THE POTENTIAL POINT OF VIEW FOR RENORMALIZATION

Abstract

For the shift σ in Σ = {0, 1}ℕ, we define the renormalization for potentials by [Formula: see text] We show that for a good H, there is a unique fixed point for [Formula: see text]. It is the Hofbauer potential V*.We show that the stable set of the Hofbauer potential, i. e. the set of potentials V such that [Formula: see text] converges to V* is characterized by the germ of these potentials close to 0∞= 000….Then, we make connections with the Manneville–Pomeau map f : [0, 1]↺. In particular we show that the lift in Σ of log f′ is in the stable set of V*.In the second part, we characterize "good" H, such that σ2◦ H = H ◦ σ.In the last part, we study the thermodynamic formalism for some special potentials in the stable set of V*. They are called virtual Manneville–Pomeau maps.