Internship proposals in mathematical physics,
Propositions de stage en physique-mathématique
Master 1, Master 2, Thèses.

A model of deterministic chaos is a dynamical system which has a sensitivity to initial conditions, i.e. two neighboring trajectories separate at an exponential rate either in the future or in the past. This property called Anosov or hyperbolicity implies a “chaotic” behavior of the trajectories: they seem almost all unpredictable. In order to understand these patterns and nevertheless to find predictive laws, one must adopt a probabilistic approach to dynamics: study the evolution of probability distributions.

See for example this vidéo, showing the evolution of a "chaotic" individual trajectory, and this vidéo showing the evolution of a probability distribution, converging to a "steady state".

f : M \to M

is the dynamical system, we have to study theopérateur de transfert

L : C^{\infty} (M) \to C^{\infty} (M)

defined by

L u : = u ○ f

. Its eigenvectors

u_{j}

defined by

L u_{j} = λ_{j} u_{j}

with

λ_{j} \in C

| λ_{j} | \leq 1

, are invariant distributions The eigenvalues are called Ruelle resonances. For a class of very chaotic models ("Axiom A") we show that the spectre of

L

is discrete (in adapted functional spaces) and that only the eigenvalue

λ_{0} = 1

is on the unit circle, associated to a spectral projector

Π_{0} = | 1 ⟩ ⟨ v_{0} |

. The other eigenvalues are

| λ_{j} | < 1

. This has the consequence that for long times

λ_{j}^{n} \overset{n \to \infty}{\to} 0

, and thus any initial distribution converges to the steady state

v_{0}

. The rate of convergence and the transient regime is controlled by the rest of the spectrum.

1.2 Objective

We consider simple and accurate models of chaotic dynamics. The goal is to compute the Ruelle spectrum numerically and to observe the distribution of the eigenvalues. We can conjecture that there is a spectral gap and try to demonstrate it. Eventually we can observe the "equipartition of the eigenspaces" and state a conjecture of type "quantum ergodicity". One could observe a band structure of the Ruelle spectrum and compare it to the spectrum of the quantum model.

The proposed models are

An $S^{1}$ extension with roof function $τ (x)$ of the cat map $(\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix})$ .
Anosov flow obtained from a suspension with variable time of the cat map model.

In each of these models, we will start by realizing a program which computes and shows the spectrum

{(λ_{j})}_{j}

. In parallel we will compute the properties of this dynamics (partially Anosov, regularity of the foliage, mixing, ergodicity,..) and the captive set. We will show that there is a discrete spectrum. We will try to prove an asymptotic spectral gap.

The student will also be asked to contribute to wikipedia on this subject.

1.3 After

the internship can be continued in a thesis, on the theme of classical chaos and quantum chaos.

Prerequisites:

Basic knowledge of spectral theory and differential geometry. Optionally, basic notions in dynamical systems theory (hyperbolic dynamics), semi-classical analysis. Motivation for physics-mathematics, and optionally motivation for numerical experimentation (e.g. C++ languages). The purpose of numerical experimentation is to illustrate the results obtained, to test new ideas or to explore new research directions.

Documentation:

Introductions:
- chaos: Vidéos by Etienne Ghys
- quantum chaos: "Introduction au chaos classique et chaos 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.)
- Exposé resonances Ruelle SFP et vidéos associées. Vidéos de spectre de Ruelle pour un modèle similaire montrant un gap spectral.
- «From classical chaos to quantum chaos», Lecturenotes for the school 22-26 April 2019 at CIRM.
- “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.
- "Semiclassical approach for the Ruelle-Pollicott spectrum of hyperbolic dynamics". May 2013. Lectures notes for summer school 13-17 May 2013 at ROMA.
Specific lectures
- Etude du «cat map». Ref: [3, p.5], [2, p.121], [5, p.13,] , [14, p.42,p.79, p.541, p.587]
- Spectre de Ruelle du cat map (et perturbations). [7, 13]. Spectre de Ruelle de flot Anosov [8].
- Livres sur les systèmes dynamiques: [5], [14],
- Livres sur la théorie spectrales et l'analyse semiclassique: [4],[#colin_livre],[15]
- Liste d'ouvrages conseillés en mathématique et physique:
- sur le C++ et l'expérimentation numérique ou programmation scientifique avec le langage Python.

1.4 Indications to start

Define
1. an extension with roof function $z = τ (x)$ of $f \in S L_{d} Z$ hyperbolic. Ex: $f = (\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}) \in S L_{2} Z$ .
2. Similarly for a flow. See suspension à temps variable de l'application du chat d'Arnold.
Find examples of matrices $f \in S L_{3} Z$ hyperbolic and study the regularity of stable/unstable distributions. For this, try $(\begin{matrix} 1 & \pm 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{matrix}) \in S L_{3} Z$ ex: $(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}) = (\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix})$ .
In case of a diffeomorphism (a),
1. Let $V \in C^{\infty} (T^{d} \times T^{1})$ , «potential function». Define the transfer operator $L$ acting on $C^{\infty} (T^{d + 1})$ and the reduced operator $L_{N}$ acting on $C^{\infty} (T^{d})$ .
2. Compute matrix elements of $L_{N}$ in Fourier basis. Tronquer la matrice et diagonaliser numériquement (taille $M \times M$ ). Dessin du spectre, et vérifier qu'il est stable par rapport à $M$ .
3. Calculer l'application symplectique $T$ sur $T^{*} T^{d}$ associée à l'opérateur de transfert $L_{N}$ . Identifier l'ensemble captif $K$ . Calculer la régularité de l'ensemble captif $K$ . Déterminer les variétés stables/instables de tout point $ρ \in K$ .
4. Par différentes methodes, montrer que L N a du spectre discret de Ruelle intrinsèque dans des espaces de Sobolev adaptés:
  1. Par les Opérateurs pseudo-différentiels (PDO) (article [13]).
  2. Par la transformation par paquets d'ondes (ou FBI), utilisant Lecturenotes,[11].
5. Application: decroissance des fonctions de correlation à $N$ fixé.
6. Formule de trace d'Atiyah Bott. Relier le spectre de Ruelle de $L_{N}$ aux orbites fermées de $f$ .
7. Suivre la même démarche que avec Tobias: forme normale globale sur l'ensemble de base et sur l'ensemble captif. Expansion asymptotique et déduire une estimation du gap spectral asymptotique (pour $N ≫ 1$ ).
8. Application: Comptage d'orbites avec poids: $N (T) : = ♯ \sum_{n \geq 1} \sum_{x = f^{n} x t . q . l (γ_{x}) \leq T} 1$ avec $l (γ_{x}) : = \sum_{k = 1}^{n} τ (f^{k} (x))$
9. Généralisation autant que possible des questions précédentes: au cas $f : M \to M$ difféomorphisme Anosov quelconque.
Dans le cas du flot (b), identifier la variété M de dimension dim M=d+1 . Expliciter une base orthonormée (ou pas) de L 2 ( M ) . Faut il considérer un opérateur Δ associé à une métrique interpolée: g( x,y,z ) = g xx ( z ) d x 2 + g yy ( z ) d y 2 + g xy ( z ) dxdy+d z 2
1. Calculer les éléments de matrice du générateur dans cette base.
2. Comptage d'orbites: $N (T) : = ♯ {p . o . γ avec repetitions s . t . | γ | \leq T}$
Relation entre les spectre du diffeomorphisme (a) et du flot (b)? (i.e. comprendre la relation générale entre le spectre d'une Section de Poincaré et le spectre du flot)
Essayer de discuter la possibilité de conjuguer l'opérateur de transfert à un opérateur sur $H : = L^{2} (T_{x,}^{1} \times T_{z}^{1})$ en moyennant sur des «fibres».

Every books can be found on libgen.
Every article can be found on libgen, or on scihub from its DOI reference.

2 Study of quantum chaos. Pre-quantum approach

2.1 Description

In the 70's, Souriau, Kostant and Kirillov laid the foundations of geometric quantization, which is a geometric construction of quantum dynamics, starting from classical dynamics (a Hamiltonian flow on a phase space). In the steps of the construction, they define an intermediate dynamics between classical and quantum that they call "pre-quantum", which is the evolution of wave functions on the phase space. We consider a classical "hyperbolic" dynamics, where all trajectories are unstable. The dynamics then has "chaotic" properties. An important field of research called "quantum chaos" is the study of the corresponding quantum dynamics (dynamic or spectral properties). For this, it seems natural to study also the pre-quantum dynamics.

We consider a simple model of hyperbolic (chaotic) dynamics: a linear hyperbolic application on the

T^{2}

torus, called "Arnold's cat map". The corresponding quantum and pre-quantum models are well established. For quantum dynamics, there is a "quantum ergodicity" theorem which states that almost all eigenmodes become "equidistributed" on the torus in the semiclassical limit (i.e. when the Chern index of the fibric tends to infinity). We also know examples of "non unique ergodicity": eigenmodes which are partially localized on unstable periodic orbits.

2.2 Objectives

Inspired by the quantum ergodicity theorem described above, the objective is to establish an analogous theorem in the pre-quantum case, for the model considered, and to discuss the non unique ergodicity.

2.3 After

the internship can be continued in a thesis, on the theme of classical chaos and quantum chaos.

2.4 Prerequisite:

Basic knowledge of spectral theory and differential geometry. Optionally, basic notions in dynamical systems theory (hyperbolic dynamics), theory of complex line fibers on a Riemann surface, with connection. Motivation for physics-mathematics, and motivation for numerical experimentation (e.g. C++ or python languages). The purpose of numerical experimentation is to illustrate the results obtained, to test new ideas or to explore new research directions.

2.5 Documentation

Introductions:
- au chaos: Vidéos de Etienne Ghys
- au chaos quantique: "Introduction au chaos classique et chaos 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.)
- Exposé resonances Ruelle SFP et vidéos associées. Vidéos de spectre de Ruelle pour un modèle similaire montrant un gap spectral.
- «From classical chaos to quantum chaos», Lecturenotes for the school 22-26 April 2019 at CIRM.
- “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.
- "Semiclassical approach for the Ruelle-Pollicott spectrum of hyperbolic dynamics". May 2013. Lectures notes for summer school 13-17 May 2013 at ROMA.
- "Introduction au chaos quantique" (Cours à l'école d'été Peyresq 2007)
Lectures spécifiques
- Etude du «cat map». Ref: [3, p.5], [2, p.121], [5, p.13,] , [14, p.42,p.79, p.541, p.587]
- Spectre de Ruelle du cat map (et perturbations). [7, 13]. Spectre de Ruelle de flot Anosov [8].
- Ergodicité quantique, [1].
- L'article , "Prequantum chaos: Resonances of the prequantum cat map" 2006.
- Livres sur les systèmes dynamiques: [5], [14],
- Livres sur la théorie spectrales et l'analyse semiclassique: [4],[#colin_livre],[15]
- Liste d'ouvrages conseillés en mathématique et physique:
- sur le C++ et l'expérimentation numérique ou programmation scientifique avec le langage Python.
- Autres documents disponibles.
Every books can be found on libgen.
Every article can be found on libgen, or on scihub from its DOI reference.

3 Ruelle spectrum of chaotic dynamics and emergence of quantum dynamics. Theoretical analysis and numerical experiments.

3.1 Description

The general motivation of this subject is to study the long time evolution of smooth probability distributions under a deterministic but chaotic classical dynamics (uniform hyperbolic dynamics). The probability distribution converges towards a measures called equilibrium or (SRB measure). We are interested to study the small fluctuations around this limit. It has been shown that they are governed by an “effective quantum dynamics”. This phenomenon called “dynamical emergence of quantum dynamics” manifests itself in the Ruelle spectrum of the classical dynamics: the spectrum has a band structure [6, 10, 9, 12].

3.2 Objectives

The objective of this Internship is to understand this mechanism in a simple model, the “cat map” and the “perturbed cat map” and also to establish a rigorous and precise numerical algorithm to compute the Ruelle spectrum of the first band, namely of the emerging quantum operator, and finally to compare with the spectrum of an ordinary quantum operator (from Weyl quantization).

A natural extension of this work is to consider next another specific model of chaotic dynamics that is the geodesic flow on a negatively curved Riemannian manifold. A specific example being a surface with constant curvature

κ = - 1

The student will also be asked to contribute to wikipedia on this subject.

3.3 Prerequisites:

Basic knowledge of spectral theory and differential geometry. Optionally, basic notions in dynamical systems theory (hyperbolic dynamics), semi-classical analysis. Motivation for physics-mathematics, and a good motivation for numerical experimentation (e.g. C++, python or other languages adapted to scientific programming). The purpose of numerical experimentation is to illustrate the results obtained, to test new ideas or to explore new research directions.

3.4 Documentation:

Introductions:
- chaos: Vidéos by Etienne Ghys
- Observation of evolution of dynamics in Sinai billard, video 1 ball, 1e6 balls, quantum wave.
- Observation of dynamics with the classical and quantum Cat map model.
- Observation of quantum wave evolutions and comparison with classical dynamics: videos
- quantum chaos: "Introduction au chaos classique et chaos 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.)
- Exposé resonances Ruelle SFP et vidéos associées. Vidéos de spectre de Ruelle pour un modèle similaire montrant un gap spectral.
- «From classical chaos to quantum chaos», Lecturenotes for the school 22-26 April 2019 at CIRM.
- “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.
- "Semiclassical approach for the Ruelle-Pollicott spectrum of hyperbolic dynamics". May 2013. Lectures notes for summer school 13-17 May 2013 at ROMA.
Specific lectures
- Etude du «cat map». Ref: [3, p.5], [2, p.121], [5, p.13,] , [14, p.42,p.79, p.541, p.587]
- L'article , "Prequantum chaos: Resonances of the prequantum cat map" 2006.
- Introduction et premiers chapitres de l'
- article [10].
- Livres sur les systèmes dynamiques: [5], [14],
- Livres sur la théorie spectrales et l'analyse semiclassique: [4],[#colin_livre],[15]
- Liste d'ouvrages conseillés en mathématique et physique:
- sur le C++ et l'expérimentation numérique ou programmation scientifique avec le langage Python.
Every books can be found on libgen.
Every article can be found on libgen, or on scihub from its DOI reference.

4 Fractal Weyl law for Axiom A flows

Extend to Axiom A flows the results of https://arxiv.org/abs/1706.09307 that concerns Anosov flows.

5 Ruelle spectrum of Baker map and prequantum Baker map

Extend to the Baker map the results of https://arxiv.org/abs/nlin/0606063 that concerns the cat map.

6 Ruelle spectrum of Pseudo Anosov maps

Extend to the Pseudo Anosov maps the results of https://arxiv.org/abs/0802.1780 that concerns Anosov maps.

Part II Microlocal Analysis

7 Quantum revival in the wave equation

There is some strange quantum revival in the model of quantum cat map. Study if this can append in the model of the wave equation.

8 Spectrum in SU(2)

Consider two elements

g_{1}, g_{2} \in S U (2)

. We denote

g_{1}^{○}, g_{2}^{○} : C^{\infty} (S U (2)) \to C^{\infty} (S U (2))

the pull back operator. Study the spectrum of the operator

g_{1}^{○} + g_{2}^{○}

in particular its spectral gap.

9 Fractal Weyl law in a simple model

Consider a a Hölder continuous function

f : x \in S^{1} \to ξ \in R

of exponent

0 < α \leq 1

. Study the spectrum of the operator

{O p}_{ℏ} (1_{{ξ \leq f (x)}})

, in particular the density of eigenvalues in

(0, 1)

in terms of the exponent

α

10 Sub-Riemaniann Laplacian and quantization of contact structures

Study the possibility of doying geometric quantization of contact structures.

Part III Music and Mathematics

11 Develop a software to perform in jazz improvisation with just intonation

See the project Music on the tonnetz.

References

1N. Anatharaman, "Le théoréme d'ergodicité quantique", (2014).

2V.I. Arnold, Geometrical methods in the theory of ordinary differential equations (Springer Verlag, 1988).

3V.I. Arnold and A. Avez, Méthodes ergodiques de la mécanique classique (Paris: Gauthier Villars, 1967).

4S. Bates and A. Weinstein, Lectures on the Geometry of Quantization vol. 8, (American Mathematical Soc., 1997).

5M. Brin and G. Stuck, Introduction to Dynamical Systems (Cambridge University Press, 2002).

6F. Faure, "Prequantum chaos: Resonances of the prequantum cat map", arXiv:nlin/0606063. Journal of Modern Dynamics 1, 2 (2007), pp. 255-285.

7F. Faure and N. Roy, "Ruelle-Pollicott resonances for real analytic hyperbolic map", Nonlinearity. https://arxiv.org/abs/nlin.CD/0601010link 19 (2006), pp. 1233-1252.

8F. Faure and J. Sjöstrand, "Upper bound on the density of Ruelle resonances for Anosov flows. A semiclassical approach", Comm. in Math. Physics, Issue 2. https://fr.arxiv.org/abs/1003.0513link 308 (2011), pp. 325-364.

9F. Faure and M. Tsujii, "Band structure of the Ruelle spectrum of contact Anosov flows", Comptes rendus - Mathématique 351 , 385-391, (2013) https://arxiv.org/abs/1301.5525link (2013).

10F. Faure and M. Tsujii, "Prequantum transfer operator for symplectic Anosov diffeomorphism", Asterisque 375 (2015), https://fr.arxiv.org/abs/1206.0282link (2015), pp. ix+222 pages.

11F. Faure and M. Tsujii, "Fractal Weyl law for the Ruelle spectrum of Anosov flows", arXiv:1706.09307 https://arxiv.org/abs/1706.09307link (2017).

12F. Faure and M. Tsujii, "Microlocal analysis and Band structure of contact Anosov flows", arxiv:2102.11196 (2021).

13F. Faure, N. Roy, and J. Sjöstrand, "A semiclassical approach for Anosov Diffeomorphisms and Ruelle resonances", Open Math. Journal. https://arxiv.org/abs/0802.1780link 1 (2008), pp. 35--81.

14A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, 1995).

15M. Zworski, Semiclassical Analysis (Amer Mathematical Society, 2012).

Internship proposals in mathematical physics, Propositions de stage en physique-mathématique Master 1, Master 2, Thèses.

Table of Contents

Part I Dynamical systems

1 Spectral approach to classical and quantum chaos

1.1 Description