Generic and dense distributional chaos with shadowing

TL;DR

This paper investigates the properties of generic and dense distributional chaos in continuous maps on compact metric spaces, establishing their equivalence under shadowing and providing illustrative examples.

Contribution

It introduces equivalent conditions for generic distributional chaos under shadowing and proves their equivalence, advancing understanding of chaos in dynamical systems.

Findings

Equivalent properties for generic uniform chaos established

Proved equivalence of chaos notions under shadowing

Provided illustrative examples of main results

Abstract

For continuous self-maps of compact metric spaces, we consider the notions of generic and dense chaos introduced by Lasota and Snoha and their variations for the distributional chaos, under the assumption of shadowing. We give some equivalent properties to generic uniform (distributional) chaos in terms of chains and prove their equivalence to generic (distributional) chaos under the shadowing. Several examples illustrating the main results are also given.