Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities

Abstract

We consider here the discrete time dynamics described by a transformation T : M → M , where T is either the action of shift T = σ on the symbolic space M = {1, 2, ..., d} N , or, T describes the action of a d to 1 expanding transformation T : S 1 → S 1 of class C 1+α ( for example x → T (x) = d x (mod 1) ), where M = S 1 is the unit circle. It is known that the infinite-dimensional manifold N of equilibrium probabilities for Hölder potentials A : M → R is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When T = σ and M = {0, 1} N such basis exists.In a different direction, we also consider the KL-divergence D KL (µ 1 , µ 2 ) for a pair of equilibrium probabilities. If D KL (µ 1 , µ 2 ) = 0, then µ 1 = µ 2 . Although D KL is not a metric in N , it describes the proximity between µ 1 and µ 2 . A natural problem is: for a fixed probability µ 1 ∈ N consider the probability µ 2 in a convex set of probabilities in N which minimizes D KL (µ 1 , µ 2 ). This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in N , a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.