Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T05:36:21.403Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological and almost Borel universality for systems with the weak specification property

Part of: Topological dynamics Ergodic theory

Published online by Cambridge University Press:  11 February 2019

DAVID BURGUET
DAVID BURGUET*
Affiliation:
LPMA - CNRS UMR 7599, Université Paris 6, 75252 Paris Cedex 05, France email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that systems with some specification properties are topologically or almost Borel universal, in the sense that any aperiodic subshift with lower entropy may be topologically or almost Borel embedded. This improves, with elementary tools, previous results of Quas and Soo [Ergodic universality of some topological dynamical systems. Trans. Amer. Math. Soc.368 (2016), 4137–4170].

Keywords

entropyweak specificationuniversal systems

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification 37B99: None of the above, but in this section 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 8 , August 2020 , pp. 2098 - 2115
Check for updates
Copyright
© Cambridge University Press, 2019

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

Burguet, D. and Downarowicz, T.. Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits. J. Dynam. Differential Equations, doi:10.1007/s10884-018-9674-y.Google Scholar
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18). Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Downarowicz, T. and Serafin, J.. Universal systems for entropy intervals. J. Dynam. Differential Equations 1–12 (2016).Google Scholar
Glasner, E. and Weiss, B.. On the Interplay Between Measurable and Topological Dynamics (Handbook of Dynamical Systems, 1B, 597648). Elsevier, Amsterdam, 2006.CrossRefGoogle Scholar
Hochman, M.. Isomorphism and embedding of Borel systems on full sets. Acta Appl. Math. 126 (2013), 187–201.CrossRefGoogle Scholar
Jung, U.. On the existence of open and bi-continuing codes. Trans. Amer. Math. Soc. 363(3) (2011), 1399–1417.CrossRefGoogle Scholar
Kwietniak, D., Łącka, M. and Oprocha, P.. A panorama of specification-like properties and their consequences. Dyn. Numbers Contemp. Math. 669 (2016), 155–186.CrossRefGoogle Scholar
Krieger, W.. On entropy and generators of measure-preserving transformations. Trans. Amer. Math. Soc. 149 (1970), 453–464.CrossRefGoogle Scholar
Krieger, W.. On the subsystems of topological Markov chains. Ergod. Th. & Dynam. Sys. 2 (1982), 195–202.CrossRefGoogle Scholar
Lind, D. A. and Thouvenot, J.-P.. Measure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations. Math. Sys. Theory 11 (1977), 275–282.CrossRefGoogle Scholar
Ornstein, D. S. and Weiss, B.. Entropy and data compression schemes. IEEE Trans. Inform. Theory 39 (1993), 78–83.CrossRefGoogle Scholar
Quas, A. and Soo, T.. Weak mixing suspension flows over shift over shifts of finite type are universal. J. Mod. Dyn. 6 (2012), 427–449.CrossRefGoogle Scholar
Quas, A. and Soo, T.. Ergodic universality of some topological dynamical systems. Trans. Amer. Math. Soc. 368 (2016), 4137–4170.CrossRefGoogle Scholar
Serafin, J.. Non-existence of a universal zero-entropy system. Israel J. Math. 194 (2013), 349–358.CrossRefGoogle Scholar
Shub, M. and Weiss, B.. Can one always lower topological entropy? Ergod. Th. & Dynam. Sys. 11 (1991), 535–546.CrossRefGoogle Scholar
Weiss, B.. Measurable dynamics. Conference in Modern Analysis and Probability (New Haven, CN, 1982) (Contemporary Mathematics, 26). American Mathematical Society, Providence, RI, 1984, pp. 395–421.CrossRefGoogle Scholar