The Chain Properties and Average Shadowing Property of Iterated Function Systems

Wu, Xinxing; Wang, Lidong; Liang, Jianhua

doi:10.1007/s12346-016-0220-1

Cited by 31 publications

(11 citation statements)

References 14 publications

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“…Precisely speaking, if (X, T n ) is chain transitive for all n > 0 then it is chain mixing, that is, for any x, y ∈ X and δ > 0 there is N > 0 such that there is a δ-chain from x to y consisting of exactly n elements for every n > N . Recently, Wu et al [19] proved that this also holds for iterated function systems. A point x ∈ X is chain recurrent if for every δ > 0 there exists a δ-chain from x to x.…”

Section: Average Tracing Of Approximate Trajectories Let {Xmentioning

confidence: 90%

On averaged tracing of periodic average pseudo orbits

Oprocha¹,

Wu²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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Section: Average Tracing Of Approximate Trajectories Let {Xmentioning

confidence: 90%

On averaged tracing of periodic average pseudo orbits

Oprocha¹,

Wu²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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“…In addition, since Wang first gave the definition of sequentially distributional chaos in [6], the research of it has had a great influence on topology (see [5,[7][8][9][10][11][12][13][14][15][16]). However, the notion about sequentially distributional chaos for group actions has not been given so far.…”

Section: Definition 13mentioning

confidence: 99%

Chaos for finitely generated semigroup actions

Wang¹,

Zhao²,

Chu³

2017

J. Nonlinear Sci. Appl.

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In this paper, we define and study Li-Yorke chaos and distributional chaos along a sequence for finitely generated semigroup actions. Let X be a compact space with metric d and G be a semigroup generated by f 1 , f 2 , · · · f m which are finitely many continuous mappings from X to itself. Then we show if (X, G) is transitive and there exists a common fixed point for all the above mappings, then (X, G) is chaotic in the sense of Li-Yorke. And we give a sufficient condition for (X, G) to be uniformly distributionally chaotic along a sequence and chaotic in the strong sense of Li-Yorke. At the end of this paper, an example on the one-sided symbolic dynamical system for (X, G) to be chaotic in the strong sense of Li-Yorke and uniformly distributionally chaotic along a sequence is given.

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“…The notion of Li-Yorke sensitivity was firstly introduced by Akin and Kolyada [1] in 2003. More recent results on chaos can be found in [2,3,7,17,18,[20][21][22][23][24][25][26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Shifts, rotations and distributional chaos

Xiang

Liang

2019

Adv Differ Equ

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Let R r 0 , R r 1 : S 1 − → S 1 be rotations on the unit circle S 1 and define f : Σ 2 × S 1 − → Σ 2 × S 1 as f (x, t) = (σ (x), R r x 1 (t)), for x = x 1 x 2 • • • ∈ Σ 2 := {0, 1} N , t ∈ S 1 , where σ : Σ 2 − → Σ 2 is the shift, and r 0 and r 1 are rotational angles. It is first proved that the system (Σ 2 × S 1 , f) exhibits maximal distributional chaos for any r 0 , r 1 ∈ R (no assumption of r 0 , r 1 ∈ R \ Q), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91-99, 2014). It is also obtained that (Σ 2 × S 1 , f) is cofinitely sensitive and (M 1 ,M 1)-sensitive and that (Σ 2 × S 1 , f) is densely chaotic if and only if r 1-r 0 ∈ R \ Q.

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The Chain Properties and Average Shadowing Property of Iterated Function Systems

Cited by 31 publications

References 14 publications

On averaged tracing of periodic average pseudo orbits

On averaged tracing of periodic average pseudo orbits

Chaos for finitely generated semigroup actions

Shifts, rotations and distributional chaos

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