We study the spectrum of transfer operators associated to various dynamical systems. Our aim is to obtain precise information on the discrete spectrum. To this end we propose a unitary approach. We consider various settings where new information can be obtained following different branches along the proposed path. These settings include affine expanding Markov maps, uniformly expanding Markov maps, non-uniformly expanding or simply monotone maps, hyperbolic diffeomorphisms. We believe this approach could be greatly generalized.
We show that all generalized Pollicott-Ruelle resonant states of a topologically transitiv $C^\infty$-Anosov flow with an arbitrary $C^\infty$ potential, have full support.
School of Mathematics and Physics, University of Queensland（昆士兰大学数学与物理学院）；School of Mathematics and Statistics, University of New South Wales Sydney（新南威尔士大学悉尼数学与统计学院）
We define Pollicott-Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, as well as for more general open hyperbolic systems related to Axiom A flows. These resonances are the poles of the meromorphic continuation of the resolvent of the generator of the flow and they describe decay of classical correlations. As an application, we show that the Ruelle zeta function extends meromorphically to the entire complex plane.
We present new methods to construct a Ruelle-Pollicott spectrum for the geodesic flow on manifolds with strictly negative curvature and a finite number of hyperbolic cusps.
Pollicott-Ruelle resonances for chaotic flows are the characteristic frequencies of correlations. They are typically defined as eigenvalues of the generator of the flow acting on specially designed functional spaces. We show that these resonances can be computed as viscosity limits of eigenvalues of second order elliptic operators. These eigenvalues are the characteristic frequencies of correlations for a stochastically perturbed flow.
Given a closed orientable hyperbolic manifold of dimension $\neq 3$ 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 numb…
The kinetic Brownian motion on the cosphere bundle of a Riemannian manifold $\mathbb{M}$ is a stochastic process that models the geodesic equation perturbed by a random white force of size $\varepsilon$. When $\mathbb{M}$ is compact with negative curvature we show that the $L^2$-spectrum of the infinitesimal generator of this process converges to the Pollicott--Ruelle resonances of $\mathbb{M}$ as $\varepsilon \rightarrow 0$.
We give a sharp polynomial bound on the number of Pollicott-Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure-Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer-Sjöstrand in scattering theory and by Faure-Sjöstrand in the theory of Anosov flows.
We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey--Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family o…
We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.
The response of a low-frequency mode of climate variability, El Ni{ñ}o-Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane-Zebiak model [ZC87], from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances , the reduced Ruelle-Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contri…
For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_τ$ over the unit cosphere bundle $S^\ast\mathcal M$, we give a precise description of those classical (Pollicott-Ruelle) resonant states on $\mathbf V_τ$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^\ast\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,σ)$ on compatible associated…
A theory of Ruelle-Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal's oscillatory components manifested by pea…
If X is a contact Anosov vector field on a smooth compact manifold M and V is a smooth function on M, it is known that the differential operator A=-X+V has some discrete spectrum called Ruelle-Pollicott resonances in specific Sobolev spaces. We show that for |Im(z)| large the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to |Im(z)|. In each isolated band the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate of the vertical line Re(z)=< D >, the space…
This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^δ)$ in strips $|\Re s| \leq K$, where $δ$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott-Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\{|\Re s | \leq |\Im s|^α\}$ is given, followed by weaker lower bound estimates in strips $\{\Re s > -C, |\Im s|\leq r\}$, and logarithmi…
We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.
