## Topological Methods in Nonlinear Analysis

### 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 $\varepsilon$-networks ($\varepsilon > 0$) whose iterations are also $\varepsilon$-networks. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 50, Number 1 (2017), 125-150.

Dates
First available in Project Euclid: 14 October 2017

https://projecteuclid.org/euclid.tmna/1507946573

Digital Object Identifier
doi:10.12775/TMNA.2017.020

Mathematical Reviews number (MathSciNet)
MR3706154

Zentralblatt MATH identifier
06850993

#### Citation

Cherkashin, Danila; Kryzhevich, Sergey. Weak forms of shadowing in topological dynamics. Topol. Methods Nonlinear Anal. 50 (2017), no. 1, 125--150. doi:10.12775/TMNA.2017.020. https://projecteuclid.org/euclid.tmna/1507946573

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