Motivation: the motivation to study these topics came after the study of quantum chaos (see below) that consists of quantization of some “chaotic classical dynamics”. In quantum chaos, the quantization is not unique in general (except for some algebraic dynamics as the linear cat map or geodesic flow on constant negative surfaces) so the questions of quantum chaos may seem to be not well defined. By studying directly the pull back operator of dynamical systems, the surprise has been to observe a “natural quantization” (and so unique). The first step is to consider the “Ruelle discrete spectrum” of the pull back operator.
“Ruelle discrete spectrum”: In the paper (2.) we simply consider expanding map on
and hyperbolic map on
, introducing the technics of “escape function” to construct “anisotropic Sobolev spaces” and reveal the “Ruelle discrete spectrum”. An other very simple “matrix model” that explains this approach is given in the appendix of paper (14.). In (3.), we show how to use microlocal analysis to extend this approach to general Anosov diffeomorphisms and to general Anosov flows in (5.). In (14.) we present a similar approach using a sligthly different technics of microlocal analysis (wave-packets transform), that is more geometric and allows to improve the escape function, and finally improve some bounds on the resolvent operator.
“Natural quantization”, “Emergence of quantum dynamics in contact Anosov dynamics”: In the paper (1.) we study the Ruelle spectrum of the contact extension of a linear symplectic Anosov map on
(also called prequantum cat map). This is a partially hyperbolic dynamical systems. We show that the long time behavior is governed by an effective operator isomorphic to the “quantum cat map”, i.e. quantum dynamics appears dynamically without an “ad hoc” quantization procedure as usual. In spectral terms, the Ruelle spectrum contains the quantum spectrum as an external band (or annulus). This phenomena exists more generally for the contact extension of any symplectic Anosov diffeomorphism (6.). Under some assumptions, this extension exists and is almost unique so intrinsic. This can also be done for contact Anosov flows 16. , announced in (7.). In (9.) we study a contact Anosov flow acting on a special line bundle
called “semiclassical bundle”, and shows that the first band of eigenvalues concentrates on the imaginary axis and are the zeroes of “the semiclassical zeta function of Gutzwiller-Voros”. This “semiclassical” zeta function somehow generalizes the Selberg zeta function (that is for constant negative curvature) to variable negative curvature and more general contact Anosov flows. In (10.) we consider the special case of hyperbolic manifolds (these are dynamical systems constructed from Lie groups as the geodesic flow on surfaces of constant negative curvature but in higher dimensions) and it is observed that the Ruelle spectrum is on vertical lines, using group theory. In (13.) we consider 3 dimensional contact Anosov flow and a different argument to show the band spectrum.
“Partially hyperbolic dynamics”. The Ruelle spectrum of a simple model that is partially hyperbolic is studied in 4., i.e. there is a neutral direction for the dynamics. We use microlocal analysis and shows the existence of a spectral gap (so exponential decay of correlations) under generic assumptions. A similar study for open partially hyperbolic dynamics is done in (8.) and (11.). It is observed that neutral directions in general implies the existence of larger and larger number of Ruelle eigenvalues in the limit where the frequency in the neutral direction is high (called semiclassical limit). In the paper 12. we study the asymptotic spectral gap and try to improve the estimates using considerations after the Ehrenfest time..
“Pseudo-Anosov dynamics”. In (15.) we study the Ruelle spectrum of linear pseudo Anosov map, i.e. there are singularities on the surface that are branching points.
2013. "Semiclassical approach for the Ruelle-Pollicott spectrum of hyperbolic dynamics". May 2013. Lectures notes for summer school 13-17 May 2013 at ROMA.
2013. "Mathematical relations between Deterministic classical Chaos and Quantum Chaos via Ruelle resonances". Slides, for S.F.P. meeting, Marseille, july 2013. Movies related to this talk.
2018.“Spectrum, traces and zeta functions in hyperbolic dynamics”. Lectures notes for the school 23-27 April 2018 at the University Cheikh Anta Diop in Dakar, Sénégal. Movies related to the notes.
2019.“From classical chaos to quantum chaos”. Lectures notes for the school 22-26 April 2019 at CIRM.
2019.Slides and Video on Youtube “Emergence of quantum dynamics in hyperbolic dynamics”. may 2019.
2019.“Micro-local analysis of hyperbolic dynamics”. Slides for the school 1-5 july 2019 at CIRM.
2019.“Micro-local analysis of hyperbolic dynamics”. Video and slides, october 18th, 2019 at MSRI.
2024.“From geodesic flow to wave dynamics on an Anosov manifold”. slides,video, february 2th, 2024 at collège de france, Paris.
The paper 1. study waves in a wave guide in the semiclassical limit of small wave lenghts. The conductivity of the wave guide (i.e. number of propagating modes) is related to ergodic properties of the rays in the guide.
In papers 2. and 3. we study the eigenfunctions of the “quantum cat map”, i.e. the quantization of a hyperbolic linear symplectic map on the torus
. The surprise is to observe that, under some specific conditions, some eigenfunctions do not equidistribute as we could expect (and as it has been conjectured before under the name Quantum Unique Ergodicity, QUE). This shows the possible existence of “scars” in some uniform hyperbolic system (the only known example up to now) and gives a counter example to the “quantum unique ergodicity conjecture, QUE” in quantum chaos. One interesting feature in this model is the periodic behavior of wave packets: they spend some time to spread along the unstable direction of the dynamics and after some while they surprisingly reconstruct along the stable direction. The period
of this phenomena is always larger than twice the Ehrenfest time,
, i.e. the regime where still mysterious interference phenomena are present in quantum chaos. Here are some movies.
In paper (4.) we try to study long time behavior of wave packets in quantum chaos, i.e. long after the Ehrenfest time. This study has been started with the the “trace formula”. After this paper we have been convinced to consider first the dynamics of pull back operators (Ruelle spectrum, previous section) before continuing quantum chaos.
2012. "Ergodicity of quantum maps". Manuscript lectures notes given in Marburg, september 2012. pdf file.
2005. "Long time semiclassical evolution of wave packets in quantum chaos. Example of non quantum unique ergodicity with hyperbolic maps" Lecture notes for I.H.P. School, june 2005. format ps.gz ou pdf
2014. “Introduction au chaos classique et quantique”(Notes d’exposé pour les journées mathématiques (journées X-UPS) à l’intention des professeurs de math de classes préparatoires, lundi 28 et mardi 29 avril 2014.)
2024. Slides of “Some aspects of geometric quantization and quantum chaos”.
2024. Slides of “Equidistribution in classical and quantum chaos”.
2022. “Gravitational lens effect revisited through membrane waves”, Stefan Catheline, Victor Delattre, Gabrielle Laloy-Borgna, F. Faure, and Mathias Fink, American Journal of Physics 90, 47 (2022).
3.2 Lecture notes
1993. "Approche géométrique de la limite semi-classique par les états cohérents", Exposé à grenoble 1993.
1997. "La quantification du champ électromagnétique" Exposé le juin 1997 à l’institut Fourier. (fichier ps).
2000. "La quantification géométrique" Exposé le 27 avril 2000 à l’institut Fourier.
2003. "Introduction à la mécanique quantique et au paradoxe E.P.R" (exposé au magistère de mathématiques, UJF, janvier 2003).
2017. "Exposé sur le théorème adiabatique en mécanique quantique. Description par l’analyse semiclassique". Exposé le 5 mai 2017 à l’institut Fourier.
A simple model with a vector bundle over
: The first appearance of this phenomena explained in (1.,2.,3.) was in the quantum spectrum of small molecules as
. This a simple model where the dynamics of the molecule is described by a bundle of
matrices over the symplectic sphere
. The eigenspaces form a rank 1 vector bundle over
with Chern index
and the above index formula predicts (in confirmation we experimental observations) that the number of eigenvalues in a cluster is
where
is the angular momentum of the molecule.
A model with a rank 2 vector bundle over
and surprising topological phenomena: In paper 4. we describe a realistic and more rich situation where the dynamics of the molecule is described by a rank 2 vector bundle over
. In that case the Chern class is polynomial with integer coefficients and algebraic topology gives more subtil rules that are discussed, as topologically coupled bands, when the polynomial can not be factorized in the ring of polynomials with integer coefficients.
Pedagogical presentation and manifestation of the Euclidean index formula in geophysics and quantum physics: In paper 6. we describe a similar phenomena that has been discovered in a different field of physics, namely geophysical waves (atmospheric or ocean surface waves near the equator). We also give a more pedagogical presentation (in the Arxiv version), and treat a very simple model in any dimension related to the Euclidean index formula of Fedosov-Hörmander.
2007. "Adiabatically coupled systems and fractional monodromy" M. Hansen, F. Faure, B. Zhilinskii, J. Phys. A: Math. Theor. 40 (2007) 13075-13089. Paper of preprint
1994. "Generic description of the degeneracies in Harper like models." F. Faure, Journal of Physics A: Math. and gen. 27, (1994), 7519-7532. Paper.
2000. "Semi-classical Quantum Hall conductivity" F. Faure, B.Parisse, Journal of Mathematical physics, 41, 62-75, (2000). Paper or preprint
2000. "Topological properties of quantum periodic Hamiltonians" F. Faure, Journal of physics A: math and general, 33 , 531-555 (2000). Paper or preprint.
5.2 Lecture notes
1992. "Structure of wave functions on the torus characterized by a topological Chern index" in Proceedings to workshop in Trieste 1992. Paper.
2000. "Rôle des indices topologiques de Chern en physique du solide et physique moléculaire. Calculs dans des modèles simples." Notes de cours à l’école d’été interdisciplinaire MÉTHODES TOPOLOGIQUES ET GÉOMÉTRIQUES: APPLICATION AUX SYSTÈMES DYNAMIQUES PHYSIQUE, CHIMIE, BIOLOGIE. Dijon, 26-30 juin 2000. (fichier ps).
2002. "Geometric and topological aspects of slow and fast coupled dynamical systems in quantum and classical dynamics". Lectures notes for lectures given in:Saclay. Spth., march-april 2002, and M.A.S.I.E. Spring School, Warwick, march 2002. Lecture notes: format ps.gz ou pdf (or 2 pages / page: format ps.gz ou pdf) or html.
2003. "Topological indices in molecular spectra". Exposé en avril 2003 au Mathematical Science Research Institute (MSRI) Berkeley Vidéo et PDF.