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Some universality results for dynamical systems


Authors: Udayan B. Darji and Étienne Matheron
Journal: Proc. Amer. Math. Soc. 145 (2017), 251-265
MSC (2010): Primary 37B99, 54H20; Secondary 54C20, 47A99
DOI: https://doi.org/10.1090/proc/13225
Published electronically: July 12, 2016
MathSciNet review: 3565377
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Abstract: We prove some ``universality'' results for topological dynamical systems. In particular, we show that for any continuous self-map $ T$ of a perfect Polish space, one can find a dense, $ T$-invariant set homeomorphic to the Baire space $ \mathbb{N}^{\mathbb{N}}$; that there exists a bounded linear operator $ U: \ell \rightarrow \ell $ such that any linear operator $ T$ from a separable Banach space into itself with $ \Vert T\Vert \leq 1$ is a linear factor of $ U$; and that given any $ \sigma $-compact family $ {\mathcal F}$ of continuous self-maps of a compact metric space, there is a continuous self-map $ U_{\mathcal F}$ of $ \mathbb{N}^{\mathbb{N}}$ such that each $ T\in {\mathcal F}$ is a factor of $ U_{\mathcal F}$.


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Additional Information

Udayan B. Darji
Affiliation: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292
Email: ubdarj01@louisville.edu

Étienne Matheron
Affiliation: Laboratoire de Mathématiques de Lens, Université d’Artois, Rue Jean Souvraz S. P. 18, 62307 Lens, France
Email: etienne.matheron@univ-artois.fr

DOI: https://doi.org/10.1090/proc/13225
Keywords: Universal, factor, $\ell_1$, Cantor space, Baire space
Received by editor(s): December 3, 2015
Received by editor(s) in revised form: March 16, 2016
Published electronically: July 12, 2016
Additional Notes: The first author would like to acknowledge the hospitality and financial support of Université d’Artois
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2016 American Mathematical Society