scholarly journals Compactness of transfer operators and spectral representation of Ruelle zeta functions for super-continuous functions

2020 ◽  
Vol 40 (11) ◽  
pp. 6331-6350
Author(s):  
Katsukuni Nakagawa ◽  
Keyword(s):  
Spectral Representation ◽  
Zeta Functions ◽  
Continuous Functions ◽  
Transfer Operators

scholarly journals Dynamical multifractal zeta-functions and fine multifractal spectra of graph-directed self-conformal constructions

2015 ◽  
Vol 36 (6) ◽  
pp. 1922-1971 ◽  
Author(s):  
V. MIJOVIĆ ◽  
L. OLSEN
Keyword(s):  
General Class ◽  
Zeta Functions ◽  
Continuous Functions ◽  
Precise Information ◽  
Multifractal Spectra ◽  
Conformal Measures

We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.

scholarly journals Some Trapezoidal Vector Inequalities for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Spectral Representation ◽  
Continuous Functions ◽  
Selfadjoint Operators

On utilising the spectral representation of selfadjoint operators in Hilbert spaces, some trapezoidal inequalities for various classes of continuous functions of such operators are given.

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.

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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