scholarly journals Zeta functions and transfer operators for piecewise monotone transformations

1990 ◽  
Vol 127 (3) ◽  
pp. 459-477 ◽  
Author(s):  
V. Baladi ◽  
G. Keller
Keyword(s):  
Zeta Functions ◽  
Piecewise Monotone ◽  
Transfer Operators

scholarly journals A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

2018 ◽  
Vol 40 (3) ◽  
pp. 612-662
Author(s):  
ALEXANDER ADAM ◽  
ANKE POHL
Keyword(s):  
Transfer Operator ◽  
Zeta Functions ◽  
Cusp Forms ◽  
Dimensional Representation ◽  
Triangle Group ◽  
Selberg Zeta Function ◽  
One Dimensional ◽  
Transfer Operators ◽  
Finite Dimensional ◽  
Automorphic Functions

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

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

1994 ◽  
Vol 14 (4) ◽  
pp. 621-632 ◽  
Author(s):  
V. Baladi ◽  
D. Ruelle
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Constant Weight ◽  
Piecewise Constant ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Interval Map ◽  
Piecewise Continuous

AbstractWe 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.

Periodic orbits and dynamical spectra (Survey)

1998 ◽  
Vol 18 (2) ◽  
pp. 255-292 ◽  
Author(s):  
VIVIANE BALADI
Keyword(s):  
Banach Spaces ◽  
Periodic Orbits ◽  
Spectral Properties ◽  
Zeta Functions ◽  
Statistical Properties ◽  
Fredholm Determinants ◽  
Rigorous Theory ◽  
Transfer Operators ◽  
Dynamical Spectra

Basic results in the rigorous theory of weighted dynamical zeta functions or dynamically defined generalized Fredholm determinants are presented. Analytic properties of the zeta functions or determinants are related to statistical properties of the dynamics via spectral properties of dynamical transfer operators, acting on Banach spaces of observables.

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