Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts

Henk Bruin; Renaud Leplaideur

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arXiv · 1010.4643

Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts

Henk Bruin·Renaud Leplaideur

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Abstract

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 $γ\mapsto \CP(-γV)$ 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.

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Henk Bruin, Renaud Leplaideur. 2012-03-20. Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts. https://arxiv.org/abs/1010.4643

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