Upper bound on the density of Ruelle resonances for Anosov flows. (English) Zbl 1260.37016
This article is a continuation of the approach initiated some years ago by the authors and collaborators in several papers [the first author and N. Roy, Nonlinearity 19, No. 6, 1233–1252 (2006; Zbl 1190.37024); the authors and N. Roy, Open Math. J. 1, 35–81 (2008; Zbl 1177.37032); the first author, Nonlinearity 24, No. 5, 1473–1498 (2011; Zbl 1233.37018)]. Namely, it deals with the use of microlocal analysis, usually confined to problems from quantum mechanics, to study a “classical” dynamical system given by a smooth map or a flow on a manifold \(X\).
The idea of using functional-analytical methods to study classical dynamical systems dates back to D. Ruelles’s work in the 70’s, who showed that, given a dynamical system \(\phi:X\rightarrow X\), the spectral properties of the composition operator (or an operator related to it) \(u\mapsto u\circ \phi\), acting on a suitable space of functions or distributions, provide a lot of statistical information about the system. In particular, the eigenvalues of this operator (called Ruelle resonances) are highly related to the decay of statistical correlation functions for large time.
These spectral methods have been extensively used and developed by many authors (V. Baladi, M. Blank, S. Gouëzel, G. Keller, K. Liverani, M. Tsujii, etc.). The originality of the approach developed by the authors of the paper under review relies on the fundamental remark that the the composition operator is a Fourier integral operator.
This fact allowed the authors to apply powerfull techniques from microlocal analysis to the study of different hyperbolic systems. In the paper under review, the authors consider an “Anosov flow” on a compact Riemannian manifold \(X\), i.e., the flow \(\phi_t\) of a smooth vector field \(V\) such that the tangent bundle \(TX\) admits a \(d\phi_t\)-invariant splitting \(TX=E_s\oplus E_u\oplus E_0\) into stable, unstable and neutral subbundles \(E_s\), \(E_u\), and \(E_0\) respectively, with the property that the neutral subbundle \(E_0(x)\) is spanned by \(V(x)\) at each point \(x\in X\).
The authors proves mainly three theorems. The first two results state that the differential operator \((-i)V\) acting on a suitable “anisotropic Sobolev space” of functions on \(X\) has, close to the real axis, a discrete spectrum consisting of eigenvalues with finite multiplicities (the Ruelle resonances). This result is very close to the one in [O. Butterley and C. Liverani, J. Mod. Dyn. 1, No. 2, 301–322 (2007; Zbl 1144.37011)]. The main result of the paper is an upper bound for the density of these resonances close to the real axis and for large real parts.
The idea of using functional-analytical methods to study classical dynamical systems dates back to D. Ruelles’s work in the 70’s, who showed that, given a dynamical system \(\phi:X\rightarrow X\), the spectral properties of the composition operator (or an operator related to it) \(u\mapsto u\circ \phi\), acting on a suitable space of functions or distributions, provide a lot of statistical information about the system. In particular, the eigenvalues of this operator (called Ruelle resonances) are highly related to the decay of statistical correlation functions for large time.
These spectral methods have been extensively used and developed by many authors (V. Baladi, M. Blank, S. Gouëzel, G. Keller, K. Liverani, M. Tsujii, etc.). The originality of the approach developed by the authors of the paper under review relies on the fundamental remark that the the composition operator is a Fourier integral operator.
This fact allowed the authors to apply powerfull techniques from microlocal analysis to the study of different hyperbolic systems. In the paper under review, the authors consider an “Anosov flow” on a compact Riemannian manifold \(X\), i.e., the flow \(\phi_t\) of a smooth vector field \(V\) such that the tangent bundle \(TX\) admits a \(d\phi_t\)-invariant splitting \(TX=E_s\oplus E_u\oplus E_0\) into stable, unstable and neutral subbundles \(E_s\), \(E_u\), and \(E_0\) respectively, with the property that the neutral subbundle \(E_0(x)\) is spanned by \(V(x)\) at each point \(x\in X\).
The authors proves mainly three theorems. The first two results state that the differential operator \((-i)V\) acting on a suitable “anisotropic Sobolev space” of functions on \(X\) has, close to the real axis, a discrete spectrum consisting of eigenvalues with finite multiplicities (the Ruelle resonances). This result is very close to the one in [O. Butterley and C. Liverani, J. Mod. Dyn. 1, No. 2, 301–322 (2007; Zbl 1144.37011)]. The main result of the paper is an upper bound for the density of these resonances close to the real axis and for large real parts.
Reviewer: Nicolas Roy (Berlin)
MSC:
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 47G30 | Pseudodifferential operators |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 35S30 | Fourier integral operators applied to PDEs |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
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