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Weak forms of shadowing in topological dynamics. / Cherkashin, Danila; Kryzhevich, Sergey.

In: Topological Methods in Nonlinear Analysis, Vol. 50, No. 1, 01.09.2017, p. 125-150.

Research output: Contribution to journal › Article › peer-review

Harvard

Cherkashin, D & Kryzhevich, S 2017, 'Weak forms of shadowing in topological dynamics', Topological Methods in Nonlinear Analysis, vol. 50, no. 1, pp. 125-150. https://doi.org/10.12775/TMNA.2017.020

APA

Cherkashin, D., & Kryzhevich, S. (2017). Weak forms of shadowing in topological dynamics. Topological Methods in Nonlinear Analysis, 50(1), 125-150. https://doi.org/10.12775/TMNA.2017.020

Vancouver

Cherkashin D, Kryzhevich S. Weak forms of shadowing in topological dynamics. Topological Methods in Nonlinear Analysis. 2017 Sep 1;50(1):125-150. https://doi.org/10.12775/TMNA.2017.020

Author

Cherkashin, Danila ; Kryzhevich, Sergey. / Weak forms of shadowing in topological dynamics. In: Topological Methods in Nonlinear Analysis. 2017 ; Vol. 50, No. 1. pp. 125-150.

BibTeX

@article{22a6915be14343288bad1c0ad70c8cd9,
title = "Weak forms of shadowing in topological dynamics",
abstract = "We consider continuous maps of compact metric spaces. It is proved that every pseudotrajectory with sufficiently small errors contains a subsequence of positive density that is point-wise close to a subsequence of an exact trajectory with the same indices. Also, we study homeomorphisms such that any pseudotrajectory can be shadowed by a finite number of exact orbits. In terms of numerical methods this property (we call it multishadowing) implies possibility to calculate minimal points of the dynamical system. We prove that for the non-wandering case multishadowing is equivalent to density of minimal points. Moreover, it is equivalent to existence of a family of ε-networks (ε > 0) whose iterations are also ε-networks. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.",
keywords = "Chain recurrence, Invariant measure, Minimal points, Shadowing, Syndetic sets, Topological dynamics, ε-networks",
author = "Danila Cherkashin and Sergey Kryzhevich",
year = "2017",
month = sep,
day = "1",
doi = "10.12775/TMNA.2017.020",
language = "English",
volume = "50",
pages = "125--150",
journal = "Topological Methods in Nonlinear Analysis",
issn = "1230-3429",
publisher = "Wydawnictwo Uniwersytetu Miko{\l}aja Kopernika",
number = "1",

}

RIS

TY - JOUR

T1 - Weak forms of shadowing in topological dynamics

AU - Cherkashin, Danila

AU - Kryzhevich, Sergey

PY - 2017/9/1

Y1 - 2017/9/1

N2 - We consider continuous maps of compact metric spaces. It is proved that every pseudotrajectory with sufficiently small errors contains a subsequence of positive density that is point-wise close to a subsequence of an exact trajectory with the same indices. Also, we study homeomorphisms such that any pseudotrajectory can be shadowed by a finite number of exact orbits. In terms of numerical methods this property (we call it multishadowing) implies possibility to calculate minimal points of the dynamical system. We prove that for the non-wandering case multishadowing is equivalent to density of minimal points. Moreover, it is equivalent to existence of a family of ε-networks (ε > 0) whose iterations are also ε-networks. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.

AB - We consider continuous maps of compact metric spaces. It is proved that every pseudotrajectory with sufficiently small errors contains a subsequence of positive density that is point-wise close to a subsequence of an exact trajectory with the same indices. Also, we study homeomorphisms such that any pseudotrajectory can be shadowed by a finite number of exact orbits. In terms of numerical methods this property (we call it multishadowing) implies possibility to calculate minimal points of the dynamical system. We prove that for the non-wandering case multishadowing is equivalent to density of minimal points. Moreover, it is equivalent to existence of a family of ε-networks (ε > 0) whose iterations are also ε-networks. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.

KW - Chain recurrence

KW - Invariant measure

KW - Minimal points

KW - Shadowing

KW - Syndetic sets

KW - Topological dynamics

KW - ε-networks

UR - http://www.scopus.com/inward/record.url?scp=85030118144&partnerID=8YFLogxK

U2 - 10.12775/TMNA.2017.020

DO - 10.12775/TMNA.2017.020

M3 - Article

AN - SCOPUS:85030118144

VL - 50

SP - 125

EP - 150

JO - Topological Methods in Nonlinear Analysis

JF - Topological Methods in Nonlinear Analysis

SN - 1230-3429

IS - 1

ER -

ID: 36098351