Equicontinuity, uniform almost periodicity, and regionally proximal relation for topological semiflows

Dai, Xiongping; Xiao, Zubiao

doi:10.1016/j.topol.2017.09.007

Cited by 7 publications

(14 citation statements)

References 12 publications

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“…If (T, X) is a flow with Q(X) = ∆ X , then by Lemma 1.10 it is equicontinuous. By using (1) of Theorem 1.13, 'pointwise almost automorphy' and 'Veech's relation' V(X), Dai and Xiao in [20] have proved the following fact (Corollary 1.16). However, we now can simply prove it by only using (1) of Lemma 1.10 and Theorem 1.13 as follows.…”

Section: Main Theoremsmentioning

confidence: 95%

“…Thus (T, X) is pointwise almost automorphic 1 so that it is equicontinuous (cf. [20,Lemma 5.2 and Proposition 5.5]). Hence Q(T, X) = P(T, X) = ∆ X .…”

Section: Distality Proximity and Regional Proximitymentioning

confidence: 99%

“…As be mentioned before, if using no (2) and (4) of Theorem 1.13, then we need a long zigzag proof for this result as in [20]. As more applications of Theorem 1.13, we will present other equivalent conditions for "equicontinuous surjective" later on.…”

Section: Main Theoremsmentioning

confidence: 99%

“…For a semiflow, using our Theorem 2.1 as an important tool, Dai and Xiao in [20] have proved the equivalence of 'almost periodic' and 'equicontinuous surjective' which shows that whether or not (T, X) is a.p. does not depend upon the topology on the phase semigroup T .…”

Section: Almost Periodic Flow and Wap Functionsmentioning

confidence: 99%

“…There has already been some applications in the recent work [20] 2 and in the proofs of Theorems 1.14 and 1.17 and Corollary 1.16. Next we shall give some other applications of Theorem 1.13 here.…”

Section: Applicationsmentioning

confidence: 99%

See 4 more Smart Citations

Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Auslander¹,

Dai²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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Let π : T × X → X with phase map (t, x) → tx, denoted (π, T, X), be a semiflow on a compact Hausdorff space X with phase semigroup T . If each t ∈ T is onto, (π, T, X) is called surjective; and if each t ∈ T is 1-1 onto (π, T, X) is called invertible and in latter case it induces π −1 : X × T → X by (x, t) → xt := t −1 x, denoted (π −1 , X, T ). In this paper, we show that (π, T, X) is equicontinuous surjective iff it is uniformly distal iff (π −1 , X, T ) is equicontinuous surjective. As applications of this theorem, we also consider the minimality, distality, and sensitivity of (π −1 , X, T ) if (π, T, X) is invertible with these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of Z-flow with compact zero-dimensional phase space.

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Section: Main Theoremsmentioning

confidence: 95%

“…Thus (T, X) is pointwise almost automorphic 1 so that it is equicontinuous (cf. [20,Lemma 5.2 and Proposition 5.5]). Hence Q(T, X) = P(T, X) = ∆ X .…”

Section: Distality Proximity and Regional Proximitymentioning

confidence: 99%

Section: Main Theoremsmentioning

confidence: 99%

Section: Almost Periodic Flow and Wap Functionsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

See 3 more Smart Citations

Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Auslander¹,

Dai²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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Devaney chaos, Li–Yorke chaos, and multi-dimensional Li–Yorke chaos for topological dynamics

Dai¹,

Tang²

2017

Journal of Differential Equations

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Let π : T × X → X, written T π X, be a topological semiflow/flow on a uniform space X with T a multiplicative topological semigroup/group not necessarily discrete. We then prove:• If T π X is non-minimal topologically transitive with dense almost periodic points, then it is sensitive to initial conditions. As a result of this, Devaney chaos ⇒ Sensitivity to initial conditions, for this very general setting.Let R + π X be a C 0 -semiflow on a Polish space; then we show:• If R + π X is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior, then it is multi-dimensional Li-Yorke chaotic; that is, there is a uncountable set Θ ⊆ X such that for any k ≥ 2 and any distinct points x 1 , . . . , x k ∈ Θ, one can find two time sequences s n → ∞, t n → ∞ with s n (x 1 , . . . , x k ) → (x 1 , . . . , x k ) ∈ X k and t n (x 1 , . . . , x k ) → (p, . . . , p) ∈ ∆ X k .Consequently, Devaney chaos ⇒ Multi-dimensional Li-Yorke chaos.Moreover, let X be a non-singleton Polish space; then we prove:• Any weakly-mixing C 0 -semiflow R + π X is densely multi-dimensional Li-Yorke chaotic.• Any minimal weakly-mixing topological flow T π X with T abelian is densely multidimensional Li-Yorke chaotic.• Any weakly-mixing topological flow T π X is densely Li-Yorke chaotic.We in addition construct a completely Li-Yorke chaotic minimal SL(2, R)-acting flow on the compact metric space R ∪ {∞}. Our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup/group T .

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Almost automorphy and Veech's relations of dynamics on compact T2-spaces

Dai

2020

Journal of Differential Equations

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Equicontinuity, uniform almost periodicity, and regionally proximal relation for topological semiflows

Cited by 7 publications

References 12 publications

Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Devaney chaos, Li–Yorke chaos, and multi-dimensional Li–Yorke chaos for topological dynamics

Almost automorphy and Veech's relations of dynamics on compact T2-spaces

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