Ruelle-Perron-Frobenius spectrum for Anosov maps.

*(English)*Zbl 1021.37015In what follows \(T\) denotes a \(C^3\) Anosov diffeomorphism defined on a smooth compact Riemannian manifolds \(\mathcal{M}\) and \(\mathcal{L}\) is the corresponding Ruelle-Perron-Frobenius operator.

The aim of the paper is to study some spectral properties of the operator \(\mathcal{L}\) when acting on a useful (from the dynamical point of view) Banach space \(B(\mathcal{M})\) (which strongly depends on the map \(T\)). Roughly speaking, \(B(\mathcal{M})\) consists of generalized functions more or less behaving like functions of bounded variation along the unstable manifolds of the Anosov map \(T\) and looking like signed measures, equipped with a weak topology, along the stable manifolds of \(T\). It turns out that \(\mathcal{L}:B(\mathcal{M})\rightarrow B(\mathcal{M})\) is quasi-compact, and from this some well-known results from the ergodic theory of Anosov diffeomorphisms (existence and properties of a Sinai-Bowen-Ruelle measure) follow quite straightforwardly.

Moreover, in the two-dimensional case it is shown that the Ruelle-Perron-Frobenius operators associated with certain smooth random perturbations of the map are close, in some sense, to the unperturbed operator. From this some spectral stability results follow, implying in particular stability of the invariant measure, of the rate of mixing and of the variance in the central limit theorem, and also spectral stability results for smooth deterministic perturbations.

Finally (still in dimension 2), an Ulam-like discretization scheme is used to reduce the computation of the spectral data of the operator to the study of a finite-dimensional matrix and then to obtain some quantitative information on the ergodic properties of \(T\).

We refer the reader to the paper itself for the precise, rather complicated statements, of the above-mentioned results. For the sake of clarity of exposition, proofs are only given in the case of multidimensional torus (with added comments explaining how to deal with the general case).

The aim of the paper is to study some spectral properties of the operator \(\mathcal{L}\) when acting on a useful (from the dynamical point of view) Banach space \(B(\mathcal{M})\) (which strongly depends on the map \(T\)). Roughly speaking, \(B(\mathcal{M})\) consists of generalized functions more or less behaving like functions of bounded variation along the unstable manifolds of the Anosov map \(T\) and looking like signed measures, equipped with a weak topology, along the stable manifolds of \(T\). It turns out that \(\mathcal{L}:B(\mathcal{M})\rightarrow B(\mathcal{M})\) is quasi-compact, and from this some well-known results from the ergodic theory of Anosov diffeomorphisms (existence and properties of a Sinai-Bowen-Ruelle measure) follow quite straightforwardly.

Moreover, in the two-dimensional case it is shown that the Ruelle-Perron-Frobenius operators associated with certain smooth random perturbations of the map are close, in some sense, to the unperturbed operator. From this some spectral stability results follow, implying in particular stability of the invariant measure, of the rate of mixing and of the variance in the central limit theorem, and also spectral stability results for smooth deterministic perturbations.

Finally (still in dimension 2), an Ulam-like discretization scheme is used to reduce the computation of the spectral data of the operator to the study of a finite-dimensional matrix and then to obtain some quantitative information on the ergodic properties of \(T\).

We refer the reader to the paper itself for the precise, rather complicated statements, of the above-mentioned results. For the sake of clarity of exposition, proofs are only given in the case of multidimensional torus (with added comments explaining how to deal with the general case).

Reviewer: Victor Jiménez López (Murcia)

##### MSC:

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

37A25 | Ergodicity, mixing, rates of mixing |

37M25 | Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) |

47A35 | Ergodic theory of linear operators |