Universal completely regular dendrites

Omiljanowski, Krzysztof; Zafiridou, Sophia

doi:10.4064/cm103-1-15

Cited by 3 publications

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“…Ważewski's universal dendrite is an example of a dendrite which is not completely regular. It is interesting that by [46], there is a universal completely regular dendrite, that is, a completely regular dendrite containing a copy of every completely regular dendrite.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Generic chaos on dendrites

Snoha¹,

Špitalský²,

Takács³

2021

Ergod. Th. Dynam. Sys.

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We characterize dendrites D such that a continuous selfmap of D is generically chaotic (in the sense of Lasota) if and only if it is generically ${\varepsilon }$ -chaotic for some ${\varepsilon }>0$ . In other words, we characterize dendrites on which generic chaos of a continuous map can be described in terms of the behaviour of subdendrites with non-empty interiors under iterates of the map. A dendrite D belongs to this class if and only if it is completely regular, with all points of finite order (that is, if and only if D contains neither a copy of the Riemann dendrite nor a copy of the $\omega $ -star).

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Generic chaos on dendrites

Snoha¹,

Špitalský²,

Takács³

2021

Ergod. Th. Dynam. Sys.

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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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Generic chaos on dendrites

Snoha¹,

Špitalský²,

Takács³

2021

Preprint

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We characterize dendrites D such that a continuous selfmap of D is generically chaotic (in the sense of Lasota) if and only if it is generically ε-chaotic for some ε > 0. In other words, we characterize dendrites on which generic chaos of a continuous map can be described in terms of the behaviour of subdendrites with nonempty interiors under iterates of the map. A dendrite D belongs to this class if and only if it is completely regular, with all points of finite order (that is, if and only if D contains neither a copy of the Riemann dendrite nor a copy of the ω-star).

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Universal completely regular dendrites

Cited by 3 publications

References 0 publications

Generic chaos on dendrites

Generic chaos on dendrites

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

Generic chaos on dendrites

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