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On $C^0$-genericity of distributional chaos

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  15 November 2021

NORIAKI KAWAGUCHI
NORIAKI KAWAGUCHI*
Affiliation:
Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan
*
e-mail: gknoriaki@gmail.com
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Abstract

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.

Keywords

distributional chaosgenericshadowingzero dimensionMycielski set

MSC classification

Primary: 37D45: Strange attractors, chaotic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 2 , February 2023 , pp. 615 - 645
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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