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

Abstract

We consider here the discrete time dynamics described by a transformation $T:M\to M$, where $T$ is either the action of shift $T=\sigma$ on the symbolic space $M=\{1,2,...,d{\}}^{\mathbb{N}}$, or, $T$ describes the action of a $d$ to $1$ expanding transformation $T:S^1\to S^1$ of class $C^{1+\alpha }$ (\,for example $x\to T(x)=dx$ (mod $1)$\,), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibrium probabilities for Hölder potentials $A:M\to \mathbb{R}$ 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 $T=\sigma$ and $M=\{0,1{\}}^{\mathbb{N}}$ such basis exists. In a different direction, we also consider the KL-divergence $D_{KL}({\mu }_1,{\mu }_2)$ for a pair of equilibrium probabilities. If $D_{KL}({\mu }_1,{\mu }_2)=0$, then ${\mu }_1={\mu }_2$. Although $D_{KL}$ is not a metric in $\mathcal{N}$, it describes the proximity between ${\mu }_1$ and ${\mu }_2$. A natural problem is: for a fixed probability ${\mu }_1\in \mathcal{N}$ consider the probability ${\mu }_2$ in a convex set of probabilities in $\mathcal{N}$ which minimizes $D_{KL}({\mu }_1,{\mu }_2)$. This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in $\mathcal{N}$, a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.

Keywords: Geodesics; infinite-dimensional Riemannian manifold; equilibrium probabilities; KL-divergence; information projections; Pythagorean inequalities; Fourier-like basis