RENORMALIZATION, FREEZING PHASE TRANSITIONS AND FIBONACCI QUASICRYSTALS

We examine the renormalization operator determined by the Fibonacci substitution within the full shift on two symbols Sigma := {0, 1}(N). We exhibit a fixed point and determine its stable leaf (under iteration of the operator acting on potentials V : E -> 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.