Nicolas Roy's Curriculum Vita - Institut für Mathematik - Humboldt ...

periodicorbits. This1-formadmitsanintegralexpressioninvolvingonlytheflowof H0,but
which is in general divergent. We give several ways to treat this divergence problem and
we explain whyit doesapparentlynot appearin the above-mentionned particular models.
[5] Asemi-classicalK.A.M. theorem
We consider a semiclassical completely integrable system defined by a pseudodifferential
operator ˆH, with small parameter ¯h, on the torus T d , whose total symbol is a classicaly
completely integrable Hamiltonian H ∈ C ∞ T ∗ T d . We study perturbed operators of the
form ˆH + ¯h κ ˆK, where ˆK is any pseudodifferential operator and κ > 0, and we show the
existence ofsemiclassical normal forms forthese operators. Thisisused toconstruct a large
number of quasimodes, in analogy with the K.A.M theorem in classical mechanics which
proves the persistency of a large number of invariant tori after perturbation. Moreover, the
first correction to the eigenvalues is related to the average of the symbol of the perturbation
ˆK. In order to prove these result, one needs to perform a very precise analysis of the
resonances and the diophantine tori with parameters dependingon ¯h.
[4] Ruelle-Pollicottresonancesfor realanalytic hyperbolicmaps
Weconsider two simplemodelsof uniformly hyperbolic dynamical systems : expansive
maponthecircleS 1 andhyperbolicmaponthetorus T 2 ,andwestudythedecayoftimecorrelation
functions, which isfundamental to establish other chaotic properties of the system.
To achieve this, we study the Ruelle transfert operator ˆM defined as the pull-back operator
ˆM (ϕ) = ϕ ◦ M, where M is a chaotic real analytic map on S 1 (resp. T 2 ) and ϕ ∈ L 2 S 1
(resp. ϕ ∈ L 2 T 2 ). We show that the Ruelle-Pollicott resonances, which describe the time
correlation functions Cϕ,φ (n) = ϕ. ˆM n (φ) dx, with n ∈ N, can be obtained as the eigenvalues
of a trace class operator on L 2 S 1 (resp. L 2 T 2 ). We show moreover that these
resonances accumulate exponentiallyfast on 0.
[6] ASemi-classicalapproachfor Anosov diffeomorphismsand Ruelle resonances
If f is an Anosov diffeomorphism on a compact manifold, the decay of the dynamical
correlation functions is governed by the so-called Ruelle resonances. It follows from the
works of Baladi & al and Liverani & al, that these resonances can be obtained by a suitable
spectral analysisofthecomposition operator (oranotherone relatedto it)calledthe "Transfer
operator". In this paper, we show how these results can be obtained by a systematic
microlocal analysis, extending the approach of the previous work [4].
[7] Thegeometryofthe spaceof fibrations
We study geometrical aspects of the space of fibrations between two given manifolds
M and B, from the point of view of Frechet geometry. As a first result, we show that any
connected component of this space is the base space of a Frechet-smooth principal bundle
with the identity component of the group of diffeomorphisms of M as total space. Second,
weprovethatthespaceoffibrationsisalsoitselfthetotalspaceofasmoothFrechetprincipal
bundle with structure group the group of diffeomorphisms of the base B.
6