On $C^0$-genericity of distributional chaos

TL;DR

This paper proves that strong distributional chaos is generically present in the space of continuous maps on compact manifolds, providing answers to previous open questions and including a counter-example related to shadowing.

Contribution

It establishes $C^0$-genericity of strong distributional chaos on compact manifolds, extending prior results and addressing open questions in the zero-dimensional case.

Findings

Strong distributional chaos is $C^0$-generic on compact manifolds.

Provides answers to questions by Li et al. and Moothathu.

Includes a counter-example on chain components under shadowing.

Abstract

Let $M$ be a compact smooth manifold without boundary. Based on results by Good and Meddaugh (2020), we prove that a strong distributional chaos is $C^0$-generic in the space of continuous self-maps (resp. homeomorphisms) of $M$. The results contain answers to questions by Li et al. (2016) and Moothathu (2011) in the zero-dimensional case. A related counter-example on the chain components under shadowing is also given.