An extension of the theory of Fredholm determinants
@article{Ruelle1989AnEO, title={An extension of the theory of Fredholm determinants}, author={David Ruelle}, journal={Publications Math{\'e}matiques de l'Institut des Hautes {\'E}tudes Scientifiques}, year={1989}, volume={72}, pages={175-193} }
Analytic functions are introduced, which are analogous to the Fredholm determinant, but may have only finite radius of convergence. These functions are associated with operators of the form ε μ(dω) ℒω, where ℒω φ(x) = ϕω(x). φ(ψω x), , φ belongs to a space of Hölder or Cr functions, ϕω is Hölder or Cr, and ψω is a contraction or a Cr contraction. The results obtained extend earlier results by Haydn, Pollicott, Tangerman and the author on zeta functions of expanding maps.
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References
SHOWING 1-10 OF 25 REFERENCES
The thermodynamic formalism for expanding maps
- Mathematics
- 1989
Letf:X↦X be an expanding map of a compact space (small distances are increased by a factor >1). A generating functionζ(z) is defined which countsf-periodic points with a weight. One can expressζ in…
Meromorphic extension of the zeta function for Axiom A flows
- MathematicsErgodic Theory and Dynamical Systems
- 1990
Abstract We prove the meromorphicity of the zeta function on shifts of finite type for Hölder continuous functions assuming that the essential spectrum of the associated Ruelle operator is contained…
Zeta functions and transfer operators for piecewise monotone transformations
- Mathematics
- 1990
Given a piecewise monotone transformationT of the interval and a piecewise continuous complex weight functiong of bounded variation, we prove that the Ruelle zeta function ζ(z) of (T, g) extends…
Zeta-functions for expanding maps and Anosov flows
- Mathematics
- 1976
Given a real-analytic expanding endomorphism of a compact manifoldM, a meromorphic zeta function is defined on the complex-valued real-analytic functions onM. A zeta function for Anosov flows is…
A complex Ruelle-Perron-Frobenius theorem and two counterexamples
- Mathematics
- 1984
Abstract In this paper a new proof of a theorem of Ruelle about real Perron-Frobenius type operators is given. This theorem is then extended to complex Perron-Frobenius type operators in analogy with…
The zeta functions of Ruelle and Selberg. I
- Mathematics
- 1986
On developpe les methodes de Ruelle et on obtient une grande classe de fonctions zeta mesomorphes d'interet geometrique. On obtient une formule pour le spectre lisse de certains flots en fonction des…
Produits Tensoriels Topologiques Et Espaces Nucleaires
- Mathematics
- 1966
© Association des collaborateurs de Nicolas Bourbaki, 1951-1954, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions…
La théorie de Fredholm
- Mathematics
- 1956
© Association des collaborateurs de Nicolas Bourbaki, 1954, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions…