Dirichlet Forms on Fractals and Products of Random Matrices

Kusuoka, Shigeo

doi:10.2977/prims/1195173187

Cited by 188 publications

(272 citation statements)

References 6 publications

(9 reference statements)

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“…In this case the operator H ± <n> has compact resolvent (cf for example, [30]) and the eigenvalues form a non-increasing sequence going to −∞. We adopt the same definition for the counting measures ν ± <n> and for the density of states as in the lattice case.…”

Section: The Continuous Casementioning

confidence: 99%

“…There is nothing special to say about the lattice case. In the continuous case, Lindström and Kusuoka constructed the self-similar Dirichlet space (cf [31], [30]) and the author proved the uniqueness of such a self-similar Dirichlet space (cf [36]). …”

Section: The Unit Intervalmentioning

confidence: 99%

“…-In this section we define a "natural" Laplace operator on the continuous sets X <n> . The problem of the construction of such an operator is not easy (cf for example, [31], [25], [30], [36]) and it is now clear that the best framework to use is the framework of Dirichlet spaces. We essentially follow the definitions of [36].…”

Section: Construction Of a Self-similar Laplacianmentioning

confidence: 99%

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Spectral properties of self-similar lattices and iteration of rational maps

Sabot¹

2003

Mémoires Soc. Math. France

121

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Abstract:In this text we consider discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets, and their continuous analogous defined on the self-similar sets themselves. We are interested in the spectral properties of these operators. The basic example is the lattice based on the Sierpinski gasket. We introduce a new renormalization map which appears to be a rational map defined on a smooth projective variety (more precisely, this variety is isomorphic to a product of three types of Grassmannians: complex Grassmannians, Lagrangian Grassmannians, orthogonal Grassmannians). We relate some characteristics of the dynamics of its iterates with some characteristics of the spectrum of our operator. More specifically, we give an explicit formula for the density of states in terms of the Green current of the map, and we relate the indeterminacy points of the map with the so-called Neumann-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map we can prove drastically different spectral properties of the operators. Our formalism is valid for the general class of finitely ramified self-similar sets (i.e. for the class of pcf selfsimilar sets of Kigami, cf [25]). Hence, this work aims at a generalization and a better understanding of the initial work of the physicists Rammal and Toulouse on the Sierpinski gasket (cf [35], [34]).Résumé: Dans ce texte, nous considérons le Laplacien discret, défini sur un réseau(1) E-mail address: sabot@ccr.jussieu.fr 1 2 construità partir d'un ensemble auto-similaire finiment ramifié, et son analogue continu défini sur l'ensemble auto-similaire lui-même. Nous nous intéressons aux propriétés spectrales de ces opérateurs. L'exemple le plus classique est celui du triangle de Sierpinski (Sierpinski gasket) et du réseau discret associé. Nous introduisons une nouvelle application de renormalisation qui se trouveêtre une application rationnelle définie sur une variété projective lisse (plus précisément, cette variété est un produit de Grassmaniennes de trois types: Grassmaniennes classiques, Grassmaniennes Lagrangiennes, Grassmaniennes orthogonales). Nous relions certaines propriétés spectrales de ces opérateurs avec la dynamique des itérés de cette application. En particulier, nous donnons une formule explicite de la densité d'états en termes du courant de Green de l'application, et nous caractérisons le spectre de Neumann-Dirichlet (qui correspond aux fonctions propresà support compact sur l'ensemble infini)à l'aide des points d'indétermination de l'application. Suivant le degré asymptotique de l'application nous pouvons prouver que les propriétés spectrales de l'opérateur sont très différentes. Notre formalisme s'appliqueà la classe des ensembles auto-similaires finiment ramifiés (ou autrement dità la classe des "pcf self-similar sets" de Kigami, cf [25]

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Section: The Continuous Casementioning

confidence: 99%

Section: The Unit Intervalmentioning

confidence: 99%

Section: Construction Of a Self-similar Laplacianmentioning

confidence: 99%

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Spectral properties of self-similar lattices and iteration of rational maps

Sabot¹

2003

Mémoires Soc. Math. France

121

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“…in [14], that any local and regular Dirichlet form defines a diffusion process on a set. The development of this theory when the underlying set is fractal started with the construction of Brownian motion on the Sierpiński gasket by Goldstein and Kusuoka in [17,33]. Since then, many results concerning both Dirichlet forms and diffusion processes on fractals have been established.…”

Section: Introductionmentioning

confidence: 99%

Weyl asymptotics for Hanoi attractors

Ruiz

Freiberg

2016

Forum Mathematicum

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“…The analysis on irregular sets such as the Sierpiński gasket has been developed since the late 80's, starting with the pioneering works [10,14]. Also, the Brownian motion on fractals has been defined (see e.g.…”

Section: Introductionmentioning

confidence: 99%