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Generic homeomorphisms have full metric mean dimension

Part of: Smooth dynamical systems: general theory Topological dynamics Connections with other structures, applications Special properties

Published online by Cambridge University Press:  28 December 2020

MARIA CARVALHO Open the ORCID record for MARIA CARVALHO [Opens in a new window] ,
FAGNER B. RODRIGUES  and
PAULO VARANDAS Open the ORCID record for PAULO VARANDAS [Opens in a new window]
MARIA CARVALHO
Affiliation:
Departamento de Matemática, Universidade do Porto, Porto, Portugal (e-mail: mpcarval@fc.up.pt)
FAGNER B. RODRIGUES
Affiliation:
Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil (e-mail: fagnerbernardini@gmail.com)
PAULO VARANDAS*
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal da Bahia, Salvador, Brazil Centro de Matemática da Universidade do Porto (CMUP), Porto, Portugal
*
e-mail: paulo.varandas@ufba.br
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Abstract

We prove that for $C^0$-generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. As an application, we show that the upper box dimension of the set of periodic points of a $C^0$-generic homeomorphism is equal to the dimension of the manifold. In the case of continuous interval maps, we prove that each level set for the metric mean dimension with respect to the Euclidean distance is $C^0$-dense in the space of continuous endomorphisms of $[0,1]$ with the uniform topology. Moreover, the maximum value is attained at a $C^0$-generic subset of continuous interval maps and a dense subset of metrics topologically equivalent to the Euclidean distance.

Keywords

metric mean dimensionpseudo-horseshoetopological dynamics

MSC classification

Primary: 37C45: Dimension theory of dynamical systems 54H20: Topological dynamics
Secondary: 37B40: Topological entropy 54F45: Dimension theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 40 - 64
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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