Academics
Minicourses
Transfer operators and spaces of distributions for hyperbolic dynamics
Nível: Mestrado / Doutorado
Programa:
For many years, transfer operators were associated to
hyperbolic dynamics by first considering a "quotient" expanding system. Due to
limited smoothness of the dynamical foliations (even if the map is very smooth)
this does not permit exploiting fully the regularity of the map.
After a
historical introduction (including a bibliography intended for beginners
entering the topic) and a motivation of the study of transfer operators, we
shall present one way [1] to introduce anisotropic spaces of distributions on
which transfer operator can studied without any need to introduce an expanding
quotient.
This approach is based on using cones in Fourier space. It yields
[1] good bounds on the essential spectral radius. More recently [2], it allowed
us to obtain a spectral interpretation of the zeroes of the dynamical
determinants. (This gives - in particular - a new proof of a difficult theorem
of Kitaev.) In the rest of the course, we shall present the results and proofs
of these two papers in a hopefully pedagogical way, giving in particular a
different proof of the result in [2].
Although
pseudodifferential operators are lurking in the background, the course will be
fully elementary. Prerequisites do not go beyond standard integration by parts,
and the Arzela-Ascoli theorem (it helps if auditors have already seen a
continuous Fourier transform).
Referências:
WITH M. TSUJII. - Anisotropic Hölder and Sobolev spaces
for hyperbolic diffeomorphisms, arxiv.org, 2005, to appear Ann. Inst.
Fourier
WITH M.
TSUJII. - Dynamical determinants and spectrum for hyperbolic diffeomorphisms,
arxiv.org, 2006, submitted for publication