Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Dynamical zeta functions for maps of the interval

Author: David Ruelle
Journal: Bull. Amer. Math. Soc. 30 (1994), 212-214
MSC (2000): Primary 58F20; Secondary 58F03
MathSciNet review: 1246470
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A dynamical zeta function $ \zeta $ and a transfer operator $ \mathcal{L}$ are associated with a piecewise monotone map $ f$ of the interval [0, 1] and a weight function $ g$. The analytic properties of $ \zeta $ and the spectral properties of $ \mathcal{L}$ are related by a theorem of Baladi and Keller under an assumption of "generating partition". It is shown here how to remove this assumption and, in particular, extend the theorem of Baladi and Keller to the case when $ f$ has negative Schwarzian derivative.

References [Enhancements On Off] (What's this?)

  • [1] V. Baladi and G. Keller, Zeta functions and transfer operators for piecewise monotone transformations, Commun. Math. Phys. 127 (1990), 459-477. MR 1040891 (91b:58196)
  • [2] V. Baladi and D. Ruelle, Some properties of zeta functions associated with maps in one dimension, in preparation.
  • [3] N. Haydn, Meromorphic extension of the zeta function for Axiom A flows, Ergodic Theory Dynamical Systems 10 (1990), 347-360. MR 1062762 (91g:58219)
  • [4] F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Related Fields 72 (1986), 359-386. MR 843500 (87k:58126)
  • [5] F. Hofbauer and G. Keller, Zeta-functions and transfer-operators for piecewise linear transformations, J. Reine Angew. Math. 352 (1984), 100-113. MR 758696 (87a:58097)
  • [6] G. Keller and T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Commun. Math. Phys. 149 (1992), 31-69. MR 1182410 (93i:58123)
  • [7] J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems, Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465-563. MR 970571 (90a:58083)
  • [8] D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), 231-242. MR 0420720 (54:8732)
  • [9] -, Analytic completion for dynamical zeta functions, Helv. Phys. Acta. 66 (1993), 181-191. MR 1218071 (94h:58141)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 58F20, 58F03

Retrieve articles in all journals with MSC (2000): 58F20, 58F03

Additional Information

Keywords: Zeta function, transfer operator, topological pressure, interval map
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society