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Periodic points of Ruelle-expanding maps  [PDF]
Maria Carvalho,Mário Alexandre Magalh?es
Mathematics , 2012,
Abstract: We prove that, for a Ruelle-expanding map, the zeta function is rational and the topological entropy is equal to the exponential growth rate of the periodic points.
A product formula of Mumford forms, and the rationality of Ruelle zeta values for Schottky groups  [PDF]
Takashi Ichikawa
Mathematics , 2014,
Abstract: In any positive genus case, we show an explicit formula of the Mumford forms expressed by infinite products like Selberg type zeta values for Schottky groups. This result is considered as an extension of the formula in terms of Ramanujan's Delta function in the genus 1 case, and is especially applied to studying the rationality of Ruelle zeta values for Schottky groups.
Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps  [PDF]
Jean-Fran?ois Arnoldi,Frédéric Faure,Tobias Weich
Mathematics , 2013, DOI: 10.1017/etds.2015.34
Abstract: We consider a simple model of an open partially expanding map. Its trapped set K in phase space is a fractal set. We first show that there is a well defined discrete spectrum of Ruelle resonances which describes the asymptotics of correlation functions for large time and which is parametrized by the Fourier component \nu on the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call "minimal captivity". This hypothesis is stable under perturbations and means that the dynamics is univalued on a neighborhood of K. Under this hypothesis we show the existence of an asymptotic spectral gap and a Fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit \nu -> infinity. Some numerical computations with the truncated Gauss map illustrate these results.
On Ruelle's Lemma and Ruelle Zeta Functions  [PDF]
Paul Wright
Mathematics , 2010,
Abstract: In this article we prove an important inequality regarding the Ruelle operator in hyperbolic flows. This was already proven briefly by Mark Pollicott and Richard Sharp in a low dimensional case, but we present a clearer proof of the inequality, filling in gaps and explaining the ideas in more detail, and extend the inequality to higher dimensional flows. This inequality is necessary to prove a proposition about the analyticity of Ruelle zeta functions.
Rationality of partial zeta functions  [PDF]
Daqing Wan
Mathematics , 2003,
Abstract: We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.
Finite rank approximations of expanding maps with neutral singularities  [PDF]
Michael Blank
Mathematics , 2007,
Abstract: For a class of expanding maps with neutral singularities we prove the validity of a finite rank approximation scheme for the analysis of Sinai-Ruelle-Bowen measures. Earlier results of this sort were known only in the case of hyperbolic systems.
Sinai billiards, Ruelle zeta-functions and Ruelle resonances: microwave experiments  [PDF]
S. Sridhar,W. T. Lu
Statistics , 2002,
Abstract: We discuss the impact of recent developments in the theory of chaotic dynamical systems, particularly the results of Sinai and Ruelle, on microwave experiments designed to study quantum chaos. The properties of closed Sinai billiard microwave cavities are discussed in terms of universal predictions from random matrix theory, as well as periodic orbit contributions which manifest as `scars' in eigenfunctions. The semiclassical and classical Ruelle zeta-functions lead to quantum and classical resonances, both of which are observed in microwave experiments on n-disk hyperbolic billiards.
The leading Ruelle resonances of chaotic maps  [PDF]
Galya Blum,Oded Agam
Physics , 2000, DOI: 10.1103/PhysRevE.62.1977
Abstract: The leading Ruelle resonances of typical chaotic maps, the perturbed cat map and the standard map, are calculated by variation. It is found that, excluding the resonance associated with the invariant density, the next subleading resonances are, approximately, the roots of the equation $z^4=\gamma$, where $\gamma$ is a positive number which characterizes the amount of stochasticity of the map. The results are verified by numerical computations, and the implications to the form factor of the corresponding quantum maps are discussed.
Linear response formula for piecewise expanding unimodal maps  [PDF]
V. Baladi,D. Smania
Mathematics , 2007, DOI: 10.1088/0951-7715/21/4/003
Abstract: The average R(t) of a smooth function with respect to the SRB measure of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz. We prove that if f_t is tangent to the topological class of f_0, then R(t) is differentiable at zero, and the derivative coincides with the resummation previously proposed by the first named author of the (a priori divergent) series given by Ruelle's conjecture.
Fractal Weyl law for skew extensions of expanding maps  [PDF]
Jean-Fran?ois Arnoldi
Mathematics , 2011,
Abstract: We consider compact Lie groups extensions of expanding maps of the circle, essentially restricting to U(1) and SU(2) extensions. The central object of the paper is the associated Ruelle transfer (or pull-back) operator $\hat{F}$. Harmonic analysis yields a natural decomposition $\hat{F}=\oplus\hat{F}_{\alpha}$, where $\alpha$ indexes the irreducible representation spaces. Using Semiclassical techniques we extend a previous result by Faure proving an asymptotic spectral gap for the family ${\hat{F}_{\alpha}}$ when restricted to adapted spaces of distributions. Our main result is a fractal Weyl upper bound for the number of eigenvalues of these operators (the Ruelle resonances) out of some fixed disc centered on 0 in the complex plane.
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