The calculus of thermodynamical formalism

• DOI: 10.4171/jems/814
• Publication Date: 2018-07-09T21:30:14Z
• Authors: Paolo Giulietti; Benoît R. Kloeckner; Artur O. Lopes; Diego Marcon
• Keywords: Formalism (music); Calculus of variations

Given an onto map T acting on a metric space \Omega and appropriate Banach of functions \mathcal X(\Omega) , one classically constructs for each potential A \in X transfer operator \mathscr L_A . Under suitable hypotheses, it is well-known that has maximal eigenvalue \lambda_A spectral gap defines unique Gibbs measure \mu_A Moreover there normalized the form B=A+f-f\circ T+c as representative class all potentials defining same measure. The goal present article to study geometry set N normalization A\mapsto B We give easy proof fact analytic submanifold analytic; we compute derivative map; last endow with natural weak Riemannian (derived from asymptotic variance) respect which gradient flow induced by pressure given potential, e.g. entropy functional. also apply these ideas recover in wide setting existence uniqueness equilibrium states, possibly under constraints.