Shadowing is generic on dendrites

Brian, Will; Meddaugh, Jonathan; Raines, Brian E.

doi:10.3934/dcdss.2019142

Cited by 10 publications

(19 citation statements)

References 10 publications

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“…A series of lemmas demonstrates that each T n (X) contains a dense open set in C(X). We proceed in a manner similar to that in [6]. For every f ∈ C(X) and every > 0, we will construct a map g ∈ C(X) with ρ(g, f ) < and find a γ > 0 so that B γ (g) ⊆ B (f ) and B γ (g) ⊆ T n (X).…”

Section: Maps Of Graphitesmentioning

confidence: 99%

“…Recently, it has been shown that these results extend to the class of continuous maps on dendrites, i.e. locally connected, uniquely arcwise connected continua [6]. This is a class of spaces which includes acyclic graphs and the Julia sets of a large class of quadratic polynomials [8].…”

mentioning

confidence: 99%

“…In Section 2, we introduce terminology and prove that graphites have particularly nice open covers. In Section 3, we use a construction similar to that of [6] to demonstrate that shadowing is generic on graphites. In Section 4, we modify this construction to achieve the same result in the space of surjections on graphites.…”

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confidence: 99%

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On genericity of shadowing in one dimension

Meddaugh

2021

Fund. Math.

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Section: Maps Of Graphitesmentioning

confidence: 99%

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confidence: 99%

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On genericity of shadowing in one dimension

Meddaugh

2021

Fund. Math.

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“…In the second section we prove that T S -bi-shadowing property is generic as well as the s-limit shadowing property is dense in the class of all continuous functions defined on any dendrite. Let us mention here that genericity of the (usual) shadowing property on dendrites was recently proved in [6]. In our proof we use a slightly different argument, motivated mainly by the methodology developed for graphs.…”

Section: Introductionmentioning

confidence: 99%

Shadowing is Generic on Various One-Dimensional Continua with a Special Geometric Structure

et al. 2019

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In the paper we use a special geometric structure of selected one-dimensional continua to prove that some stronger versions of the shadowing property are generic (or at least dense) for continuous maps acting on these spaces. Specifically, we prove that (i) the periodic $$\mathscr {T}_{S}$$TS-bi-shadowing property, where $$\mathscr {T}_{S}$$TS means some class of continuous methods, is generic as well as the s-limit shadowing property is dense in the space of all continuous maps (and all continuous surjective maps) of any topological graph; (ii) the $$\mathscr {T}_{S}$$TS-bi-shadowing property is generic as well as the s-limit shadowing property is dense in the space of all continuous maps of any dendrite; (iii) the $$\mathscr {T}_{S}$$TS-bi-shadowing property is generic in the space of all continuous maps of chainable continuum that can by approximated by arcs from the inside. The results of the paper extend ones obtained over the last few decades by various authors (see, e.g., Kościelniak in J Math Anal Appl 310:188–196, 2005; Kościelniak and Mazur in J Differ Equ Appl 16:667–674, 2010; Kościelniak et al. in Discret Contin Dyn Syst 34:3591–3609, 2014; Mazur and Oprocha in J Math Anal Appl 408:465–475, 2013; Mizera in Generic Properties of One-Dimensional Dynamical Systems, Ergodic Theory and Related Topics, III, Springer, Berlin, 1992; Odani in Proc Am Math Soc 110:281–284, 1990; Pilyugin and Plamenevskaya in Topol Appl 97:253–266, 1999; and Yano in J Fac Sci Univ Tokyo Sect IA Math 34:51–55, 1987) for both homeomorphisms and continuous maps of compact manifolds, including (in particular) an interval and a circle, which are the simplest examples of one-dimensional continua. Moreover, from a technical point of view our considerations are a continuation of those carried out in the earlier work by Mazur and Oprocha in J. Math. Anal. Appl. 408:465–475, 2013.

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“…The interest in the study of chaotic behaviour on dendrites has been increasing for a few years. For examples, researches focus on shadowing [17], distributional chaos [18], induced map [19], and relations between some chaotic notions [20][21][22][23]. It turns out that most of the results on the interval do not necessarily hold true on dendrites.…”

Section: Introductionmentioning

confidence: 99%

Some Chaos Notions on Dendrites

Fadel

Dzul-Kifli

2019

Symmetry

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Transitivity is a key element in a chaotic dynamical system. In this paper, we present some relations between transitivity, stronger and alternative notions of it on compact and dendrite spaces. The relation between Auslander and Yorke chaos and Devaney chaos on dendrites is also discussed. Moreover, we prove that Devaney chaos implies strong dense periodicity on dendrites while the converse is not true.

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Shadowing is generic on dendrites

Abstract: We show that shadowing is a generic property for continuous maps on dendrites.

Cited by 10 publications

References 10 publications

On genericity of shadowing in one dimension

On genericity of shadowing in one dimension

Shadowing is Generic on Various One-Dimensional Continua with a Special Geometric Structure

Some Chaos Notions on Dendrites

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