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On $C^0$ -genericity of distributional chaos
Part of
- Dynamical systems with hyperbolic behavior
NORIAKI KAWAGUCHI
Journal:

Ergodic Theory and Dynamical Systems / Volume 43 / Issue 2 / February 2023

Published online by Cambridge University Press:

15 November 2021, pp. 615-645

Print publication:

February 2023
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Let M be a compact smooth manifold without boundary. Based on results by Good and Meddaugh [Invent. Math. 220 (2020), 715–736], we prove that a strong distributional chaos is $C^0$ -generic in the space of continuous self-maps (respectively, homeomorphisms) of M. The results contain answers to questions by Li, Li and Tu [Chaos 26 (2016), 093103] and Moothathu [Topology Appl. 158 (2011), 2232–2239] in the zero-dimensional case. A related counter-example on the chain components under shadowing is also given.