Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-03T00:13:38.136Z Has data issue: false hasContentIssue false
  • English
  • Français

Entropy, transverse entropy and partitions of unity

Published online by Cambridge University Press:  19 September 2008

Rémi Langevin  and
Pawel G. Walczak
Rémi Langevin
Affiliation:
Université de Bourgogne, Laboratoire de Topologie, BP 138, 21004 Dijon, France
Pawel G. Walczak
Affiliation:
Uniwersytet Lódzki, Instytut Matematyki, ul. Banacha 22, 90–238 Lódź, Poland
Get access Rights & Permissions [Opens in a new window]

Abstract

The topological entropy of a transformation is expressed in terms of partitions of unity. The transverse entropy of a flow tangential to a foliation is defined and expresed in a similar way. The geometric entropy of a foliation of a Riemannian manifold is compared with the transverse entropy of its geodesic flow.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 3 , September 1994 , pp. 551 - 563
Copyright
Copyright © Cambridge University Press 1994

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

[AKM]Adler, R.L., Konheim, A.G. and McAndrew, M.H.. Topological entropy Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
[Bo]Bowen, R.E.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
[F]Frings, H.. Generalized entropy for foliations. Thesis, Düsseldorf (1991).Google Scholar
[GLW1]Ghys, E., Langevin, R. and Walczak, P.G.. Entropie mesurée et partitions de l'unité. C. R. Acad. Sci. Paris, Sér. I, 303 (1986), 251–254.Google Scholar
[GLW2]Ghys, E., Langevin, R. and Walczak, P.G.. Entropie géométrique des feuilletages. Acta Math. 160 (1988), 105–142.CrossRefGoogle Scholar
[Gr]Gromov, M.. Entropy, homology and semialgebraic geometry (after Y. Yomdin). In: Séminaire Bourbaki 1985–86. No. 663.Google Scholar
[HH]Hector, G. and Hirsch, U.. Introduction to the Geometry of Foliations. Parts A and B. Vieweg & Sohn: Braunschweig-Wiesbaden, 1981 and 1983.CrossRefGoogle Scholar
[Hu]Hurder, S.. Ergodic theory of foliations and a theorem of Sacksteder. Dynamical Systems. Proc. Special Year at Univ. of Maryland, 1986/87. Springer Lecture Notes in Mathematics, vol. 1342. Springer: Berlin-Heidelberg—New York, 1988. 291–328.Google Scholar
[K]Klingenberg, W.. Riemannian Geometry. Walter de Gruyter: 1982.Google Scholar
[LWa]Langevin, R. and Walczak, P.G.. Entropie d'une dynamique. C. R. Acad. Sci. Paris, Sér. I 312 (1991), 141–144.Google Scholar
[LWi]Langevin, R. and Wirtz, B.. Functional definitions of entropy. Preprint. Dijon: 1994.Google Scholar
[Wai]Walczak, P.G.. A relation between dynamics and geometry of foliations. Unpublished.Google Scholar
[Wa2]Walczak, P.G.. Dynamics of the geodesic flow of a foliation. Ergod. Th. & Dynam. Sys. 8 (1988), 637–650.CrossRefGoogle Scholar
[W]Walters, P.. An Introduction to Ergodic Theory. Springer: Berlin-Heidelberg—New York, 1982.CrossRefGoogle Scholar