Generic and dense distributional chaos with shadowing

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.

Keywords:

• Generic chaos
• dense chaos
• distributional chaos
• shadowing property

2020 Mathematics Subject Classifications:

• Primary 37B05
• 37B65
• Secondary 37B10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by JSPS KAKENHI [Grant Number JP20J01143].