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.