# Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

@article{Arnoldi2015AsymptoticSG, title={Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps}, author={Jean‐François Arnoldi and F. Faure and Tobias Weich}, journal={Ergodic Theory and Dynamical Systems}, year={2015}, volume={37}, pages={1 - 58} }

We consider a simple model of an open partially expanding map. Its trapped set ${\mathcal{K}}$ in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component $\unicode[STIX]{x1D708}$ in the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call ‘minimal captivity’. This hypothesis…

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## References

SHOWING 1-10 OF 85 REFERENCES

Spectral problems in open quantum chaos

- Physics, Mathematics
- 2011

We present an overview of mathematical results and methods relevant for the spectral study of semiclassical Schrödinger (or wave) operators of scattering systems, in cases where the corresponding…

Ruelle?Perron?Frobenius spectrum for Anosov maps

- Mathematics
- 2002

We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle–Perron–Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows…

RUELLE-PERRON-FROBENIUS SPECTRUM FOR ANOSOV MAPS MICHAEL BLANK, GERHARD KELLER, AND CARLANGELO LIVERANI

- Mathematics
- 2002

We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle Perron Frobenius transfer operator to Anosov di eomorphisms on com pact manifolds This allows to…

Fractal Weyl law for skew extensions of expanding maps

- Mathematics
- 2012

We consider compact Lie group extensions of expanding maps of the circle, essentially restricting to SU(2) extensions. The main objective of the paper is the associated Ruelle transfer (or pull-back)…

Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces

- Mathematics
- 1999

The purpose of this paper is to show how the methods of Sjj ostrand for proving the geometric bounds for the density of resonances 28] apply to the case of convex co-compact hyperbolic surfaces. We…

On the thermodynamic formalism for the gauss map

- Mathematics
- 1990

AbstractWe study the generalized transfer operator ℒβf(z)=
$$\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{z + n}}} \right)^{2\beta } \times f\left( {1/\left( {z + n} \right)} \right)} $$
of the…

Distribution of Resonances for Open Quantum Maps

- Physics
- 2006

We analyze a simple model of quantum chaotic scattering system, namely the quantized open baker’s map. This model provides a numerical confirmation of the fractal Weyl law for the semiclassical…

Upper Bound on the Density of Ruelle Resonances for Anosov Flows

- Physics, Mathematics
- 2011

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper…

Calculating Hausdorff dimension of Julia sets and Kleinian limit sets

- Mathematics
- 2002

<abstract abstract-type="TeX"><p>We present a new algorithm for efficiently computing the Hausdorff dimension of sets <i>X</i> invariant under conformal expanding dynamical systems. By locating the…