RENORMALIZATION, THERMODYNAMIC FORMALISM AND QUASI-CRYSTALS IN SUBSHIFTS.

Henk Bruin, (1) , Renaud Leplaideur (2)

1 Department of Mathematics
2 LM - Laboratoire de mathématiques de Brest

We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator $\CR$ acting on the space of potentials are such that the pressure function $\gamma \mapsto \CP(-\gamma V)$ exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of $\CR$, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.

Mots clés: Renormalization; Phase transition; Thermodynamic formalism; Substitution

Domaines: Systèmes dynamiques [math.DS]

Henk Bruin,, Renaud Leplaideur. RENORMALIZATION, THERMODYNAMIC FORMALISM AND QUASI-CRYSTALS IN SUBSHIFTS.. 2010. ⟨hal-00528299v3⟩