Geodesic
Renormalization, freezing phase transitions and Fibonacci quasicrystals
[Renormalisation, transitions de phase congelantes et quasi-cristal de Fibonacci]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 739-763.

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We examine the renormalization operator determined by the Fibonacci substitution within the full shift on two symbols Σ:={0,1}ℕ. We exhibit a fixed point and determine its stable leaf (under iteration of the operator acting on potentials V:Σ→ℝ), which is completely determined by the germ near the attractor of the substitution. Then we study the thermodynamic formalism for potentials in this stable leaf, and prove they have a freezing phase transition at finite temperature, with ground state supported on the attracting quasi-crystal associated to the Fibonacci substitution.

Nous étudions les relations entre renormalisation, substitutions et transitions de phase : nous montrons que la substitution de Fibonacci dans le shift plein à deux symboles Σ:={0,1}ℕgénère un opérateur de renormalisation sur les potentiels V:Σ→ℝ. Nous montrons que cet opérateur possède un point fixe, uniquement déterminé par son germe proche de l'attracteur associé à la substitution de Fibonacci. Nous déterminons aussi la feuille stable de ce point fixe. Dans un second temps, nous montrons que tous les potentiels dans cette feuille stable présentent une transition de phase congelante. En particulier, cela donne un nouvel exemple d'obtention d'un état fondamental porté par un quasi-cristal avant le zéro absolu.

Publié le :
DOI : 10.24033/asens.2257
Classification : 37A35, 37A60, 37D20, 37D35, 47N10
Keywords: Thermodynamic formalism, equilibrium states, phase transition, substitution, Fibonacci numbers.
Mots-clés : Formalisme thermodynamique, états d'équilibre, transition de phase, substitution, nombres de Fibonacci. 
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     title = {Renormalization, freezing phase transitions and {Fibonacci} quasicrystals},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {739--763},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Bruin, Henk; Leplaideur, Renaud. Renormalization, freezing phase transitions and Fibonacci quasicrystals. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 739-763. doi : 10.24033/asens.2257. http://geodesic.mathdoc.fr/articles/10.24033/asens.2257/

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