In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems.

<jats:title>Abstract</jats:title><jats:p>The kinetic Brownian motion on the sphere bundle of a Riemannian manifold <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {M}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is a stochastic process that models a random perturbation of the geodesic flow. If <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {M}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>-spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.</jats:p>

<jats:title>Abstract</jats:title><jats:p>Given a closed orientable hyperbolic manifold of dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$\ne 3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≠</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numbers. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers. </jats:p>

<jats:p>We consider a simple model of an open partially expanding map. Its trapped set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline1" /><jats:tex-math>${\mathcal{K}}$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline2" /><jats:tex-math>$\unicode[STIX]{x1D708}$</jats:tex-math></jats:alternatives></jats:inline-formula> in 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 in a neighborhood of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline3" /><jats:tex-math>${\mathcal{K}}$</jats:tex-math></jats:alternatives></jats:inline-formula>. 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0143385715000346_inline4" /><jats:tex-math>$\unicode[STIX]{x1D708}\rightarrow \infty$</jats:tex-math></jats:alternatives></jats:inline-formula>. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.</jats:p>