Open AccessZero-dimensional and symbolic extensions of topological flows
Author(s) -
David Burguet,
Ruxi Shi
Publication year - 2022
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2021148
Subject(s) - extension (predicate logic) , mathematics , zero (linguistics) , flow (mathematics) , symbolic dynamics , pure mathematics , topology (electrical circuits) , smoothness , topological conjugacy , mathematical analysis , combinatorics , geometry , computer science , philosophy , linguistics , programming language
A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [ 6 ] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time- \begin{document}$ t $\end{document} map admits an extension by a subshift for any \begin{document}$ t\neq 0 $\end{document} . Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on \begin{document}$ \{0,1\}^{\mathbb Z} $\end{document} with a roof function \begin{document}$ f $\end{document} vanishing at the zero sequence \begin{document}$ 0^\infty $\end{document} admits a principal symbolic extension or not depending on the smoothness of \begin{document}$ f $\end{document} at \begin{document}$ 0^\infty $\end{document} .