RESUM´ E´ · 2016. 9. 2. · RESUM´ E ´ On se propose d ... Brossard [11] pour le cas continu...

1

RÉSUMÉ

On se propose d’étudier les propriétés spectrales l’opérateur de Schrödinger avec po-tentiel aléatoire dans un ruban. En fait les modèles étudiés sont de deux types:

• Le cas indépendant: Les potentiels en chaque site sont supposés former une famillede variables aléatoires indépendantes et équidistribuées.

• Le cas Markovien: Les potentiels en chaque site sont gouvernés par un processusou une chaine de Markov selon que le modèle est discret ou continu.

Il s’agit donc d’étendre à ces modèles les résultats prouvés antérieurement dans le casunidimensionnel, c’est à dire la propriété de localisation exponentielle et les résultatsde régularté de la distribution d’état. Le premier ingrédient utilisé est la simplicité duspectre de Lyapunov pour un produit de matrices symplectiques. Le second outil estfourni par la théorie spectrale des opérateurs associés à la transformation de Laplace surcertaines frontières du groupe symplectique. Il s’en suit que le spectre est purementponctuel dans le cas où la loi des potentiels est absolument continue. La preuve delocalisation dans le cas général résulte des propriétés de décroissance de la fonctionde Green établies dans le cas multidimensionnel. Enfin les résultats précédents sontutilisés pour prouver certaines propriétés de régularité de la distribution d’états à savoircontinuité Höldérienne et dérivabilité.

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SOURCES

Cette thèse résulte essentiellement de la compilation d’articles publiés par l’auteur.Certaines démonstrations ont été actualisées et complétées pour tenir compte d’ uncertain nombres de développements récents.

1. Problèmes probabilistes liés a l’étude des opérateurs aux différences aléatoires.Publié en 1982 dans Ann. Inst. Elie Cartan 7, Marches aléatoires et processusstochastiques sur les groupes de Lie, 80-95.

2. Singularité du spectre de l’opérateur de Schrödinger aléatoire dans un ruban ouun demi-ruban.Publié en 1983 dans Ann. Inst. H. Poincaré A 38 385-399

3. Localisation pour l’opérateur de Schrödinger aléatoire dans un ruban.Publié en 1984 dans Ann. Inst. H. Poincaré A 40 97-116

4. Computations of the sum of positive Lyapunov exponents for the Llyod model ina strip.Publié en 1986 dans Lectures Notes in Math. 1186 258-264

5. Localization for the Anderson model on a strip with singular potentials.Ecrit en collaboration avec A.Klein & A.Speis. A paraitre dans J. FunctionalAnal. 27

6. Regularity of the density of states in the Anderson model on a strip for potentialswith singular continuous distributions.Ecrit en collaboration avec A.Klein & A.Speis. Publié en 1990 dans J. Statist.Phys. 57, 65-88

Les résultats obtenus dans ces travaux ont étés en partie repris et développés dans deuxlivres consacrés à l’étude des propriétés spectrales de l’opérateur de Schrödinger avecpotentiel aléatoire:

• Products of random matrices with applications to Schrödinger operators.Ouvrage écrit en collaboration avec P. Bougerol, publié en 1985 dans Progress inprobability and statistics (Birkhäuser)

• Spectral properties of Random Schrödinger Operators.Ouvrage écrit en collaboration avec R. Carmona, publié en 1990 dans Probabilityand Its Applications (Birkhäuser)

Cette thèse s’appuie essentiellement sur des extraits de ce dernier ouvrage qui avaientété rédigés par l’auteur, en tenant compte de certains résultats postérieurs à sa pub-lication. Ceci explique que la plus grande partie de ce travail soit rédigée en langueanglaise (tout au moins l’auteur espère-t-il l’indulgence du lecteur, vu son niveau deconnaissance rudimentaire de cette langue. . . ).

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INTRODUCTION

Cette thèse a pour but d’illustrer l’importance et la nécessité de l’utilisation de la théoriedes produits de matrices aléatoires pour l’étude spectrale des opérateurs de Schrödingerà potentiels aléatoires. L’exposé ne concerne que la dimension un et le ruban, mais il estfort probable qu’en dimension supérieure, on doive encore faire appel a ses résultats pourmontrer que la théorie de la localisation multidimensionnelle actuellement disponibleest bien fondée. En effet son système d’hypothèses requiert le contrôle de la fonction deGreen sur une boite initiale suffisamment grande et ceci ne peut résulter, en dehors decas extrèmes (très grande énergie ou très grand désordre), que de l’étude des exposantsde Lyapounov dans un ruban de grande largeur.

Nous avons choisi de faire une présentation simultanée des cas discrets et continus(en fait sur le ruban il faut entendre par cas continu, le cas de plusieurs équations detype Schrödinger continues couplées, le cas du Laplacien bidimensionnel restant, pourl’instant, un problème complétement ouvert). L’utilisation d’un formalisme commun,clarifie de façon significative les preuves originales du modèle continu. En effet lespropriétés ergodiques, et plus généralement les propriétés asymptotiques des systèmescontinus se déduisent facilement du cas discret (tout au moins tant que l’on s’intéresseà des propriétés presque sûres ou génériques, car il est facile de trouver des exemplesdéterministes où ces deux types d’opérateurs sont aussi dissemblables que possible. . . ).

Du point de vue mathématique le problème la localisation dans les systèmes désordonnésest le suivant: les opérateurs autoadjoints utilisés comme hamiltoniens dans ces systèmesont une tendance à avoir un spectre purement ponctuel, contrairement au cas des mi-lieux périodiques où ce spectre est absolument continu. La première preuve rigoureusede ce phénomène a été fournie par Goldsheid, Molcanov & Pastur en 1977 [38]. Dans cetarticle les auteurs étudient le cas unidimensionnel continu, le potentiel étant généré parun mouvement Brownien. A la suite de ce travail H. Kunz et B. Souillard ont établi unrésultat identique dans le cas unidimensionnel discret, les potentiels en chaque site for-mant une famille de variables aléatoires indépendantes et équidistribuées. Ces preuvesont été ensuite simplifiées et généralisées par R. Carmona [14][15], G.Royer [93], J.Brossard [11] pour le cas continu et par l’auteur dans le cas discret [71] puis de nou-veau par F. Delyon, H. Kunz & B. Souillard [23] dans le cas de potentiels aléatoires“perturbés” par un bruit déterministe. D’autre part de très nombreux travaux ont étésconsacrés à la distribution d’états (“integrated density of states”) en dimension un etplus particulièrement au problème de sa régularité en tant que mesure. En particulieril faut citer les travaux de Le Page [80],[81], Craig & Simon [22], et Campanino & Klein[13] que nous généraliserons au cas du ruban.

Bien sûr chacun s’accorde à penser que l’extension à un ruban des questions précédentes

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ne doit pas faire apparaitre de phénomènes qualitativement nouveaux. Malheureuse-ment une bonne partie des techniques utilisées à une dimension ne s’adaptent qu’auprix d’un effort important. De façon näıve ont peut concevoir qu’il est certainementplus délicat de manipuler des matrices plutôt que des nombres réels. D’un point de vuemathématique la difficulté nouvelle est la suivante:

• En dimension un, le groupe qui apparâıt naturellement est le groupe unimodulaireSL(2,R) dont l’unique frontière est la droite projective. Celle-ci s’identifie peuou prou à la droite réelle et une bonne partie de la démonstration peut doncêtre effectuée sans référence explicite à la théorie des frontières des groupes semi-simples.

• Il en va tout diffèremment sur le ruban où maintenant le groupe sur lequel ondoit travailler est le groupe symplectique SP (ℓ,R) dont les diffèrentes frontièress’identifient à des variétés Lagrangiennes. De plus il faut maintenant contrôler lapositivité d’une famille d’exposants de Lyapunov et non plus d’un seul. Dans cecas une approche plus intrinsèque du problème s’avère nécessaire.

Signalons enfin l’existence d’autres approches du problème de la localisation dans lecas de potentiels absolument continus, citons en particulier:

• La théorie des perturbations de rang un des opérateurs autoadjoints utilisée parSimon et Wolff [96]

• La méthode de Kotani[68] que l’on peut assimiler à une extension du théorèmede Fubini au cas des transitions de probabilités.

Dans ces travaux l’on n’utilise plus du tout l’approche par approximation de type“limite thermodynamique” le long de boites croissant vers l’espace tout entier. L’aspectconstructif de la preuve présentée ici n’apparâıt donc plus, en particulier l’on perd toutcontrôle de la vitesse de convergence des approximations de la mesure spectrale et parvoie de conséquence de la régularité éventuelle de la distribution d’états.

Nous introduisons maintenant quelques notations utiles pour la description des chapitresqui suivent.

Soit ℓ un entier positif, d = 2ℓ ,et T = R ou Z. Le ruban de hauteur ℓ est défini commel’ensemble {(i, t); i = 1 . . . ℓ, t ∈ T} c’est à dire la somme de ℓ copies de T . L’opérateurde Schrödinger sur le ruban peut alors être considéré comme la juxtaposition de ℓopérateurs unidimensionnels couplés. Soit {V (i, t); i = 1 . . . ℓ, t ∈ T} une famille denombres réels. On définit la matrice V (n) d’ordre ℓ× ℓ par:

Vij(t) =

V (i, t) si i = j−1 si |i− j| = 10 si |i− j| > 1

5

L’opérateur de SchrödingerH sur le ruban de hauteur ℓ opère sur les suites {ψ(t); t ∈ T}de vecteurs à ℓ composantes complexes (ψ1(t), · · · , ψℓ(t)) par la formule:

Hψ(t) = −ψ(t+ 1) − ψ(t− 1) + V (t)ψ(t) cas discretHψ(t) = −ψ̈(t) + V (t)ψ(t) cas continu

En fait, dans le cas discret, toute la théorie s’applique sans grandes modifications à desopérateurs plus généraux de la forme:

[Hψ](n) = B−1n (−ψ(n+ 1) − ψ(n − 1) + V (n)ψ(n))

où {Bn;n ∈ Z} est une suite de matrices symétriques définies positives dont le spectrereste contenu dans un intervalle [a, b] avec a > 0.

Dans le cas aléatoire on considère que les {Vω(i, t); i = 1 . . . ℓ, t ∈ T} forment unefamille de variables aléatoires réelles de même loi, définies sur un espace probabilisé(Ω,P). Les deux modèles les plus étudiés correspondent au cas de variables aléatoiresindépendantes ou en dépendance Markovienne. Dans le cas aléatoire on note par Hωl’opérateur correspondant à la trajectoire ω.

6

I PRODUITS DE MATRICES ALEATOIRES

Ce chapitre contient l’essentiel des outils et résultats généraux utilisés par la suite. Nousavons choisi de regrouper l’approche via le théorème d’Osseledec empruntée à Ledrap-pier [77], qui traite des systèmes ergodiques bilatères généraux et celle de Guivarc’h &Raugi [44] qui poursuivant les travaux de Furstenberg & Kesten [31], [30], s’intéressentplus particulièrement au cas des systèmes indépendants. La première présentations’applique immédiatement au cas des systèmes multiplicatifs Markoviens étudiés parBougerol dans [7], [8] tandis que la seconde permet d’obtenir des résultats plus précisen ce qui concerne la simplicité du spectre de Lyapounov. Nous avons modifié un cer-tain nombre d’énoncés de [19] afin de tenir compte du résultat récent de Goldsheid etMargulis sur la fermeture de Zariski des matrices de Schrödinger qui n’était pas publiéau moment de l’écriture de ce livre. L’étude spectrale de la transformation de Laplacesur les espaces de fonctions höldériennes est empruntée à Le Page [79].

Les principales contributions de l’auteurs à ce chapitre sont les suivantes:

• L’étude des relations d’entrelacement des opérateurs de Fourier Laplace sur SP (ℓ,R)associées à la multiplication par un cocycle de Radon-Nikodym et leur conséquencessur les propriétés spectrales de ces opérateurs construits à partir de noyaux dePoisson. (Sections 3 , 5.3).

• La vérification des conditions assurant la validité des théorèmes généraux dansle cas de matrices de Schrödinger dans un ruban, par exemple l’existence d’unedensité pour les matrices de transfert sur le groupe SP (ℓ,R) (section 4.c). Cettepreuve a été reprise par Glaffig [34] et étendue au cas où seuls les potentiels de laligne supérieure du ruban admettent une densité, les autres étant constants.

• L’étude des variétés Lagrangiennes, le calcul de leurs noyaux de Poisson et l’établissementde formules donnant le plus petit exposant de Lyapunov positif (section 6).

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II L’OPERATEUR DE SCHRÖDINGER DETERMINISTE

Ce chapitre contient une bonne partie de ce qu’il est convenu d’appeler le “folklore” dusujet. . . . Son originalité tient dans le calcul barycentres de mesures spectrales dans desboites par rapport a une loi de Cauchy sur les “conditions au bord” aussi bien dans lecas continu que dans le cas discret. Cette technique semble apparâıtre pour la premièrefois dans Atkinson [3] pour le réseau unidimensionnel. Ce procédé utilisé par l’auteurd’abord en dimension un [71] puis sur un ruban [73] permet d’obtenir des conditionsde localisation très simples. Il a été depuis repris de nombreuses fois car il permet detransformer une famille de mesures discrètes en une mesure absolument continue, parune opération de barycentre. Tout ceci peut apparâıtre surprenant dans la mesure oùle but final de la manœuvre est, dans beaucoup de cas, de prouver que la limite de cesmesures spectrales est purement ponctuelle. Le choix de la mesure de Cauchy commerépartition des conditions au bord reste arbitraire, la seule raison de celui-ci étantqu’il fournit les calculs et les expressions finales les plus simples. En effet, l’utilisationsystématique de la notion de noyau de Poisson pour un espace Riemannien symétriquepermet de simplifier notablement certains calculs antérieurs fastidieux effectués parl’auteur [73] et R. Carmona [15]. Ces approximations sont ensuite très utiles pourdonner des critères de localisations directement reliés aux propriétés asymptotiques desproduits de matrices de transfert (ou de l’opérateur d’évolution dans le cas continu).Le critère du cas Markovien unidimensionnel est emprunté à Carmona [14] et celui duruban figurait déjà dans les travaux de l’auteur [73].

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III L’OPERATEUR DE SCHRÖDINGER ALEATOIRE

L’essentiel du travail de préparation ayant déjà été effectué dans les deux chapitresprécédents il ne reste plus qu’à mettre en œuvre ces divers outils. On établit d’abord lasingularité du spectre sous des hypothèses assez générales. La technique de démonstrationest très largement inspirée des preuves fournies par Ishii [52] et surtout Pastur [90].En fait la singularité du spectre est une conséquence immédiate de l’hyperbolicité dusystème dynamique pour toute valeur de l’énergie et pour presque tout ω. Cette pro-priété découle directement de la simplicité du spectre de Lyapunov pour un produitaléatoire de matrices de SP (ℓ,R) ou bien pour le système multiplicatif engendré parle propagateur dans le cas continu. La positivité du plus petit exposant de Lyapunovnon négatif est assurée par le résultat de Goldsheid & Margulis (on peut aussi utiliserles résultats antérieurs de Guivarc’h & Raugi dans le cas où la loi des potentiels admetune densité ou bien lorsque son support contient un ouvert puisqu’on a prouvé dans lechapitre I qu’il en est de même d’une puissance de convolution de la loi des matricesde transfert sur SP (ℓ,R)).

La preuve de localisation pour des potentiels absolument continus repose essentielle-ment sur l’utilisation systématique des propriétés spectrale des opérateurs de FourierLaplace sur le groupe symplectique. Dans le cas unidimensionnel cette approche avaitdéjà été utilisée par Goldsheid, Molcanov & Pastur de façon plus ou moins explicite.La trame de la preuve présentée ici est aussi très proche du travail de Kunz et Souillard[70] mais utilise les opérateurs associés à des noyaux de Poisson plutôt que ceux utiliséspar Kunz et Souillard. Ces derniers correspondent à des cocycles singuliers et n’ontpas d’équivalents pour le ruban.

De façon plus précise, si l’on dénote par B la Lagrangienne Lℓ−1,ℓ les approximationsde la mesure spectrale calculées au chapitre I font apparâıtre les opérateurs Tt,λ opérantsur l’ensemble des fonctions continues sur B par:

Tt,λf(b) =

∫f(gb)(χ(g, b))t/2 dµλ(g)

où µλ est la loi des matrices de transfert et le cocycle χ est précisé ci-dessous.Une condition suffisante de localisation est alors que pour chaque intervalle borné deR il existe des constantes C1, C2 et ρ < 1 telles que pour n ∈ N et λ ∈ I on ait :

‖T n1,λ‖ ≤ C1ρn , ‖T n2,λ‖ ≤ C2Le cocycle χ(g, (ȳ, x̄)) sur Lℓ−1,ℓ est défini par:

χ(g, (ȳ, x̄)) = rℓ−1,ℓ(g, (ȳ, x̄))r−1ℓ−1(g, x̄) = (

‖y‖‖gy‖ )

−2(‖x‖‖gx‖)

2

9

Dans cette formule rℓ−1 est le noyau de Poisson de Lℓ−1 soit:

rℓ−1(g, ȳ) =dg−1mℓ−1dmℓ−1

(ȳ) = (‖y‖‖gy‖ )

ℓ+2

et rℓ−1,ℓ est le noyau de Poisson de Lℓ−1,ℓ soit:

rℓ−1,ℓ(g, (ȳ, x̄)) = (‖y‖‖gy‖)

ℓ(‖x‖‖gx‖)

2

Des opérateurs analogue apparaissent dans le cas Markovien. Dans le cas où la loi despotentiels possède une densité on peut prouver que ces opérateurs sont compact surC(B) et leurs propriétés spectrales résultent du chapitre I.

On pourrait déduire des estimations précédentes la décroissance exponentielle des fonc-tions propres comme dans Carmona [14] ou Royer [93] mais la méthode de Kotanidonne ce résultat beaucoup plus simplement et nous n’aborderons donc pas ce sujetici. Signalons enfin que contrairement au cas unidimensionnel, le fait que les valeurspropres soient non dégénérées n’est pas automatique. Voir à ce propos la preuve deDelyon, Levy, Souillard utilisant le “Kotani’s trick” [24].

Dans le cas de potentiels quelconques, il s’agit d’adapter la preuve de localisation donnéedans le modèle unidimensionnel par Carmona, Klein et Martinelli [18]. En fait cettepreuve consiste à vérifier que les hypothèses d’un théorème, impliquant la localisationen toutes dimensions, sont satisfaites dans le cas du ruban pour tout désordre et touteénergie. Ce théorème a été établi par Dreifus et Klein [28] d’aprés un travail original deMartinelli et Scoppola [83]. Il peut sembler curieux de recourir à une analyse sophis-tiquée du modèle multidimensionnel pour conclure dans ce cadre, à priori plus simple.Ceci tendrait à prouver que le sujet est loin d’être clos...Soit Λ la boite {(k, i); i = 1 . . . ℓ, k ∈ [−n,+n]}, les conditions à vérifier sont les suiv-antes:

1. Pour toute énergie λ il existe des constantes χ > 0, ζ > 0 telles que

P{λ0 6∈ Σ(HΛ) et ‖GΛλ0(0,±n)‖ ≤ e−nχ} ≥ 1 − e−ζ√n

pour n assez grand.

2. Pour tout intervalle borné I il existe τ > 0 tel que:

P{d(λ , Σ(HΛ)) ≤ e−√n} ≤ e−τ

√n

pour λ ∈ I et n assez grand.

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La première condition va résulter de la théorie générale des produits de matrice lorsquel’on s’est assuré que les conditions standards sont applicables, à savoir forte irréductibilitéet contractivité sur toutes les variétés Lagrangiennes de la loi des matrices de transfert.Ce point résulte à nouveau, dans le cas d’une distribution quelconque des potentiels,du théorème de Goldsheid et Margulis.

La seconde condition est plus difficile à mettre en évidence et requiert une démonstrationde la continuité Hölderienne de la densité d’états sur le ruban. Ce résultat est dû à LePage en dimension un [80] et est étendu au ruban dans l’annexe (A). On peut aussitrouver une extension du même type dans un article récent du même auteur [81].

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IV LA DISTRIBUTION D’ETATS

On se restreint ici au cas d’un ruban discret. L’existence de la distribution d’étatsdans un réseau ne pose pas de problèmes techniques aussi délicats que dans le cascontinu. Cette probabilité, notée N , est définie comme la limite vague presque sûredes répartitions empiriques NΛ(ω) des valeurs propres de HΛ(ω) lorsque la boite Λtend vers le ruban entier (en fait, vu la stationnarité il suffit que l’un des cotés de laboite tende vers l’infini). Dans le cas de potentiels bornés cette convergence ne posepas de problèmes et résulte du théorème de Birkhoff (voir par exemple [9]) car toutesles probabilités NΛ(ω) sont portées par un compact fixe. Dans le cas général ergodiqueoù l’on ne suppose que l’intégrabilité de log(1 + |V |) il faut utiliser des méthodes detroncature (voir Fukushima [29]). De plus on obtient que N est égale à l’espérancede la mesure spectrale. Le fait que les sous espaces propres de H soient de dimension≤ ℓ implique que la mesure N est continue (cette propriété reste vraie dans le casmultidimensionnel voir Delyon & Souillard [25]). La fonction de répartition de N étanttoujours continue, on peut se poser le problème d’une régularité supériure:

• Continuité Höldérienne

• Existence d’une densité, et dans ce cas déterminer l’ordre de dérivabilité de celle-ci.

Ces problèmes ont déjà été étudiés en dimension un par de nombreux auteurs et nousnous proposons ici de les étendre au ruban.

L’une des extensions les plus simples concerne la preuve de la formule de Thouless pourle ruban. Cette étape est cruciale car cette formule permettra de prouver la régularitéde N en utilisant les propriétés de la transformation de Hilbert. La preuve donnée icine diffère de la preuve originale de Craig & Simon que par l’utilisation systématiquedu formalisme Lagrangien ce qui supprime la partie algébrique de leur travail.

Cas de potentiels absolument continus

L’un des plus anciens résultats établissant l’existence d’une densité d’états est celui deWegner [104]. Nous reproduisons ici la preuve originale à ceci près que l’on remplacel’astuce consistant à intégrer la fonction de Green sur une boite par un lemme plusgénéral d’intégration de mesures spectrales d’une matrice symétrique. Ceci permetl’extension de cette preuve à des Hamiltoniens plus généraux ne faisant plus intervenirque les seuls plus proches voisins. Ce résultat suppose l’existence d’une densité bornéepour les potentiels.On peut utiliser les opérateurs de Fourier Laplace et leurs propriétés démontrées au

12

chapitre précédent pour obtenir une formule de représentation de la densité d’états entermes de fonctions propres de ces opérateurs ce qui fournit la continuité de la densitéd’états. Une telle formule avait déjà été obtenue en dimension un par Kunz & Souillard[70], l’originalité tient ici à l’apparition de certaines “torsions” de la mesure invariantesur la frontière maximale. La preuve se trouve dans l’annexe (C) et donne prétexte àune étude plus détaillée des opérateurs de Fourier Laplace.En supposant toujours les potentiels absolument continus, Le Page a prouvé dans [81]que le plus grand exposant de Lyapunov est C∞. Ceci implique immédiatement lamême régularité de la densité d’états en dimension un via la formule de Thouless.Malheureusement l’extension de ce résultat au ruban n’est pas immédiate car cecisupposerait l’existence d’une densité sur la puissance extérieure

∧ℓ(Rd) ce qui n’estpas le cas. Nous reprenons donc les arguments de [81] en les adaptant à notre situationpour obtenir le même résultat sur le ruban.

Cas de potentiels singuliers

Dans ce cas (par exemple pour des lois de Bernoulli) on a seulement la locale continuitéHöldérienne de la fonction de répartition de N . Ceci a été prouvé en dimension un parLe Page [80]. Pour l’étendre au ruban on doit prouver ce type de continuité pour lasomme des exposants de lyapunov positifs ce qui est fait dans l’annexe (A) en vue de lalocalisation pour des potentiels singuliers. En fait, dans une publication ultérieure, LePage [81] obtient le même résultat par une méthode légèrement différente qui consiste àne considérer que le plus grand exposant et à passer ensuite aux puissances extérieures(Il n’y a pas ici la même objection que pour la dérivabilité).Si maintenant, tout en englobant le cas de certains potentiels singuliers, on imposequelques restrictions sur ceux-ci, par exemple en demandant que la transformée deFourier de leur loi soit Cp0 pour un certain p, alors on peut obtenir un ordre de dérivalité[(p+ 1)/2] pour la fonction de répartition de la distribution d’états. Ceci a été prouvéen dimension un par Campanino & Klein [13] et l’extension au ruban se trouve dansl’annexe (B). Dans les deux cas l’on utilise les techniques de la “super-symétrie” et celademande quelques mots d’explications. Cette théorie (voir [?], [?], [?]), très utiliséeen physique des milieux désordonnés, peut laisser perplexes certains mathématicienslorsque les Grasmaniennes sont de dimension infinie, certains résultats n’ayant guèrede justification théorique. Par contre, en dimension finie, ce formalisme est tout à faitrigoureux et nous n’utiliserons cette approche que dans ce cas.

Tout d’abord le lien entre la distribution d’états et la fonction de Green est évidentpuisque le théorème spectral implique que pour ℑm(z) 6= 0 on a:

E{trace(Gz(0, 0))} =∫dN(t)

t− zoù Gz est la matrice ℓ× ℓ de Green définie au chapitre II. Si l’on se souvient que Gz est

13

limite des GΛz lorsque la boite Λ tend vers le ruban tout entier, l’on est conduit à tenterde trouver une formule intéressante pour le calcul de l’inverse de la matrice HΛ − zI.Soit A = U + iV , où U et V sont des matrices symétriques d’ordre r avec U définiepositive. Un simple calcul d’intégrale gaussienne permet d’écrire:

1

detA=

∫exp(−

r∑

i,j=1

A(i, j)ϕi.ϕj)r∏

j=1

d2ϕjπ

A−1(a, b)detA

=

∫ϕa.ϕb exp(−

r∑

i,j=1

A(i, j)ϕi.ϕj)

r∏

j=1

d2ϕjπ

où les ϕi i = 1 . . . r sont r vecteurs de R2, la notation ϕi.ϕj désigne le produit scalaire

usuel et d2ϕ signifie l’intégration par rapport aux deux coordonnées de ϕ. Dans cetteformule la présence du déterminant de A est gênante. Une astuce souvent utilisée en

physique sous le nom “replica trick” consiste à integrer sur n copies ϕ(k)i k = 1 . . . n , i =

1 . . . r et l’on obtient:

A−1(a, b) = (detA)n∫ϕ(1)a .ϕ

(1)b exp(−

r∑

i,j=1

n∑

k=1

A(i, j)ϕ(k)i .ϕ

(k)j )

r∏

j=1

n∏

k=1

d2ϕ(k)j

π

et ensuite faire tendre n vers 0 !!. . . Ce procédé est pour le moins audacieux et pourl’éviter on introduit le formalisme de la super-symétrie pour prendre en compte ledéterminant.

Pour i = 1 . . . r soient ψi, ψ̄i des vecteurs de Rs avec s suffisamment grand pour que

ces 2r éléments soient linéairement indépendants. On note par G l’algèbre extérieureengendrée par les ψi, ψ̄i et pour simplifier l’écriture on notera le produit extérieurcomme un produit ordinaire. Pour X ∈ G et pour i donné on peut écrire de façonunique X = Y +Zψiψ̄i de telle façon que les éléments Y et Z de G ne contiennent plusle produit extérieur ψiψ̄i. On définit alors une application lináire de G dans lui-même,que l’on note sous forme intégrale:

∫Xd(ψiψ̄i) = −Z

Plus généralement si F est une fonction suffisament dérivable de Rd dans R on noteraF (X1, . . . ,Xd) l’expression obtenue en remplaçant dans le développement de Taylorde F les monômes en X1, . . . ,Xd par les produits extérieurs correspondants. Un teldéveloppement est donc toujours nul à partir d’un certain rang et l’on peut donc écriredes pseudo intégrales du genre

∫F (X1, . . . ,Xd)

r∏

i=1

d(ψiψ̄i)

14

qui n’est en fait qu’une forme linéaire. Avec ces notations on vérifie immédiatementque l’on a:

∫exp(−

r∑

i,j=1

A(i, j)ψiψ̄j)r∏

k=1

d(ψkψ̄k) = detA

∫ψaψ̄b exp(−

r∑

i,j=1

A(i, j)ψiψ̄j)

r∏

k=1

d(ψkψ̄k) = (detA)A−1(a, b)

En utilisant les notations condensées

Φi = (ϕi, ψi, ψ̄i), dΦi =d2ϕiπ

d(ψiψ̄i), Φi.Φj = ϕi.ϕj +1

2(ψiψ̄j + ψjψ̄i)

et en réunissant les deux types d’intégrales on obtient:

∫ψaψ̄b exp(−

r∑

i,j=1

A(i, j)Φi.Φj)

r∏

k=1

dΦk = A−1(a, b)

∫ψ(1)a ψ̄

(1)b exp(−

r∑

i,j=1

n∑

k=1

A(i, j)Φ(k)i .Φ

(k)j )

r∏

k=1

n∏

j=1

dΦ(j)k = A

−1(a, b)

La dernière formule étant obtenue en considérant n copies du champ des Φi. On peutdonc appliquer ces formules au cas du calcul du noyau de Green dans une boite avecr = |Λ|. Soit z = λ + iν avec ν > 0. En prenant A = ν + i(HΛ − λ) on obtient pourdeux sites a et b de la boite Λ:

GΛz (a, b) = i

∫ψaψ̄b exp(−i

∑

x,y∈Λ(HΛ − z)(x, y)Φx.Φy) dΦ

L’espérance de la fonction de Green ne fait donc apparaitre les potentiels qu’au traversde la transformée de Fourier de leur loi et c’est ce qui fait l’intérêt de cette formule.De plus on peut réécrire cette expression comme résultant de l’itération d’un certainopérateur Tz opérant sur des espaces de fonctions “super-symétriques” que l’on vadécrire ci dessous.

• Le “super-espace” Lr est défini comme l’ensemble des variables Φ = (Φ1, . . . ,Φr).

• Une “super-fonction” est une application de Lr dans G de la forme:

F (Φ) =∑

α

Fα(ϕ1, . . . , ϕr) Xα

où les Fα sont des fonctions complexes définies sur R2r et Xα des éléments de G.

La différentiabilité d’une telle fonction fait référence à celle de ses composantesFα.

15

• Une application de Lr dans lui-même est appellée “super-symétrie” si elle conservele pseudo produit scalaire

ΦΦ′ =r∑

k=1

Φk.Φ′k

Une super-fonction est dite super-symétrique si elle est invariante par toutes lessuper-symétries de Lr.

Ce formalisme est présenté dans la section 3 de l’annexe (B) avec une complicationsupplémentaire due au fait que l’on considère simultanément plusieurs copies du super-espace. On montre alors dans la section 4, en utilisant les résultats obtenus pour lesproduits de matrices aléatoires sur le ruban, que les oérateurs Tz opérant sur certainsespaces de Hilbert de fonctions super-symétriques différentiables ont des propriétésspectrales intéressantes. En particulier ce sont des opérateurs compacts et leur spectreconsiste en la valeur propre simple 1 et une partie contenue dans un disque de rayonstrictement plus petit que 1. Ceci permet de passer à la limite lorsque la taille desboites tend vers l’infini dans le calcul de l’espérance de la fonction de Green puis defaire tendre z vers l’axe réel en récupérant les propriétés de différentiabilité cherchées.

Chapter 1

Products of Random Matrices

Contents :

1. General Ergodic Theorems

2. Matrix Valued Systems

3. Group Action on Compact Spaces

(a) Definitions and Notations(b) Laplace Transform on the Space of Continuous functions(c) Laplace Transform on the Space of Hölder Continuous functions

4. Products of Independent Random Matrices

(a) The Upper Lyapunov Exponent(b) The Lyapunov Spectrum(c) Schrödinger Matrices

5. Markovian Multiplicative Systems

(a) The Upper Lyapunov Exponent(b) The Lyapunov Spectrum(c) Laplace Transform

6. Boundaries of the Symplectic Group

7. Notes and Complements

Part of the one dimensional or quasi-one dimensional theory of localization can be reduced to

the study of products of random matrices. One of the most important result in this direction

17

18 CHAPTER 1. PRODUCTS OF RANDOM MATRICES

is the extension to matrix valued random variables of the strong law of large numbers. Un-

fortunately the identification of the limit (called the Lyapunov exponent) in this theorem is

more complicated than in the classical case of real valued random variables. In particular this

limit can no longer be written as a single expectation. Moreover its determination involves the

computation of some invariant measure on the projective space.

1.1 General Ergodic Theorems

Let (Ω,F ,P) be a probability space and T be the set of time parameter that we willchoose to be the set of non negative integers (discrete case) or non negative real numbers(continuous case). Endowed with a semi-group {θt ; t ∈ T} of measure preservingtransformations of Ω the system (Ω,F , θt,P) will hence’ forth be called a dynamicalsystem. (In the continuous case a dynamical system is often called a “dynamical flow”).An invariant random variable is a measurable function Z such that Z ◦ θt = Z for allt ∈ T . A dynamical system (Ω,F , θt,P) is said to be ergodic if every invariant randomvariable is constant P-a.e.

Definition 1.1.1 A subadditive process is a measurable real valued process {X(t) ; t ∈T} such that :

X(0) = 0 , X(t+ s) ≤ X(t) +X(s) ◦ θt s, t ∈ T

If the inequality is replaced by an equality in the above definition then X(t) is called anadditive process.

The Kingman subadditive ergodic theorem stated below is one of the basic tools in thischapter .

Theorem 1.1.2 Let {X(n) ; n ≥ 0} be a subadditive process with discrete time suchthat :

(i) X(n) is integrable for all n

(ii) The sequence { 1nE{X(n)} ; n ≥ 1} is bounded below

Then there exists an integrable invariant random variable Z such that the sequence1nX(n) converges P-a.e. and in expectation to Z. Moreover E{Z} = infn≥1 1nE{X(n)}.

Proof:There are a lot of proofs available for the subadditive ergodic theorem. The originalone can be found in Kingman [58], and certainly the simplest one has been given byJ.M. Steele[98]

1.1. GENERAL ERGODIC THEOREMS 19

In the additive case we remark that X(n) =∑n−1

k=0 X(1) ◦ θk thus if X(1) is integrablethen conditions (i) and (ii) are satisfied and the Kingman Theorem reduces to theBirkhoff Theorem. It is well known that Theorem 1.1.2 is false for continous time(even in the additive case) but it becomes true with some extra regularity assumptionson the process {X(t) ; t > 0}

Corollary 1.1.3 Let {X(t) ; t ≥ 0} be a subadditive process with continuous time suchthat

(i) X(t) is integrable for all t ∈ T ,

(ii) The set {1tE{X(t)} ; t > 0} is bounded below,

(iii) The process X(t) is separable and M = sup0≤s0 1tE{X(t)}.

Proof:For n = 0, 1, 2 . . . we can apply Theorem 1.1.2 to the process X(n). If n is the integralpart of t then we can write :

X(n+ 1) −X(n + 1 − t) ◦ θt ≤ X(t) ≤ X(n) +X(t− n) ◦ θn

We remark that |X(n + 1 − t) ◦ θt| ≤ M ◦ θn and |X(t − n) ◦ θn| ≤ M ◦ θn. TheBorel-Cantelli lemma implies that 1nM ◦ θn converges P-a.e. to zero and this yields theresult.

The Birkhoff Theorem admits an important improvement which essentially says thatwhen the limit is not obviously zero then it has to be positive and this property willbe very useful in order to prove that the limit is positive in the ergodic case.

Theorem 1.1.4 Let {X(n) ; n ≥ 1} be an integrable additive process with discrete timesuch that X(n) converges P-a.e. to +∞. Then the limit random variable given by theBirkhoff ergodic theorem has a positive expectation.

Proof:Suppose that the conclusion is false and let us denote by Z the limit. We know that Zis a nonnegative random variable thus if we suppose that it as a nul expectation then Zis zero P-a.e. For a positive real number ǫ we define the sequence of random variables

An = m(

n⋃

k=1

[X(k) − ǫ , X(k) + ǫ] )


where m(I) is the Lebesgue measure of the set I. Using the convergence of 1nX(n) to

zero one can conclude that 1nE{An} also converges to zero. We now write :

An −An−1 ◦ θ ≥ 2ǫ 1{∩nk=2{|X(k) −X(1)| > 2ǫ}}E{An} − E{An−1} ≥ 2ǫ P{∩nk=2{|X(k) −X(1)| > 2ǫ}}

≥ 2ǫ P{∩∞k=2{|X(k) −X(1)| > 2ǫ}}

From lim 1nE{An} = 0 one obtains that P{∩∞k=2{|X(k) − X(1)| > 2ǫ}} = 0 and bystationarity that

P{∪∞n=1 ∩∞k=1 {|X(k + n) −X(n)| > 2ǫ}} = 0

This last relation contradicts the convergence of X(n) to ∞.

1.2 Matrix Valuated Systems

Let GL(d,R) be the group of linear automorphisms of Rd. For a matrix g in GL(d,R)g′ denotes the transpose, g−1 the inverse, tr(g) the trace and det(g) the determinant.Every matrix g ∈ GL(d,R) has a polar decomposition g = k(g)a(g)u(g) where u(g)and k(g) are orthogonal matrices and a(g) is a diagonal matrix with diagonal entriesa1(g) ≥ a2(g) ≥ . . . ≥ ad(g) > 0. The matrix a(g) is well defined since the scalars ai(g)are the root-squares of the eigenvalues of g′g (but in general the matrices k(g) and u(g)are not uniquely defined). We denote by Λp(Rd) the vector space of alternating p linearforms on the dual of Rd. For u1, u2, . . . , up in R

d and f1, f2, . . . , fp in the dual of Rd let

m be the matrix defined bymi,j = fi(uj). The formula u1∧u2∧. . .∧up (f1, f2, . . . , fp) =det(m) defines an element u = u1 ∧ u2 ∧ . . . ∧ up of Λp(Rd) called a decomposable p-vector. For a p-vector u we also denote by u the d × p matrix of the coordinates ofu1, u2, . . . , up in the usual basis of R

d.

Lemma 1.2.1 It is well known that :

(i) The decomposable p vectors constructed with the vectors of a basis of Rd span thelinear space Λp(Rd)

(ii) The p-vector u1 ∧ u2 ∧ . . . ∧ up is nonzero iff the system (u1, u2, . . . , up) is linearlyindependent

(iii) The linear extension to the whole of the space Λp(Rd) of the formula < u1 ∧ u2 ∧. . . ∧ up , v1 ∧ v2 ∧ . . . ∧ vp >= det(u′v) defines an inner product on Λp(Rd).

1.2. MATRIX VALUATED SYSTEMS 21

In the sequel we will always use the norm on Λp(Rd) associated to the dot productdefined in Lemma 1.2.1(iii), that is ‖u‖2 = det(u′u).For g ∈ GL(d,R) we let Λpg be the linear automorphism of Λp(Rd) defined by Λpg(u1∧u2 ∧ . . . ∧ up) = gu1 ∧ gu2 ∧ . . . ∧ gup. We note that Λpmn = ΛpmΛpn thus if k isan orthogonal matrix so is Λpk. We can deduce from these properties that ‖Λpg‖ =a1(g)a2(g) . . . ap(g) and this implies ‖Λpg‖ ≤ ‖g‖p.

When d = 2ℓ there is a subgroup of GL(d,R) of particular interest called the symplecticgroup of order ℓ and denoted by SP (ℓ,R).

Definition 1.2.2 The symplectic group SP (ℓ,R) is the set of matrices g in GL(d,R)satisfying :

g′Jg = J with J =(

0 Iℓ−Iℓ 0

)where Iℓ is the identity matrix of order ℓ.

We remark that for ℓ = 1 the symplectic group is the unimodular group SL(2,R) butthat SP (ℓ,R) is a proper subgroup of SL(2ℓ,R) for greater values of ℓ. One can foundmore details about the symplectic group in section 6 of this chapter. The proof of thefollowing Lemma is an easy exercise :

Lemma 1.2.3 Let g be a matrix in SP (ℓ,R). Then one has :

(i) det(g) = 1 and g′ ∈ SP (ℓ,R)

(ii) If λ is an eigenvalue of g so is 1/λ

(iii) In the polar decomposition of g we have a(g)i = a(g)−1d−i+1 for any i = 1 . . . d

For a sequence γ1 ≥ γ2 ≥ . . . ≥ γd of real numbers we let λ1 > λ2 > . . . > λr be theset of its distinct values . We call multiplicity of λi the number of occurence of λi inthe sequence (γ1, γ2, . . . , γd). The next Theorem is called the “Deterministic” Oseledectheorem.

Theorem 1.2.4 Let {g(n) ; n ≥ 0} be a sequence in GL(d,R) and let assume that thereexist numbers γ1 ≥ γ2 ≥ . . . ≥ γd such that the product U(n) = gn−1 . . . g0 satisfies forp = 1, 2, . . . , d :

(i) limn→∞ 1n log ‖Λpg(n)‖ = 0,

(ii) limn→∞ 1n log ‖ΛpU(n)‖ = γ1 + . . . + γp.Let us denote by r the number of distinct Lyapunov exponents, then there exists astrictly decreasing family of subspaces of Rd such that


(i) Rd = V1 ⊃ V2 ⊃ . . . ⊃ Vr ⊃ Vr+1 = {0},

(ii) v ∈ V i \ V i+1 ⇐⇒ limn→∞ 1n log ‖U(n)v‖ = λi i = 1, 2, . . . , r,

(iii) dim V i − dim V i+1 = multiplicity of λi i = 1, 2, . . . , r.

Proof:The original proof is in Oseledec [86] and a simpler one can be found in Ledrappier[77].

In the particular case of products of symplectic matrices Lemma 1.2.3(iii) implies thatγi = −γd−i+1 for i = 1, 2, . . . , d and since a symplectic matrix has a determinant equalto one we obtain γ1 ≥ γ2 ≥ . . . ≥ γℓ ≥ 0

We now introduce the important concept of multiplicative process. Such processes aredefined and studied in full generality for example in Bougerol [8],[7] but we will use asimpler definition adapted to our situation.

Definition 1.2.5 Let {U(t) ; t ∈ T} be a process taking its values in the group GL(d,R)and defined on the dynamical system (Ω,F , θt,P). We say that it is a mutiplicativeprocess if :

• U(0) = I , U(t+ s) = (U(s) ◦ θt)U(t) s, t ∈ T

Such a process is said to be regular if :

• In the discrete case ( log+ ‖U(1)‖ + log+ ‖U−1(1)‖) is integrable.

• In the continuous case it is a separable stochastic process and the random variableM = ( sup0≤t≤1 log

+ ‖U(t)‖ + sup0≤t≤1 log+ ‖U−1(t)‖) is integrable.

We note that | log ‖g‖ | ≤ log+ ‖g‖ + log+ ‖g−1‖ whenever g ∈ GL(d,R) and in theparticular case of matrices in SP (ℓ,R) we have ‖g‖ = ‖g−1‖ ≥ 1. Moreover, theintegrability conditions in (ii) are also satisfied by the exterior powers of the matricesU(t) since ‖Λpg‖ ≤ ‖g‖p. In the discrete case a mutiplicative process can be written asa product of matrices of GL(d,R) since if we denote by g(n) the matrix U(1) ◦ θn thenU(n) = g(n− 1) . . . g(0).The application of the deterministic Oseledec’s theorem to multiplicative processes isoften refered as the “Random” Oseledec theorem :

Theorem 1.2.6 Let {U(t) ; t ∈ T} be a regular multiplicative process defined on anergodic dynamical system. Then there exists an invariant set Ω0 of full P measure suchthat for ω ∈ Ω0 one has :


(1) limt→∞ 1t log ‖ΛpU(t)‖ = limt→∞ 1tE{log ‖Λ

pU(t)‖} = γ1+. . .+γp for p = 1, 2, . . . , d.( The numbers γ1 ≥ γ2 ≥ . . . ≥ γd are called the Lyapunov exponents of the pro-cess U(t)).

(2) Let r be the number of distinct Lyapunov exponents, then there exists a strictlydecreasing sequence of measurable subspaces of Rd denoted by V iω such that

(i) Rd = V1ω ⊃ V2ω ⊃ . . . ⊃ Vrω ⊃ Vr+1ω = {0},(ii) v ∈ V iω \ V i+1ω ⇐⇒ limt→∞ 1t log ‖U(t)v‖ = λi i = 1, 2, . . . , r,(iii) dim V iω − dim V i+1ω = multiplicity of λi i = 1, 2, . . . , r,(iv) V iθtω = U(t)V iω.

Proof:

For p = 1, . . . , d the process X(t) defined by log ‖ΛpU(t)‖ is subadditive and the in-tegrability conditions in the Definition 1.2.5 make it possible to apply the Kingmansubadditive ergodic Theorem to the process X(t). This yields conclusion (1) and if wedefine g(n) = U(1) ◦ θn then U(n) = g(n − 1) . . . g(0) and we obtain the conclusions2 (i) and (ii) and (iii) from Theorem 1.2.4 at least for discrete time. The same resultfollows for continuous time if we remark that for n ≤ t ≤ n+ 1 one has :

| log ‖U(t)v‖ − log ‖U(n)v‖ |

= | log ‖U(t)v‖‖U(n)v‖|

= | log ‖(U(t− n) ◦ θn)U(n)v‖‖U(n)v‖ |

≤ max(| log ‖U(t− n) ◦ θn‖ | , | log ‖U−1(t− n) ◦ θn‖ |) ≤M ◦ θn

and the Borel-Cantelli lemma yields the result. It remains only to check the measura-bility assumptions on the spaces V iω and conclusion 2(iv); these are consequence of therelation V iω = {v ∈ Rd; limn→∞ 1n log ‖U(n)v‖ ≤ λi}.

Dynamical systems for which {θt ; t ∈ T} form a group of invertible mappings of Ω willbe called invertible dynamical systems. For such systems one can apply the precedingtheorems in the “two directions” of T and it turns out that the limits at +∞ and−∞ are closely related. We will add the superscript ± to the Lyapunov exponentsand subspaces defined in Theorem 1.2.6. We still consider limits for t → ±∞ but thenormalization factor is now 1| t | rather than

1t .

Proposition 1.2.7 Let us denote by (Ω,F , θt,P) an invertible ergodic dynamical sys-tem and by U(t) a regular multiplicative process. Then:


(i) −γp = −+γd−p+1 p = 1 . . . d,(ii) −λi = −+λr−i+1 i = 1 . . . r,(iii) Rd = +V iω ⊕ −Vr−iω i = 1 . . . r − 1.The spaces ±V iω are defined for ω in an invariant set Ω0 of full P-measure

Proof:Because U(−t) = U−1(t)◦θ−t the process U(t) is multiplicative. This implies ap(U(−t)) =a−1d−p+1(U(t)◦θ−t). The invariance of the probability measure P yields E{log ap(U(−t))} =−E{log ap(U(t))} proving (i) and (ii). To prove (iii) we remark that we have dim±V iω =(# of exponents ≤± λi) and (ii) implies that the sum of the dimensions of the subspacesappearing in (iii) is equal to d. To prove that the intersection of these subspaces is thezero vector, we pick a unit vector v in +V iω ∩ −Vr−iω and we consider the randomvariables Z±i,n(v) =

1n log(‖U(n)v‖/‖v‖) −± λi. We can write v = (U(−n) ◦ θn)U(n)v

and obtain :0 ≤ n(−λr−i ++ λi + Z−r−i,−n ◦ θn(U(n)v) + Z+i,n(v))

We know from (ii) that −λr−i = −+λi+1 and from Theorem 1.2.6 that U(n)v ∈− Vr−iθnω.Since Z±i,n(v) converges in probability to zero for v ∈± V iω we get a contradiction.

The decomposition (iii) is often called a “splitting of Rd”

This proposition has an important consequence for multiplicative processes with valuesin SP (ℓ,R). In this case there is no need to put a superscript ± to the Lyapunovexponents since −γp = −+γd−p+1 =+ γp and the “splitting of Rd” has a simpler inter-pretation.

Corollary 1.2.8 Let (Ω,F , θt,P) be an invertible ergodic dynamical system and U(t)be a regular multiplicative process with values in SP (ℓ,R). Then with probability onefor any non-zero vector v ∈ Rd we have :

max( limt→−∞

1

|t| log ‖U(t)v‖ , limt→+∞1

tlog ‖U(t)v‖) ≥ γℓ)

Proof:If s is the number of non negative distinct Lyapunov exponents then r = 2s and weapply (ii) and (iii) of Proposition 1.2.7 with i = s. We then remark that if v 6∈± Vsωthen the above limits are greater than γℓ.

When the lowest non negative exponent γℓ is positive this corollary says that for anynonzero vecror v the process U(t)v has an “hyperbolic behavior”,i.e. that it is exponen-tially growing in at least a direction of T . One of the essential problems in the sequel


will be to prove that γℓ is positive under suitable hypothesis. We will see in the nextsections that this can be done for independent and Markovian models. Obviously theproblem is much simpler in the case d = 2 where γℓ is also the upper exponent !

We end this section with the classical construction of a product dynamical systemassociated with (Ω,F , θt,P) and a regular multiplicative process U(t). This will turnout to be very useful when we try to obtain an explicit formula for the upper Lyapunovexponent γ1. Let P (R

d) be the projective space of Rd, that is the set of directions inRd and let B be its Borel sigma-algebra . For a non-zero vector v in Rd we denote by v̄

the image of v in P (Rd). GL(d,R) acts on P (Rd) by the formula gv̄ = gv and we set :

Ω̃ = Ω × P (Rd), F̃ = F × B, θ̃t(ω, v̄) = (θtω,U(t, ω)v̄)

It is easy to check that we have the semigroup property θ̃t+s = θ̃t ◦ θ̃s and that thefunction v →֒ log(‖U(t)v‖/‖v‖) is actually a function of v̄.

Proposition 1.2.9 Let us assume that (Ω,F , θt,P) is an ergodic dynamical systemand that P̃ is a θ̃t-invariant probability measure with projection P on Ω. Then :

(i) The function t −→∫

log(‖U(t, ω)v‖‖v‖−1) dP̃(ω, v̄) is additive.

(ii) If P̃{(ω, v̄) ; ω ∈ Ω0, v ∈ V2ω} = 0 (notations of Theorem 1.2.6) then:

γ1 =

∫log

‖U(1, ω)v‖‖v‖ dP̃(ω, v̄)

(iii) If (Ω̃, F̃ , θ̃t, P̃) is ergodic and P̃{(ω, v̄) ; ω ∈ Ω0, v 6∈ V2ω} > 0 then the formula(ii) holds

Proof:Let X(t, ω, v̄) = log(‖U(t, ω)v‖‖v‖−1) be defined on the dynamical system (Ω̃, F̃ , θ̃t, P̃).It is readily seen thatX(t) is an additive process and the invariance of P̃ gives conclusion(i). To prove (ii) we remark that X(1) is integrable thus the Birkhoff ergodic Theorem

implies that the sequence 1nX(n) converges P̃-a.e. and in expectation to an invariantrandom variable Z. The Oseledec Theorem 1.2.6 implies that for P-almost all ω andfor all v in Rd − V2ω this sequence converges to γ1 thus this sequence converges to thesame limit P̃-a.e. It follows that Z = γ1 =

∫Z(ω, v̄) dP̃(ω, v̄) and we obtain (ii). The

same proof works for (iii) since we already know that Z is constant thus it is enoughto have the above convergence on a set of positive P̃-measure.

We will see in the next sections how to construct measures P̃ satisfying (ii) or (iii) inthe independent and Markovian cases.


1.3 Group Action on Compact Spaces

1.3.1 Definitions and Notations

Let G be a locally compact σ-compact metric group with unit e and let B be a metrictopological space.

Definition 1.3.1 We say that G acts on B if one can associate continuously to each(g, b) in G×B an element gb of B such that :

(i) g1(g2b) = (g1g2)b ∀g1 , g2 ∈ G , b ∈ B

(ii) eb = b ∀b ∈ B

We define the “pseudo-convolution” of a probability measure µ on G and a measure νon B as the unique measure µ ∗ ν on B determined by :

µ ∗ ν(f) =∫f(gb) dµ(b) dν(g)

where f is any bounded measurable function on B. We use the same symbol than forthe ordinary convolution on G since when B = G the two definitions coincide. We alsoremark that (µ1 ∗ µ2) ∗ ν = µ1 ∗ (µ2 ∗ ν).

From now on we fix a probability measure µ on G. The nth power of convolution of µon G is denoted by µn. We also denote by Tµ the smallest closed sub-semi-group of Gcontaining the support of µ and by Gµ the smallest closed sub-group of G containingthe support of µ. A measure ν on B is said to be µ invariant if µ∗ν = ν. If B is compactthen it is readily seen that that there always exist invariant probability measures (Take

any weak limit point of the sequence 1n∑n

j=1 µn∗m where m is any probability measure

on B). Uniqueness of the invariant measure will be one of the problems discussed inthe next section.

On abelian groups the harmonic analysis of µ is carried out using the Fourier-Laplacetransform associated to the set of exponentials on G. Unfortunately general groups(Simple Lie Groups for instance) have very few exponentials. This is why we need tointroduce cocycles generalizing the notion of exponentials.

Definition 1.3.2 A continuous map σ from G× B with values into (0 , ∞) is calleda cocycle if

σ(g1g2, b) = σ(g1, g2b)σ(g2, b) g1 , g2 ∈ G, b ∈ B

1.3. GROUP ACTION ON COMPACT SPACES 27

We remark that if B is reduced to a single point then a cocycle is nothing but anexponential. Products and real powers of cocycles are again cocycles and σ(e, b) = 1for any b ∈ B. A cocycle σ is said µ-integrable if the function σ̄(g) = supb∈B σ(g, b) isµ-integrable. One remarks that if we set σ̃(g) = supb∈B | log σ(g, b)| then exp(σ̃(g)) is µintegrable iff σ and σ−1 are µ integrable. The following lemma is an easy consequenceof the definition of a cocycle.

Lemma 1.3.3 Let σ be a µ integrable cocycle, then the sequences an and bn defined by:

an =

∫σ̄(g) dµn(g) and bn = sup

{b ∈ B}

∫σ(g, b) dµn(g)

satisfy an+m ≤ an am and bn+m ≤ bn bm.

We will often use the following Proposition :

Proposition 1.3.4 Let ν be an invariant probability measure and let σ be a cocyclesuch that σ̃(g) is µ integrable. Then the sequence 1n log σ(U(n), b) converges P ⊗ ν-almost surely to some random variable Z. If we assume furthermore that the se-quence σ(U(n), b) converges to +∞ for P ⊗ ν-almost all (ω , b), then the expectation∫Z(ω, b) dP(ω)dν(b) is strictly positive.

Proof:

Let (Ω̃, F̃ , θ̃t, P̃) be the product dynamical system defined by :

Ω̃ = Ω ×B, P̃ = P ⊗ ν, θ̃(ω, b) = (θ(ω), ω0b).

The process X(n) = log σ(U(n), b) is additive on (Ω̃, F̃ , θ̃t, P̃) and the Birkhoff ergodictheorem implies that n−1X(n) converges P̃-almost surely. The last assertion is an easyconsequence of Proposition 1.1.4

Proposition 1.3.5 Let σ be a cocycle such that

(i) There exists a positive number τ such that σt is integrable for |t| ≤ τ ,

(ii) There exists an integer N such that supb∈B∫

log σ(g, b) dµN < 0.

Then there exists a positive real number α such that for 0 < t ≤ α there exist positiveconstants Ct < +∞ and ρt < 1 such that :

supb∈B

∫σt(g, b) dµn(g) ≤ Ct ρnt n = 1, 2, . . .


Proof:

The inequality exp(t| log σ|) ≤ σt+σ−t implies that for any integer p then supb∈B∫

exp t| log σ(g, b)| dµpand supb∈B

∫| log σ(g, b)|p dµp are finite. From the inequality :

σt ≤ 1 + t log σ + t2

2| log σ| exp(t| log σ|) 0 ≤ t ≤ τ

one can deduce that if we set bn(t) = supb∈B∫σ(g, b)t dµn(g) then for 0 ≤ t ≤ τ then

aN (t) is less than

1 + t supb∈B

∫log σ(g, b) dµN (g) +

t2

2supb∈B

∫| log σ(g, b)| et| log σ(g,b)| dµN (g)

The hypothesis (ii) implies that for t sufficiently small one has bN (t) < 1 and theconclusion follows from the subadditivity of the sequence log bn(t) which implies thatthis sequence converges to its infimum.

We now define the operators extending the notion of Fourier-Laplace transform of ameasure on a commutative group. For a cocycle σ and a complex number z we defineformally the operator Tσ,z on the set B(B) of bounded measurable function on B bythe formula :

Tσ,zf(b) =

∫σz(g, b)f(gb) dµ(g)

Tσ,z is a bounded operator on B(B) when |σz| is µ-integrable. It is readily seen thatT nσ,z is defined by the same formula but with µ replaced by µ

n. When z = 1 we write Tσinstead of Tσ,z and we only write T when the cocycle σ is equal to the constant one. Theoperator Tσ is non negative thus ‖Tσ‖ = supb∈B

∫σ(g, b) dµ and Tf(b) =

∫f(gb) dµ(g)

defines a Markov kernel on B.

Proposition 1.3.6 Let τ be a positive real number such that the cocycle σt is µ inte-grable for |t| ≤ τ and let us denote by ρ(t) the logarithm of the spectral radius of Tσ,tacting on B(B). Then for |t| ≤ τ we have :

(i) The function t →֒ ρ(t) is convex and ρ(0) = 0,

(ii) ρ(t) ≥ t∫

log σ(g, b) dµ(g) dν(b) for any invariant probability measure ν on B.

(iii) If the function ρ admits a derivative at the origin then its value is given by :

dρ(t)

dt|t=0 =

∫log σ(g, b) dµ(g) dν(b)

(Hence this integral is independent from the choice of ν)


Proof:

The spectral radius can be computed from the formula :

ρ(t) = lim1

nlog sup

b∈B

∫σt(g, b) dµn(g)

Hence (i) is a consequence of the Hölder inequality. We now remark that for an invariantmeasure ν the quantity

∫σt(g, b) dµn(g) dν(b) is additive in n and thus (ii) is a simple

consequence of Jensen’s inequality for the logarithm function. The function ρ(t) −t∫

log σ(g, b) dµ(g) dν(b) is non negative so that its derivative at the origin, if any hasto be zero proving conclusion (iii).

The next proposition extends a well known property of the Fourier-Laplace transform.

Proposition 1.3.7 If there exists τ > 0 such that the cocycle σt is µ-integrable for|t| ≤ τ , then z →֒ Tσ,z is an analytic function from a neighbourhood of the origin to theBanach space of bounded operators on B(B).

Proof:

Note that if we set Dnf(b) =∫

( log σ(g, b))nf(gb) dµ(g) then we have the upper bound‖Dn‖ ≤ supb∈B

∫| log σ(g, b)|n dµ(g) ≤

∫σ̃n(g) dµ(g) and thus the series

∑ 1n! |z|n ‖Dn‖

is convergent for |z| ≤ τ .

When µ is a “smooth” measure on G we will see in subsection 2 that these operatorshave nice properties on the space C(B) of continuous functions but this is no longer thecase for general µ and this is why we will introduce the spaces Lα of Hölder continuousfunctions in subsection 3.

1.3.2 Laplace Operators on the Space of Continuous functions

Throughout this subsection we assume that B is a compact metric space.

If σ is a µ-integrable cocycle then Tσ is a bounded operator on C(B). We denote byT ∗σ its transpose. It acts on the measures on B and T̆σ denotes the operator associatedto the measure µ̆ image of µ by g →֒ g−1.If m is a measure on B we denote by

• gm the image of m by the map b →֒ gb,

• fm the measure of density f with respect to m for f ∈ C(B),


• < f, h >m the continuous bilinear form∫fh dm defined on the Banach space

C(B).

Let us assume that the support of m is equal to B and that for any g ∈ G the measuresgm and m have the same null sets. Then if the density r(g, b) =

dg−1mdm (b) is a

continuous function of the couple (g, b) it is a cocycle called a m-Radon-Nikodymcocycle.

Proposition 1.3.8 Let σ be a µ-integrable cocycle and let r be a m-Radon-Nikodymcocycle. Let us assume that rσ−1 is µ-integrable. Then, for any positive integer k andany complex number λ we have :

(i) < T̆σf , h >m=< Trσ−1h , f >m f, h ∈ C(B)

(ii) [ker(T̆σ − λI)k]m ⊂ ker(T ∗rσ−1 − λI)k

Proof:If f , h ∈ C(B), then :

[T ∗rσ−1(fm)](h) =∫f(b)r(g, b)σ−1(g, b)h(gb) dµ(g) dm(b)

=

∫f(b)h(g−1b)σ−1(g−1, b) dµ̆(g) dgm(b)

=

∫f(gb)σ(g, b)h(b) dµ̆(g) dm(b)

= [(T̆σf)m](h)

The assertion (ii) is a simple consequence of (i). Notice that (i) implies T ∗rm = m.

We now assume that the action of G on B admits a continuous lifting i.e. that thereexists a continuous map s : B →֒ G and a fixed element b0 ∈ B such that s(b)b0 = bfor any b ∈ B. This implies that a sequence bn in B converges to b iff there exists asequence gn in G converging to e such that bn = gnb. In others words, the uniformstructure on B induced by the action of G is identical to its initial uniform structureof compact space.

Proposition 1.3.9 Suppose that µ has a density with respect to a Haar measure on Gand that σ is an integrable cocycle. Then Tσ is a compact operator on C(B).

Proof:Let dg be a right Haar measure on G and dµ = φdg . One can write for u ∈ G

Tσf(ub) =

∫f(gb)σ(g, b)σ−1(u, b)φ(gu−1) dg


and :

|Tσf(ub) − Tσf(b)| ≤ ‖f‖∫σ(g, b)|φ(g) − σ−1(u, b)φ(gu−1)| dg

If φ is a continuous function with compact support then we can conclude that Tσmaps a bounded set of C(B) in an equicontinuous set and the compactness followsfrom the Arzela-Ascoli theorem. In the general case there exists a sequence φn(g) ofcontinuous functions with compact support converging to φ(g)σ̄(g) in L1(dg). Takingψn(g) = σ̄

−1(g)φn(g) and denoting by nTσ the operator associated to the measureµn = ψn dg we obtain :

‖nTσ − Tσ‖ ≤∫

|φ(g)σ̄(g) − ψn(g)| dg

We can conclude that Tσ is compact as limit in norm of compact operators.

One remarks that the above proof is somewhat simpler in the following case. Assumethat B is a homogeneous space under the action of a subgroup K ⊂ G and that σ isa K invariant cocycle (that is σ(k, b) = 1 for k ∈ K and b ∈ B). Then in the aboveproof one can chooses u ∈ K and the term σ(u, b) disappears. Actually this will be thegeneral situation for G = SP (ℓ,R), K the orthogonal subgroup of G and B a boundaryof the symplectic group.

For a compact operator T on a Banach space it is known that T and T ∗ have the samespectrum. More precisely, the dimensions of the kernels of (T − λI)k and (T ∗ − λI)kare the same for any positive integer k and non-zero complex number λ. This yields :

Proposition 1.3.10 Let σ be a cocycle and let r be a m-Radon-Nikodym cocycle suchthat σ and rσ−1 are µ and µ̆ integrable. If µ has a density with respect to a Haarmeasure on G, then for any positive integer k and non zero complex number λ we have:

(i) (ker(T̆σ − λI)k)m = ker(T ∗rσ−1 − λI)k

(ii) dim(ker(T̆σ − λI)k) = dim ker(Trσ−1 − λI)k

Proof:It is enough to apply Proposition 1.3.8 to the cocycles σ and rσ−1 replacing µ by µ̆ forthe latter and to take in account the above remark.

An important consequence of proposition 1.3.10 is that when µ has a density on G thenfor a µ and µ̆ integrable m-Radon-Nikodym cocycle r the operators T and Tr have thesame spectral properties . Thus we can “shift” the spectral properties of T which arewell known for a Markov operator to the operator Tr which is much more difficult tostudy.


Definition 1.3.11 A Markov operator T on the Banach space C(B) is said aperiodicif 1 is the only eigenvalue of modulus one of T .

The following result can be understood as a generalization of the Peron-Frobeniustheorem for positive operators :

Proposition 1.3.12 Let r be a µ and µ̆ integrable m-Radon-Nikodym cocycle. Let usassume that µ has a density, that µ and µ̆ admit unique invariant measures ν and ν̆ andthat T and T̆ are aperiodic. Then the invariant measures µ and µ̆ admit continuousdensities ψ and ψ̆ with respect to m and we have the decompositions :

(i) T nf = ν(f) +Qnf

(ii) T nr f = m(f)ψ +Qnr f

the operators Q and Qr having spectral radius strictly less than one. It follows that thespectral radius of Tr is equal to one and that the sequence ‖T nr ‖ is bounded.

Proof:The unicity of the invariant measure added to the compactness of the operators T givenby Proposition 1.3.9 implies that the eigenvalue 1 is of geometric multiplicity one. Thealgebraic multiplicity is also one since the equation T ∗τ = τ + ν has obviously nosolution τ in the space of measures on B (otherwise the total mass of ν would be zero!). This implies immediately the decomposition of T .From Proposition 1.3.10 we deduce that T and Tr have the same spectral properties.The relation ν̆ = ψm is then a consequence of the equality ker(T̆ ∗−I) = (ker(Tr−I))m.

Proposition 1.3.13 Under the hyphothesis of Proposition 1.3.12 if we denote by ρ(t)the logarithm of the spectral radius of Tr,t then we have :

(i) ρ(t) is a convex function on [−1 1] with ρ(0) = ρ(1) = 0.

(ii) If there exists an integer N with supb∈B∫

log r(g, b) dµN (g) < 0, then ρ(t) < 0 fort ∈ (0 , 1)

(iii)dρ(t)dt

|t=0 =∫

log r(g, b) dµ(g) dν(b)

Thus if this integral is negative then ρ(t) < 0 for t ∈ (0 , 1)

Proof:As already noticed in the proof of Proposition 1.3.6 the convexity of ρ(t) is a conse-quence of Hölder’s inequality and (i) follows from Proposition 1.3.8. Conclusion (ii) is


a consequence of Proposition 1.3.5 which implies that for a small positive t we haveρ(t) < 0. We know that the spectrum of T splits in the simple eigenvalue 1 and apart contained in a disk of radius strictly less than one. Furthermore Proposition 1.3.9(stated on the space C(B) rather than on B(B) ) implies that z →֒ Tr,z is analytic andthe analytic perturbation theory (See Kato [54]) yields that ρ(t) is smooth at the originand (iii) follows from Proposition 1.3.6(iii).

One can apply the preceeding results in the particular case of the action of GL(d,R)on the projective space B = P (Rd). In this situation there exists a probability measurem invariant under all the rotations of SO(d,R). It is called the Cauchy measure andit satisfies :

r(g, v̄) =dg−1mdm

(v̄) =

( ‖v‖‖g v‖

)d

This cocycle is µ and µ̆ integrable whenever the random variables ‖g‖d and ‖g−1‖d areµ-integrable.

Using the notations and definitions that will be introduced in section 6 we now state aproposition which will allow us to check the hypotheses of Proposition 1.3.12.

Proposition 1.3.14 Assume that the probability measure µ has a density on the sym-plectic group SP (ℓ,R), then:

• There exists a unique µ-invariant probability measure for the operator T actingon each boundary LI .

• The operator T acting on C(LI) is aperiodic.

Proof:The unicity of the invariant measure will follow from Proposition 1.4.25 and 1.6.2,moreover Proposition 1.3.9 yields the compactness of T . To obtain the aperiodicity itis enough to prove this property for the “maximal boundary” associated to I = {1 . . . ℓ}that we will call B. Let f ∈ C(B) with Tf = λf , ‖f‖ = 1 , |λ| = 1 and let us pickb ∈ B with |f(b)| = 1. The equation T nf = λnf first implies that |f(gb)| = |f(b)| forg ∈ Tµ. Secondly the same eigenvalue equation yields that f(gb) = λnf(b) when g isin the support of µn say Sn. The absolute continuity of µ implies that it is impossiblethat all the sets Sn be disjoint since their Cauchy measure is bounded below (SeeTutubalin[100] for a computation of the Cauchy measure on B using the Haar measureof G) and it follows that λ is a root of the unity. The unicity of the invariant measurethen implies that λ = 1.

The decomposition in Proposition 1.3.12 part (i) does not hold in general if we do notassume the existence of a density for µ. If we only assume that µ is “spread out”,i.e.


there exists an integer n such that µn is not orthogonal to the Haar measure of G,then the operator T is quasi-compact (See Brunel and Revuz [12]). Such an operatorhas a finite number of eigenvalues of modulus one and their multiplicities are finite.Moreover the rest of the spectrum is contained in a disk of radius strictly less than one.Unfortunately this property is not true for the operators Tσ.

1.3.3 The Laplace Operators on the Space of Hölder Continuous Func-tions

We still assume that B is a compact metric space.

We denote by δ the distance of B. For a positive real number α we define the space ofHölder continuous functions of order α say Lα as the set of continuous functions f onB such that :

mα(f) = sup(a,b)∈B×B ; a6=b

|f(a) − f(b)|δα(a, b)

< +∞

Endowed with the norm ‖f‖α = ‖f‖∞ +mα(f) the linear space Lα is a Banach space.Moreover the map α →֒ Lα is nonincreasing and the identity map from Lα to Lβ iscontinuous for β ≤ α.

Let σ be a cocycle defined on the set {(a, b) ∈ B ×B a 6= b} by :

σ(g, (a, b)) =δ(ga , gb)

δ(a , b)

The following proposition is essentially due to Le Page [79] :

Proposition 1.3.15 Assume that the cocyle σ defined above satisfies :

(i) There exists a positive number τ such that σt is integrable for |t| ≤ τ

(ii) sup{(a,b)∈B a6=b}∫

log σ(g, (a, b)) dµN (g) < 0 for some integer N .

Then there exists a positive real number α0 such that for 0 < α ≤ α0 we have :

(i) T is a bounded operator on Lα,

(ii) There exist constants Cα


Proof:We can write

|Tf(a) − Tf(b)|δα(a, b)

≤ mα(f)∫σ̄α(g) dµ(g)

The obvious bound ‖Tf‖∞ ≤ ‖f‖∞ and hypothesis (i) yield conclusion (i). We knowfrom Proposition 1.3.5 that for α sufficiently small then

∫σα(g, (a, b)) dµn(g) ≤ Cα ρnα

hence we obtain for such an α :

|T nf(a) − T nf(b)|δα(a, b)

≤ mα(f)∫σα(g, (a, b)) dµn(g)

≤ mα(f)Cα ρnαLet now ν be an invariant probability measure then :

|T nf(a) − ν(f)| = |∫f(ga) dµn −

∫f(gb) dµn(g) dν(b)|

≤∫

|f(ga) − f(gb)| dµn(g) dν(b)

≤ mα(f)∫σα(g, (a, b)) dµn(g) dν(b)

≤ mα(f)Cα ρnα sup(a,b)∈B×B

δα(a, b)

This proves conclusion (ii).

This result gives the exponential convergence of T n to the rank one operator N(f) =ν(f). In some sense it is an operator form of the classical Dœblin Theorem for Markovchains.

Corollary 1.3.16 Under the hypothesis of Proposition 1.3.15 for α sufficiently smallthe operator T on Lα has eigenvalue 1 and the rest of the spectrum is contained in adisk of radius strictly less than one. Moreover T admits the decomposition :

T nf = ν(f) +Qnf f ∈ LαWhere ν is the invariant measure and Q an operator of spectral radius strictly less than1 on Lα.

We will see in the next section that one can check the hypotheses of Proposition 1.3.15without any smoothness assumption on µ. Using the theory of analytic perturbationsit is possible to extend this property to the operators Tσ,z for small values of z, in orderto obtain central limit and large deviations theorems (See Bougerol-Lacroix [9]). Wewill not dwell on this kind of result there.


1.4 Products of Independent Random Matrices

This section contains the essential results of the theory of products of independent ran-dom matrices. Such products yield one of the most interesting example of multiplicativeprocess. Following Ledrappier [77] we obtain in the first subsection some immediateconsequences of Proposition 1.2.9. We also give the classical result of Furstenberg [30]establishing the positivity of γ1. Under stronger hypotheses we obtain in the secondsubsection the simplicity of the Lyapunov spectrum. This last part is essentially bor-rowed from Guivarc’h-Raugi [44].

If µ is a probability measure on the linear group GL(d,R) we define the followingergodic dynamical system :

• Ω is the set of sequences ω = (ω0, ω1, . . .) with ωi ∈ G

• P is the infinite product measure P = µ⊗ µ . . .

• θ is the shift operator θ(ω) = (ω1, ω2, . . .)

The coordinates functions g(n, ω) = ωn form an independent sequence of identicallydistributed matrix valued random variables and the ergodicity of the dynamical systemfollows from the classical zero-one law. One sets U(0) = I and U(n) = g(n− 1) . . . g(0)for n ≥ 1. We will always assume the integrability condition :

∫{log+ ‖g‖ + log+ ‖g−1‖} dµ(g)

1.4. PRODUCTS OF INDEPENDENT RANDOM MATRICES 37

Definition 1.4.2 We say that the probability measure µ is strongly irreducible if thesupport of µ is strongly irreducible.

One remarks that µ is strongly irreducible iff Gµ is strongly irreducible.

Definition 1.4.3 A probability measure ν on P (Rd) is said to be a proper measure iffor any proper subspace V of Rd we have ν(V ) = 0.

Lemma 1.4.4 If µ is strongly irreducible then any invariant probability measure ν isproper.

Proof:Let us assume that the conclusion of the Lemma is false and let d0 be the smallestdimension of the subspaces V for which ν(V ) 6= 0 and α be the supremum of the νmeasure of the subspaces of dimension d0. The set of subspaces V of dimension d0for which ν(V ) = α is finite. Let W be the union of these subspaces. The invarianceequation implies that for µ-almost all g we have gW = W . This proves the Lemma.

Proposition 1.4.5 If µ is strongly irreducible then for any invariant probability mea-sure ν we have the formula :

γ1 =

∫log

‖gv‖‖v‖ dµ(g) dν(v̄)

Proof:Let ν be an invariant measure. Then the probability measure P̃ = P ⊗ ν is invariantfor the shift operator θ̃ introduced in Proposition 1.2.9. By the previous lemma ν isproper and the result now follows from Proposition 1.2.9(ii).

Lemma 1.4.6 Let Φ be the function defined by Φ(v̄) =

∫log

‖gv‖‖v‖ dµ(g). Then Φ is

a continuous function on P (Rd) and we have the relation :∫

Φ(gv̄) dµk(g) = E{log ‖U(k + 1)v‖‖U(k)v‖ }

Proof:The first conclusion is a consequence of the Lebesgue theorem and the second is theresult of an easy computation.

Proposition 1.4.7 If µ is strongly irreducible then for any non zero-vector v ∈ Rd wehave :

limn→∞

1

nlog

‖U(n)v‖‖v‖ = γ1 P-a.e


Proof:For a nonzero v ∈ Rd any limit point ν of the sequence of probability measures νn,v =n−1

∑n−1k=0 µ

k∗δv̄ is invariant and we can write with the notations of the previous Lemma:

γ1 =

∫Φ(ū)dν(ū)

= lim

∫Φ(ū) dνn,v(ū)

= limn→∞

1

nE

{log

‖U(n)v‖‖v‖

}

≤ E{

lim supn→∞

1

nlog

‖U(n)v‖‖v‖

}

The last inequality follows from the Fatou’s lemma and the bound

| log ‖U(n)v‖‖v‖ | ≤ log+ ‖U(n)‖ + log+ ‖U−1(n)‖

Moreover, we know from the Oseledec theorem that there exists an invariant set Ω0of full P-measure for wich the above lim sup is actually a limit and is equal to oneof the Lyapunov exponents. This yields the result. Note nevertheless that the setof convergence in this proposition depends upon the choice of v and that there is nocontradiction with the Oseledec Theorem !

Actually the last two propositions can be proved under the weaker hypothesis of mereirreducibility defined below.

Definition 1.4.8 A subset S of GL(d,R) is said to be not irreducible if there exists aproper subspace V of Rd such that gV = V for all g ∈ S.

Definition 1.4.9 We say that the probability µ is irreducible if its support is irre-ducible.

One sees easily that it is the same than to say that Gµ is irreducible and that if Gµ is aconnected subgroup of GL(d,R) then the irreducibility implies the strong irreducibility.

Lemma 1.4.10 If µ is irreducible then for any invariant probability measure ν andany proper subspace V of Rd we have ν(V ) < 1.

Proof:


Let us assume that the conclusion is false and let d0 be the smallest dimension of thesubspaces V for which ν(V ) = 1. For a subspace V of dimension d0 and such thatν(V ) = 1, the invariance equation :

ν(V ) =

∫ν(gV ) dµ(g)

implies that for µ almost all g we have gV = V and this proves the Lemma.

As a consequence if we only assume that µ is irreducible one can prove Proposition 1.4.5and hence Proposition 1.4.7 using a theorem of Breiman [10] which asserts that when νis an extremal invariant probability measure then the dynamical system (Ω̃, F̃ , θ̃t, P̃) isergodic. Then one can apply Proposition 1.2.9(iii) to an extremal invariant measure toobtain the formula for γ1 and then integrate on the set of invariant extremal measures.

Theorem 1.4.11 Assume that the probability measure µ is strongly irreducible andthat Gµ is a non compact subgroup of SL(d,R), then γ1 > 0.

Proof:We only give a sketch of the original proof of Furstenberg [30] since we will prove againthis result in the next subsection. First we remark that the strong irreducibility of µimplies that there is no probability measure π on P (Rd) such that gπ = π for µ-almostall g. Actually we can replace the hypothesis of the theorem by this property andthe irreducibility of µ. If we assume that the invariant measure ν appearing in theformula of Proposition 1.4.5 has a continuous strictly positive density with respect tothe Cauchy measure m on P (Rd) then we have the cocycle relation :

( ‖v‖‖gv‖

)d=dg−1mdm

=dg−1mdg−1ν

dg−1νdν

dν

dm

This yields

γ1 = −1

d

∫log

(dg−1νdν

(b)

)dν(b)dµ(g)

If the conclusion of the Theorem was false Jensen’s inequality would imply that ν isleft invariant by µ-almost all g and this gives a contradiction. It still remains to extendthis proof to a general ν and this is the hard part of the work ! (See also Ledrappier[77] for a complete proof )

When µ is supported by the symplectic group one of the crucial questions is the pos-itivity of the exponent γℓ. Unfortunately when d > 2 Theorem 1.4.11 says nothingabout this problem thus we need the stronger results proved in the next sub-section.We also remark that we cannot replace the strong irreducibility by the irreducibility inthis Theorem.


1.4.2 The Lyapunov Spectrum

Definition 1.4.12 A subset T of GL(d,R) is said to be contractive if there exists asequence hn in T such that hn‖hn‖−1 converges to a rank one matrix.

Definition 1.4.13 The measure µ is said to be contractive if Tµ is contractive.

An equivalent definition is inf{g∈Tµ}(a2(g)a−11 (g)) = 0 (recall that a(g) is the diago-

nal part in the polar decomposition of g). One remarks that contractivity cannot berelated directly to the group Gµ. This explains why it will be difficult to check thisproperty (since semi-groups are in general less tractable than groups). Nevertheless inthe particular case of SL(2,R) one has a very simple criterion :

Proposition 1.4.14 If the probability measure µ is supported by SL(2,R) then µ iscontractive if and only if Gµ is not compact.

Proof:

It is known that a compact semi-group is actually a group (See [48]). Hence one hasonly to check that the sequence hn‖hn‖−1 has a rank one limit point iff the sequence‖hn‖ is unbounded.

We also remark that if µ is strongly irreducible and contractive then the image measureµ′ of µ under the map g →֒ g′ has the same properties. We will denote by Rn theright product Rn = g(1)g(2) . . . g(n). We are now in position to state and prove thefundamental result of Guivarc’h-Raugi :

Theorem 1.4.15 Let us assume that µ is strongly irreducible and contractive. Thenthere exists a random variable Z with values in the projective space P (Rd) such that forP-almost all ω every limit point of the sequence Rn(ω)‖Rn(ω)‖−1 is a rank one matrixand the direction of its range is precisely given by Z(ω).

Proof:

Let ν be an invariant probability measure on B = P (Rd) and let f be a continuousfunction on B. The random variables Zn =

∫f(Rnb) dν(b) form a bounded martin-

gale with respect to the family Fn of σ-algebras generated by the random variablesg1, . . . , g(n). Taking a countable dense subset of the set of continuous functions on Bwe can conclude that the sequence Rnν is P-almost surely converging to some probabil-ity measure νω. Let λ be the probability measure

∑∞n=0 2

−n−1µn. An easy computation


shows that if Zgn =∫f(Rngb) dν(b) then

n=∞∑

n=1

E{(Zn+r − Zn)2} ≤ 2r‖f‖2 and∞∑

n=0

∫(Zn − Zgn)2 dλ(g)

2

‖U(n)v‖2‖U(n)‖2 = < v , u

′(U(n))e1 >2 +

k=n∑

k=2

a2k(U(n))

a21(U(n))< v , u′(U(n))ek >

2


It follows from (*) and (**) that

limn→∞

‖U(n)v‖‖U(n)‖ = | < v, z

′ > |

where z is a unit vector of direction Z ′. The law of Z ′, which is the invariant measure forµ′, is proper hence for a fixed non zero v the set {ω ; v is orthogonal to the direction of Z ′}is of zero P-measure and this yields conclusion (i). Conclusion (ii) is a consequence of(i) and (*) once we remark that ‖Λ2(U(n))‖ = a1(U(n))a2(U(n)).

Theorem 1.4.17 Let assume that µ is strongly irreducible and contractive. Then onehas:

(i) There exists a unique invariant probability measure ν and ν is proper.

(ii) γ1 =∫

log‖gv‖‖v‖ dµ(g) dν(v̄)

(iii) 1nE

{log

‖U(n)v‖‖v‖

}converges uniformly to the exponent γ1 with respect to v̄ ∈

P (Rd).

Proof:Recall that a probability measure m on P (Rd) is said to be proper if m(V ) = 0 forany proper subspace V of Rd. Theorem 1.4.15 implies that for any proper probabilitymeasure m on P (Rd) the sequence Rn(ω)m converges weakly to the Dirac measureat the point Z(ω). Let ν be an invariant probability measure and f be a continuousfunction on P (Rd). The sequence

∫f(Rn(ω)v̄) dν(v̄) is a bounded martingale with

expectation ν(f). We know from Lemma 1.4.4 that ν is proper, thus ν has to be equal tothe law of the random variable Z proving (i). The inequalities Cv‖U(n)‖ ≤ ‖U(n)v‖ ≤‖U(n)‖ yield immediately (ii). To prove (iii) it is enough to show that for any sequencev̄n converging to a point v̄0 then the sequence n

−1E{log ‖U(n)vn‖ ‖vn‖−1} convergesto γ1. Any weak limit point of the sequence νn = n

−1 ∑n−1k=0 µ

n ∗ δv̄n is µ invariant.Hence this sequence converges weakly to the unique invariant measure ν. If we apply

this sequence of probability measures to the continuous function v̄ →֒∫

log‖gv‖‖v‖ dµ(g)

we obtain the result.

Theorem 1.4.18 Let us assume that µ is strongly irreducible and contractive. Thenwe have the strict inequality γ1 > γ2.

Proof:Let B1 be the set of norm one matrices of R

d and B2 the set of norm one matrices ofΛ2(Rd). Following the lines of the paper of Furstenberg and Kesten [31] Theorem I.4.1,


one can prove that there exist probability measures ν1 and ν2 on B1 and B2 such that:

limn→∞

1

nlog ‖U(n)M‖ = γ1 P ⊗ ν1 − a.e. (1)

limn→∞

1

nlog ‖Λ2U(n)N‖ = γ1 + γ2 P ⊗ ν2 − a.e. (2)

Notice that the group G acts on B = B1 ×B2 by g.(M,N) = (g.M, g.N) and so if wedefine the cocycle σ by σ(g, (M,N)) =

‖gM‖2‖Λ2(g)N‖ , it is easy to check that any limit

point ν of the sequence n−1∑n

k=1 µn ∗ (ν1 ⊗ ν2) is invariant and admits ν1 and ν2 as

projections on B1 and B2. Lemma 1.4.16(ii) then implies that

limσ(U(n), (M,N)) ≥ lim ‖U(n)M‖2

‖Λ2(U(n))‖ = +∞ P ⊗ ν − a.e.

From Proposition 1.3.4, the sequence n−1 log σ(U(n), (M,N)) converges P ⊗ ν-a.e.to arandom variable having a positive expectation. We now remark that (1) and (2) implythat this limit is P⊗ν-almost surely equal to 2γ1−(γ1+γ2) = γ1−γ2 and this completesthe proof.

It is worth to mention that Theorem 1.4.18 implies that γ1 is strictly positive whenµ is carried by SL(d,R) since in this case the sum of the Lyapunov exponent is nul.In the particular case of SL(2,R), Proposition 1.4.14 shows that this result is actuallyequivalent to the Furstenberg Theorem 1.4.11. But the most important consequence ofTheorem 1.4.18 is the contractive action of the sequence U(n) on the projective spaceP (Rd). One defines the projective distance δ on P (Rd) by the formula

δ(x̄, ȳ) =‖x ∧ y‖‖x‖ ‖y‖ x , y ∈ R

d

The following proposition shows that the distance between two columns of the matrixU(n) goes to zero exponentially fast.

Proposition 1.4.19 If µ is strongly irreducible and contractive then for non-zero vec-tors x, y ∈ Rd x 6= y one has :

(i) limn→∞

1

nlog

δ(U(n)x̄, U(n)ȳ)

δ(x̄, ȳ)< 0 P − a.e.

(ii) there exists an integer N such that sup

x, y ∈ Rd

∫log

δ(gx̄, gȳ)

δ(x̄, ȳ)dµN < 0


Proof:From the definition of δ we obtain

δ(U(n)x̄, U(n)ȳ)

δ(x̄, ȳ)=

‖U(n)x ∧ U(n)y‖‖x ∧ y‖

‖x‖‖U(n)x‖

‖y‖‖U(n)y‖

≤ ‖Λ2(U(n))‖ ‖x‖‖U(n)x‖‖y‖

‖U(n)y‖

The convergence of the sequence 1n log‖U(n)x ∧ U(n)y‖

‖x ∧ y‖ can be deduced from the Os-eledec Theorem applied on Λ2(Rd) and (i) follows from Theorem 1.4.18. To prove (ii)we write :

1

n

∫log

δ(gx̄, gȳ)

δ(x̄, ȳ)dµn

≤ 1n

E{log ‖Λ2(U(n))‖} − 1n

E{log ‖U(n)x‖‖x‖ } −1

nE{log ‖U(n)y‖‖y‖ }

It only remains to apply again Theorem 1.4.18 together with the uniform convergenceproved in Proposition 1.4.17(iii).

The uniform contraction in mean of the projective space by the sequence U(n) ex-pressed by this proposition can be viewed as a subsitute for the Dœblin condition forMarkov chains (which is in general not satisfied under the hypothesis of this section).A consequence of this property is the exponential speed of convergence of the iteratesT n of the kernel of the Markov chain U(n)v̄ to the invariant measure ν.

Proposition 1.4.20 Let us assume that µ is strongly irreducible and contractive andthat there exists a positive τ such that ‖g‖t is µ integrable for −τ ≤ t ≤ τ . T

RESUM´ E´ · 2016. 9. 2. · RESUM´ E ´ On se propose d ... Brossard [11] pour le cas continu...

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