Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T09:46:21.530Z Has data issue: false hasContentIssue false
  • English
  • Français

Common extensions and hyperbolic factor maps for coded systems

Published online by Cambridge University Press:  19 September 2008

Doris Fiebig
Doris Fiebig
Affiliation:
Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

The classification of dynamical systems by the existence of certain common extensions has been carried out very successfully in the class of shifts of finite type (‘finite equivalence’, ‘almost topological conjugacy‘). We consider generalizations of these notions in the class of coded systems. Topological entropy is shown to be a complete invariant for the existence of a common coded entropy preserving extension. In contrast to the shift of finite type setting, this extension cannot be made bounded-to-1 in general. Common extensions with hyperbolic factor maps lead to a version of almost topological conjugacy for coded systems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 3 , June 1995 , pp. 517 - 534
Copyright
Copyright © Cambridge University Press 1995

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

REFERENCES

[A]Ashley, J.. Bounded-to-1 factors of an aperiodic shift of finite type are 1-to-1 almost everywhere also. Ergod. Th. & Dynam. Sys. 10 (1990), 615–625.Google Scholar
[AM]Adler, R. and Marcus, B.. Topological entropy and equivalence of dynamical systems. Mem. Amer. Math. Soc. 219 (1979).Google Scholar
[B]Blanchard, F.. Extensions à fibre constante. Ergod. Th. & Dynam. Sys. 11 (1991), 7–17.CrossRefGoogle Scholar
[BH]Blanchard, F. and Hansel, G.. Systèmes codés. Theoret. Comput. Sci. 44 (1986), 17–49.CrossRefGoogle Scholar
[B-M]Bertrand-Mathis, A.. Specification and synchronisation. Preprint.Google Scholar
[DGS]Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic Theory on Compact Spaces. Springer Lecture notes in Mathematics 527. Springer: Berlin, Heidelberg, New York, 1976.CrossRefGoogle Scholar
[F]Fiebig, D.. Common closing extensions and finitary regular isomorphism for synchronized systems. Amer. Math. Soc. Conm. Symbolic Dynamics and its Application. 135 (1992), 125–138.CrossRefGoogle Scholar
[FF]Fiebig, D. and Fiebig, U.. Covers for coded systems. Amer. Math. Soc. Conm. Symbolic Dynamics and its Application 135 (1992), 139–180.CrossRefGoogle Scholar
[G]Grillenberger, C.. Constructions of strictly ergodic systems I. Given entropy. Z. Wahrscheinlichkeitstheorie.verw.Gebiete 25 (1973), 323–334.CrossRefGoogle Scholar
[P]Parry, W.. A finitary classification of topological Markov chains and sofic systems. Bull. London Math. Soc. 9 (1977), 86–92.CrossRefGoogle Scholar
[T]Tuncel, S.. Markov measures determine the zeta function. Ergod. Th. & Dynam. Sys. 7 (1987), 303–311.CrossRefGoogle Scholar