Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-29T20:44:22.430Z Has data issue: false hasContentIssue false
  • English
  • Français

Reconstructing a minimal topological dynamical system from a set of return times

Part of: Topological dynamics

Published online by Cambridge University Press:  03 September 2021

KAMIL BULINSKI Open the ORCID record for KAMIL BULINSKI [Opens in a new window]  and
ALEXANDER FISH
KAMIL BULINSKI*
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales, Australia (e-mail: alexander.fish@sydney.edu.au)
ALEXANDER FISH
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales, Australia (e-mail: alexander.fish@sydney.edu.au)
*
e-mail: kamil.bulinski@sydney.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate to what extent a minimal topological dynamical system is uniquely determined by a set of return times to some open set. We show that in many situations, this is indeed the case as long as the closure of this open set has no non-trivial translational symmetries. For instance, we show that under this assumption, two Kronecker systems with the same set of return times must be isomorphic. More generally, we show that if a minimal dynamical system has a set of return times that coincides with a set of return times to some open set in a Kronecker system with translationarily asymmetric closure, then that Kronecker system must be a factor. We also study similar problems involving nilsystems and polynomial return times. We state a number of questions on whether these results extend to other homogeneous spaces and transitive group actions, some of which are already interesting for finite groups.

Keywords

topological dynamicsreturn timesKronecker systems

MSC classification

Primary: 37B20: Notions of recurrence
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2723 - 2739
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albertson, M. O. and Collins, K. L.. Symmetry breaking in graphs. Electron. J. Combin. 3(1) (1996), Article no. 18, 17pp.CrossRefGoogle Scholar
Gutman, Y.. Taken’s embedding theorem with a continuous observable . Ergodic Theory. Ed. Assani, I.. De Gruyter, Berlin, 2016, pp. 134–141.CrossRefGoogle Scholar
Host, B. and Kra, B.. Nilpotent Structures in Ergodic Theory (Mathematical Surveys and Monographs, 236). American Mathematical Society, Providence, RI, 2018, 427pp.CrossRefGoogle Scholar
Pavlov, R.. Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys. 28(4) (2008), 1291–1322.CrossRefGoogle Scholar
Smith, S. M., Tucker, T. W. and Watkins, M. E.. Distinguishability of infinite groups and graphs. Electron. J. Combin. 19(2) (2012), Article no. 27, 10pp.CrossRefGoogle Scholar
Takens, F.. Detecting strange attractors in turbulence. Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980) (Lecture Notes in Mathematics, 898). Eds. Rand, D. and Young, L. S.. Springer, Berlin, 1981, pp. 366–381.Google Scholar
Weyl, H.. Über die Gleichverteilung von Zahlen Mod Eins. Math. Ann. 77(3) (1916), 313–352 (in German).CrossRefGoogle Scholar