Dynamical zeta functions for maps of the interval

Ruelle, David

doi:10.1090/S0273-0979-1994-00489-6

Dynamical zeta functions for maps of the interval
HTML articles powered by AMS MathViewer

by David Ruelle PDF

Bull. Amer. Math. Soc. 30 (1994), 212-214 Request permission

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

V. Baladi and G. Keller, Zeta functions and transfer operators for piecewise monotone transformations, Comm. Math. Phys. 127 (1990), no. 3, 459–477. MR 1040891

Some properties of zeta functions associated with maps in one dimension

Nicolai T. A. Haydn, Meromorphic extension of the zeta function for Axiom A flows, Ergodic Theory Dynam. Systems 10 (1990), no. 2, 347–360. MR 1062762, DOI 10.1017/S0143385700005587
Franz Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Relat. Fields 72 (1986), no. 3, 359–386. MR 843500, DOI 10.1007/BF00334191
Franz Hofbauer and Gerhard Keller, Zeta-functions and transfer-operators for piecewise linear transformations, J. Reine Angew. Math. 352 (1984), 100–113. MR 758696, DOI 10.1515/crll.1984.352.100
Gerhard Keller and Tomasz Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), no. 1, 31–69. MR 1182410
John Milnor and William Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563. MR 970571, DOI 10.1007/BFb0082847
David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), no. 3, 231–242. MR 420720, DOI 10.1007/BF01403069
David Ruelle, Analytic completion for dynamical zeta functions, Helv. Phys. Acta 66 (1993), no. 2, 181–191. MR 1218071