Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts

TL;DR

This paper investigates the thermodynamic formalism of renormalizable dynamical systems generated by the Thue-Morse substitution and Feigenbaum maps, exploring phase transitions and the emergence of quasi-crystals in the zero-temperature limit.

Contribution

It extends previous work by analyzing phase transitions in systems with Cantor set attractors and demonstrates the possibility of reaching quasi-crystals as ground states.

Findings

Existence of phase transitions in certain fixed points of the renormalization operator.

Identification of conditions under which quasi-crystals emerge as ground states.

Demonstration of complex phase behavior in systems with Cantor set attractors.

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 $\gamma \mapsto \CP(-\gamma V)$ 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.