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We examine the renormalization operator determined by the Fibonaccisubstitution. We exhibit a fixed point and determine its stable leaf (underiteration of the operator). Then, we study the thermodynamic formalism for po-tentials in this stable leaf, and prove they have a freezing phase transition,with ground state supported on the attracting quasi-crystal associated to theFibonacci substitution

Renormalization, Freezing Phase Transitions and Fibonacci Quasicrystals