On averaged tracing of periodic average pseudo orbits

Abstract

We propose a definition of average tracing of finite pseudo-orbits and show that in the case of this definition measure center has the same property as nonwandering set for the classical shadowing property. We also show that the average shadowing property trivializes in the case of mean equicontinuous systems, and that it implies distributional chaos when measure center is nondegenerate.

Sets of invariant measures and Cesaro stability

“…Moreover, for an a priori selected observable, we can say that its average value along the given trajectory does not drastically change if the system is slightly perturbed. In a sense, the weak shadowing, we obtain, is close to statistical shadowing, introduced by M. Blank [17], see also [18].…”

Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

“…A dynamical system has the shadowing property if every sufficiently precise trajectory is closed to some exact trajectory. The shadowing property has been developed intensively in recent years, and many authors obtained results about chaos and stability by studying the various type of shadowing (see [1,11,17,19,20,22,24,25,[27][28][29]). Wu et al [25] introduced the notion of M α -shadowing and proved that a dynamical system has the average shadowing property if and only if it has the M α -shadowing property for any α ∈ [0, 1).…”

“…Wu et al [25] introduced the notion of M α -shadowing and proved that a dynamical system has the average shadowing property if and only if it has the M α -shadowing property for any α ∈ [0, 1). Oprocha and Wu [17] proved that a dynamical system with the average shadowing of periodic pseudo-orbits property is distributionally chaotic if and only if it has a distal pair. Lewowicz [14] introduced the concept of persistence for dynamical systems which is weaker than that of topological stability.…”

Generic and dense distributional chaos with shadowing

“…Shadowing is a feature of topologically hyperbolic dynamical systems, and its implications for chaos is a subject of ongoing research [2,5,6,7,10] (see [1,12] for a general background). One of the definitions of chaos is the generic chaos proposed by Lasota (see [13]).…”