Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-19T03:47:15.942Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral properties of a class of operators associated with maps in one dimension

Published online by Cambridge University Press:  19 September 2008

David Ruelle
David Ruelle
Affiliation:
Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France
Get access Rights & Permissions [Opens in a new window]

Abstract

Let f be a piecewise monotone map of the interval [0,1] to itself, and g a function of bounded variation on [0, 1]. Hofbauer, Keller and Rychlik have studied operators on functions of bounded variation, where

Among other things, they show that the essential spectral radius of is in many cases strictly smaller than the spectral radius; there exist therefore isolated eigenvalues of finite multiplicity. The purpose of the present paper is to prove similar results for a more general class of operators forming an algebra (and therefore containing sums of operators like ). An analogous extension was presented by Ruelle for operators associated with expanding maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 4 , December 1991 , pp. 757 - 767
Copyright
Copyright © Cambridge University Press 1991

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]Baladi, V. & Keller, G., Zeta functions and transfer operators for piecewise monotone transformations. Commun. Math. Phys. 127 (1990), 459–477.CrossRefGoogle Scholar
[2]Haydn, N.. Meromorphic extension of the zeta function for Axiom A flows. Ergod. Th. & Dynam. Sys. 10 (1990), 347–360.CrossRefGoogle Scholar
[3]Hofbauer, F. & Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.CrossRefGoogle Scholar
[4]Hofbauer, F. & Keller, G.. Zeta-functions and transfer-operators for piecewise linear transformations. J. reine angew. Math. 352 (1984), 100–113.Google Scholar
[5]Milnor, J. & Thurston, W.. On iterated maps of the interval I. The kneading matrix II. Periodic points. Preprint. Princeton University 1977.Google Scholar
[6]Nussbaum, R. D.. The radius of the esssntial spectrum. Duke Math. J. 37 (1970), 473–478.CrossRefGoogle Scholar
[7]Preston, C.. What you need to know to knead. Preprint.CrossRefGoogle Scholar
[8]Ruelle, D.. The thermodynamic formalism for expanding maps. Commun. Math. Phys. 125 (1989), 239–262.CrossRefGoogle Scholar
[9]Ruelle, D.. An extension of the theory of Fredholm determinants. Preprint.CrossRefGoogle Scholar
[10]Rychlik, M.. Bounded variation and invariant measures. Studia Math. 76 (1983), 69–80.CrossRefGoogle Scholar
[11]Tangerman, F.. Meromorphic continuation of Ruelle zeta functions. Boston University Thesis, 1986 (unpublished).Google Scholar