Multishadowing in topological dynamics

TL;DR

This paper introduces the concept of multishadowing in topological dynamics, showing that pseudotrajectories can be approximated by a finite set of exact orbits, and provides criteria for networks with specific shadowing properties.

Contribution

It develops a framework for multishadowing, establishing conditions for the existence of networks where iterations preserve shadowing properties, and explores connections with ergodic and topological dynamics.

Findings

Every pseudotrajectory with small errors contains a shadowed subsequence.

Systems with multishadowing have finite orbit sets approximating pseudotrajectories.

Criteria for ε-networks with iterative ε-network properties are provided.

Abstract

An approach to find a weak form of shadowing is developed. We consider homeomorphisms of a compact metric space. It is proved that every pseudotrajectory with sufficiently small errors contains at least one subsequence that can be shadowed by a subsequence of an exact trajectory with same indices. We study systems with so-called multishadowing property that is any pseudotrajectory can be shadowed by a finite number of exact orbits. Criteria for existence of $\varepsilon$-- networks whose iterations are $\varepsilon$ -- networks are given. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.