w (1) ] P × [w (2) ] P ⊂ (σ 2 × σ 3 ) −s ([u (1) ] P × [u (2) ] P ) ∩ ([v (1) ] P × [v (2) ] P ).…”

mentioning

confidence: 92%

“…, x) has a dense orbit in X m under the action of f × f 2 × · · · × f m . Following [26], we will say that (X, f ) is multi-transitive if it satisfies (1) and that (X, f ) is -transitive if it satisfies (2). It is shown in [26] that weak mixing, multi-transitivity and -transitivity are equivalent for minimal systems.…”

Section: Introductionmentioning

confidence: 98%

“…Fix open cylinders [u (1) ] P , [u (2) ] P , [v (1) ] P and [v (2) ] P of P . Without loss of generality, we assume that there is k ≥ 2 such that t := |u (i) | = |v (i) | = 2 2k−2 for any i = 1, 2, where |u| denotes the length of u.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Point transitivity, -transitivity and multi-minimality

Chen¹,

Li²,

Lü³

2014

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

show abstract

“…is clearly open, and it is dense by (2). Then by Baire category theorem, Y is a dense G δ subset of X .…”

Section: Proofmentioning

confidence: 95%

“…. , x) is a transitive point of (X r , f (a) ); (2) -transitive if it is -transitive with respect to (1, 2, . . .…”

Section: Proofmentioning

confidence: 99%

“…which implies that [w (1) ] P × [w (2) ] P ⊂ (σ 2 × σ 3 ) −s ([u (1) ] P × [u (2) ] P ) ∩ ([v (1) ] P × [v (2) ] P ).…”

mentioning

confidence: 92%

Section: Introductionmentioning

confidence: 98%

mentioning

confidence: 99%

See 3 more Smart Citations

Point transitivity, -transitivity and multi-minimality

Chen¹,

Li²,

Lü³

2014

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

Mixing Properties in Coded Systems

Epperlein

Kwietniak

Oprocha

2019

New Trends in One-Dimensional Dynamics

Self Cite

View full text Add to dashboard Cite

We show that topological mixing, weak mixing and total transitivity are equivalent for coded systems. We provide an example of a mixing coded system which cannot be approximated by any increasing sequence of mixing shifts of finite type, has only periodic points of even period and each set of its generators consists of blocks of even length. We prove that such an example cannot be a synchronized system. We also show that a mixing coded systems has the strong property P .

show abstract

Multi-recurrence and van der Waerden systems

Kwietniak

Oprocha

et al. 2016

Sci. China Math.

Self Cite

View full text Add to dashboard Cite

ABSTRACT. We explore recurrence properties arising from dynamical approach to the van der Waerden Theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.

show abstract

Dynamics of spacing shifts

Cited by 18 publications

References 18 publications

Point transitivity, -transitivity and multi-minimality

Point transitivity, -transitivity and multi-minimality

Mixing Properties in Coded Systems

Multi-recurrence and van der Waerden systems

Contact Info

Product

Resources

About