Publications / Research

Frédéric Faure
Maitre de conférence en physique, Université Grenoble Alpes.

Table of Contents


1 Hyperbolic dynamical systems, Classical chaos

1.1 Introduction

This short text presents the list of papers given below, following a thematic description. This series of papers concern the hyperbolic dynamical systems, i.e. “chaotic dynamics” with high sensitivity to initial conditions, also called more precisely “Anosov dynamics”. The point of view here (following David Ruelle, R. Bowen and others), is to consider not evolution of individual trajectories that look like random, but the evolution of a probability distribution that appears to be more predictable. The tools are spectral analysis of the linear pull back operator: the flow (or the map) acting on functions (or sections of bundles) composition. Since chaotic dynamics generates oscillations with high frequencies that get larger and larger with time, it appears necessary to use microlocal analysis and symplectic geometry on cotangent bundle to study these operators.

1.2 Publications

  1. 2006. "Prequantum chaos: Resonances of the prequantum cat map", F. Faure, Journal of Modern Dynamics, Vol.1 No.2, 255-285, (2007), Paper, preprint 2006.
  2. 2006. "Ruelle-Pollicott resonances for real analytic hyperbolic map", F. Faure and N. Roy, Nonlinearity Vol. 19, 1233-1252, (2006), Paper, or preprint 2006
  3. 2008. "Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances" F. Faure, N. Roy and J. Sjöstrand, Open Math. Journal, vol. 1, 35--81, (2008). Paper, or preprint 2008
  4. 2009. "Semiclassical origin of the spectral gap for transfer operators of partially expanding map" F. Faure. Nonlinearity Vol. 24, 1473-1498, (2011), Paper, or preprint 2009. Movies related to this paper.
  5. 2010. "Upper bound on the density of Ruelle resonances for Anosov flows" F. Faure and J. Sjöstrand, Comm. in Math. Physics, vol. 308, 325-364, (2011) Paper. or preprint 2010
  6. 2012. "Prequantum transfer operator for Anosov diffeomorphism" F. Faure and M. Tsujii. Astérisque 375 (2015). preprint 2012. article.
  7. 2013. "Band structure of the Ruelle spectrum of contact Anosov flows" F. Faure and M. Tsujii. Comptes rendus - Mathématique 351 , 385-391, (2013). paper, preprint 2013.
  8. 2013. "Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps", J.F. Arnoldi, F.Faure, T. Weich. Ergodic Theory of Dynamical Systems (2015). preprint 2013
  9. 2013. "The semiclassical zeta function for geodesic flows on negatively curved manifolds", F. Faure and M. Tsujii, Invent. math. (2017) 208: 851. paper. preprint 2013
  10. 2014. "Power spectrum of the geodesic flow on hyperbolic manifolds", S. Dyatlov, F. Faure, C. Guillarmou, Analysis & PDE 8 (2015), 923–1000. paper, preprint 2014
  11. 2014. "Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum" S. Barkhofen, F. Faure, T. Weich, Nonlinearity Vol. 27, 1829, (2014), preprint 2014
  12. 2015. "Asymptotic spectral gap for open partially expanding maps" F.Faure, T. Weich. Communications in Mathematical Physics 2017, 356(3), 755-822. Paper. preprint 2015
  13. 2017. "Horocyclic invariance of Ruelle resonant states for contact Anosov flows in dimension 3" C. Guillarmou, F.Faure Math. Res. Lett., Volume 25, Number 5, 1405–1427, 2018, paper,preprint 2017.
  14. 2017. "Fractal Weyl law for the Ruelle spectrum of Anosov flows", F. Faure and M. Tsujii. Annales Henri Lebesgue 6 (2023) 331-426, Paper, preprint 2017. Slides. Video talk.
  15. 2018. "Ruelle spectrum of linear pseudo-Anosov maps”, F. Faure, S. Gouëzel, E. Lanneau, Journal de l’École polytechnique — Mathématiques, Tome 6 (2019) , pp. 811-877. paper, preprint 2018. talk on Youtube by Sebastien Gouezel at I.C.M. 2018.
  16. 2021. "Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum", F. Faure and M. Tsujii. preprint 2021. Slides (long). Slides (short, ICMP 2021). Video talk.
    Slides, Roscoff March-2022.

1.3 Lecture notes and slides


2 Quantum chaos

2.1 Introduction

This short text presents the list of papers given below, following a thematic description. This series of papers concern “quantum chaos”, this means the dynamics of a linear PDE for which, in the limit of small wave length h1 , called semiclassical limit, the Hamiltonian dynamics that describes motion of wave packets or rays, is chaotic, e.g. hyperbolic. In quantum chaos, there is a special time called Ehrenfest time t E defined as the typical time when small wave length h are amplified to macroscopic size 1 under the chaotic dynamics with Lyapunov exponent λ >0 : this gives e λ t E h1 t E 1 λ ln ( 1 h ) . Notice that even if h is very small, t E is not very large due to ln ( 1 h ) .

2.2 Publications

  1. 2002. "Propagating modes in a periodic wave guide in the semi-classical limit", F.Faure, J. Phys. A: Math. Gen. 35, 1339-1356, (2002). Paper
  2. 2003. "Scarred eigenstates for quantum cat maps of minimal periods", F. Faure, S. Nonnenmacher and S. De Bièvre, Communications in Mathematical Physics ,Vol. 239, 449-492, (2003). Paper, or preprint 2002
  3. 2003. "On the maximal scarring for quantum cat map eigenstates", F. Faure, S. Nonnenmacher , Communications in Mathematical Physics, Vol 245, 201 - 214, (2004). Paper, or preprint 2003
  4. 2006. "Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula", F. Faure, Annales de l'Institut Fourier, Vol. 57 No.7, p. 2525-2599 (2007)., Paper, or preprint 2006 ., Movies related to this paper.

2.3 Lecture notesSlides


3 General Quantum physics

3.1 Publications


3.2 Lecture notes


4 Topological phenomena for slow-fast dynamical systems and molecular physics

4.1 Introduction

This short text presents the list of papers given below, following a thematic description. Before explaining each paper that corresponds to simple and specific examples we describe the abstract general phenomena.
Description of the general phenomena:
This series of papers concern some “topological phenomena in quantum mechanics”, and more precisely the manifestation of the Atiyah-Singer index formula in quantum physics of slow-fast dynamical systems also called adiabatic systems when a quantum physical system consists of sub-systems that interact together but each with different time scale. Example: small molecules (with few atoms) are quantum objects but quite complicated because electrons are light so move fast, they interact together but also with the atoms that are heavier so slower. These atoms have vibration motions that are still faster then the whole rotationnal motion of the molecule that is still slower. Geometricaly we can describe the global dynamics with different time scale as a bundle F where the fast dynamics takes place in the fibers whereas the slow dynamics takes place on the base.
Of course an isolated molecule should be considered within a quantum description. Experimentally, by spectroscopy and fluorescence, physicists observe the energy levels (i.e. eigenvalues of the Hamiltonian quantum operator). They observe that energy levels of the molecules are discrete but form some groups or clusters (called energy bands). The exact number of energy levels in a cluster is given by the index formula that expresses this number in terms of the topology of a specific vector bundle F that describes the dynamics of the molecule.
More informations:
We give now more precise informations. As we have said, this clustering structure is due to the coexistence of motions of the molecules with different time of scale: the rotation of the whole molecule is slower than the faster oscillations of the atoms. In heuristic terms (from the uncertainty principle in Fourier transform), we have time scales T rotation. T vibration . that corresponds to separation of energy levels Δ E rot . = T rot Δ E vib . = T vib . Using microlocal analysis we can describe with good accuracy the slow rotationnal motion using a classical mechanics i.e. dynamics on a symplectic manifold M rot . The fast motion is described in quantum mechanics by a finite rank operator H vib ( x ) that depends on the point x M rot . This gives a bundle of operators over M rot . . Each eigenspace defines a finite rank vector bundle F over M rot . Each eigenspace F corresponds to an observed cluster in the spectrum. The Atiyah-Singer index formula gives that the exact number of eigenvalues N in a cluster is N = M rot Todd ( T M rot ) Ch ( FL ) where Todd ( T M rot ) is the Todd class of the tangent bundle of M rot , L is the line bundle over M rot whose curvature is the symplectic form ( L is called the prequantum line bundle) and Ch ( . ) is the Chern class.

4.2 Publications

  1. 2000. "Topological Chern indices in molecular spectra" F. Faure, B. Zhilinskii, Physical Review Letters , 85, 960-963 , (2000). Paper, or preprint.
  2. 2001. "Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum" F. Faure, B. Zhilinskii, Letters in Math. Physics vol 55, 219-238, (2001). Paper or preprint.
  3. 2002. "Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology" F. Faure and B. Zhilinskií, Acta Appl. Math. 70, 265-282 (2002). Paper in ps.gz .
  4. 2002. "Topologically coupled energy bands in molecules" F. Faure and B. Zhilinskií, Phys. Letters A, 302, p.242-252, 2002. Paper, or preprint.
  5. 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
  6. 2019. "Manifestation of the topological index formula in quantum waves and geophysical waves" F. Faure, Annales Henri Lebesgue 6 (2023) 449-492, Paper, preprint.

5 Semi-classical and topological analysis of the quantum Hall effect

5.1 Publications

  1. 1993. "Mécanique quantique sur le tore et dégénérescences dans le spectre". F. Faure, Séminaires d'analyse spectrale de l'Institut Fourier, Vol n°11, 1993. Paper.
  2. 1994. "Generic description of the degeneracies in Harper like models." F. Faure, Journal of Physics A: Math. and gen. 27, (1994), 7519-7532. Paper.
  3. 2000. "Semi-classical Quantum Hall conductivity" F. Faure, B.Parisse, Journal of Mathematical physics, 41, 62-75, (2000). Paper or preprint
  4. 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


6 Mathematics and music

Our main project is presented here: Music and maths project..

6.1 Music on the tonnetz


6.2 Lecture notes and slides


6.3 Softwares

6.3.1 Music with just intonation and Analysis of sounds


6.4 External links


7 Thèses (Thesis)


8 Liens, External links

8.1 In Grenoble

8.2 In General