Multishadowing in topological dynamics

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.