Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-12T17:36:12.441Z Has data issue: false hasContentIssue false
  • English
  • Français

On topological entropy of maps

Published online by Cambridge University Press:  19 September 2008

Mike Hurley
Mike Hurley
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106-7058, USA (email: mgh3@po.cwru.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce an ‘entropy’ hm(f) for a continuous mapping of a compact metric space to itself which is denned in terms of (n, ∈)-separated subsets of inverse images of individual points. This invariant is compared with the inverse-image entropy h_(f) introduced recently by Langevin and Walczak. The two main results are: (1) the inequality hm(f) ≤ h(f) ≤ hm(f) + h_(f) relating hm, h_ and the topological entropy h(f); (2) if pseudo-orbits are used in place of orbits in the definition of hm then the quantity that results is equal to the topological entropy. We actually establish an inequality that at least formally is slightly stronger than (1) by defining a variant of h_ which we call hi; it is trivial to show that hi ≤ h_, and we show that h ≤ hi + hm, from which (1) follows.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 3 , June 1995 , pp. 557 - 568
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

[1]Barge, M. and Swanson, R.. Pseudo-orbits and topological entropy. Proc. Amer. Math. Soc. 109 (1990), 559–566.CrossRefGoogle Scholar
[2]Bowen, R.. Entropy-expansive maps. Trans. Amer. Math. Soc. 164 (1972), 323–331.CrossRefGoogle Scholar
[3]Burton, R., Goulet, M. and Nitecki, Z.. Preimage entropy of semi-subshifts. In preparation.Google Scholar
[4]Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic Theory on Compact Spaces. Springer Lecture Notes in Mathematics 527. Springer: New York, 1976.CrossRefGoogle Scholar
[5]Ghys, E., Langevin, R. and Walczak, P.. Entropie géométrique des feuilletages. Acta. Math. 160 (1988), 105–142.CrossRefGoogle Scholar
[6]Misiurewicz, M.. Remark on the definition of topological entropy. pp. 65–67 of Dynamical Systems and Partial Differential Equations. Lara-Carrero, L. and Lewowicz, J., eds. Equinoccio: Caracas, 1986.Google Scholar
[7]Langevin, R. and Przytycki, F.. Entropie de l'image inverse d'une application. Bull. Soc. Math. France. 120 (1992), 237–250.Google Scholar
[8]Langevin, R. and Walczak, P.. Entropie d'une dynamique. C.R. Acad. Sci. Paris, 312 (1991), 141–144.Google Scholar
[9]Nitecki, Z.. Preimage entropy for maps, lecture at the Workshop on Dynamical Systems held at the Pennsylvania State University, 10 1992.Google Scholar
[10]Nitecki, Z. and Przytycki, F.. Preimage entropy of mappings. Preprint.CrossRefGoogle Scholar