Inheritance of shadowing for dynamical semigroups

TL;DR

This paper investigates whether the shadowing property in dynamical semigroups is inherited from individual generators, revealing that inheritance is generally not guaranteed and highlighting differences from non-autonomous systems.

Contribution

It extends the single-perturbation shadowing approach to semigroups, providing a constructive method to analyze shadowing and exploring inheritance properties.

Findings

Shadowing for generators does not necessarily imply shadowing for the entire semigroup.

Inheritance of shadowing is generally negative, contrary to initial expectations.

Shadowing behavior differs significantly between semigroups and non-autonomous systems.

Abstract

We extend the single-perturbation approach (developed in our earlier publications for the case of a single map) to the analysis of the shadowing property for semigroups of endomorphisms. Our approach allows to give a constructive representation for a true trajectory which shadows a given pseudo-trajectory. One of the main motivations is the question of inheritance: does the presence of shadowing for all generators of a semigroup imply shadowing for the semigroup and vice versa. Somewhat surprisingly, the answer to these questions is generally negative. Moreover, the situation with shadowing turns out to be quite different in a semigroup and in a non-autonomous system, despite the fact that the latter can be represented as a single branch of the former.