Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T03:31:26.890Z Has data issue: false hasContentIssue false
  • English
  • Français

An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps

Published online by Cambridge University Press:  19 September 2008

V. Baladi  and
D. Ruelle
V. Baladi
Affiliation:
CNRS, UMR 128, UMPA, ENS Lyon, 46, llée d'ltalie, F-69364 Lyon Cedex 07, France
D. Ruelle
Affiliation:
Institut des Hautes Études Scientifiques, F-91440 Bures-sur-Yvette, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor—Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 4 , December 1994 , pp. 621 - 632
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

[1]Baladi, V. and Keller, G.. Zeta functions and transfer operators for piecewise monotone transformations. Comm. Math. Phys. 127 (1990), 459–477.CrossRefGoogle Scholar
[2]Hofbauer, F. and Keller, G.. Zeta-functions and transfer-operators for piecewise linear transformations J. reine angew. Math. 352 (1984), 100–113.Google Scholar
[3]Keller, G. and Nowicki, T.. Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. Comm. Math. Phys. 149 (1992), 31–69.CrossRefGoogle Scholar
[4]Mayer, D.H.. The Ruelle—Araki Transfer Operator in Classical Statistical Mechanics. Springer Lecture Notes in Physics 123. Springer: Berlin, 1980.Google Scholar
[5]Milnor, J. and Thurston, W.. Iterated maps of the interval. Dynamical Systems (Maryland 1986–1987). Springer Lecture Notes in Mathematics 1342. ed. Alexander, J.C.. Springer: Berlin—Heidelberg—New York, 1988.Google Scholar
[6]Mori, M.. Fredholm determinant for piecewise linear transformations. Osaka J. Math. 27 (1990), 81–116.Google Scholar
[7]Mori, M.. Fredholm matrices and zeta functions for piecewise monotonic transformations. Dynamical Systems and Related Topics (Nagoya 1990) Adv. Ser. Dyn. Syst., 9. World Scientific: River Edge, NJ, 1991). pp. 388–400.Google Scholar
[8]Preston, C.. What you need to know to knead. Adv. Math. 78 (1989), 192–252.CrossRefGoogle Scholar
[9]Rand, D.. The topological class of Lorenz attractors. Math. Proc. Cambridge Philos. Soc. 83 (1978), 451–460.CrossRefGoogle Scholar
[10]Ruelle, D.. Analytic completion for dynamical zeta functions. Helv. Phys. Acta 66 (1993), 181–191.Google Scholar
[11]Williams, R.F.. The structure of Lorenz attractors. Inst. Hautes Etudes Sci. Publ. Math. 50 (1979), 73–100.CrossRefGoogle Scholar