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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 symbolicspace $M=\{1,2,...,d\}^\mathbb{N}$, or, $T$ describes the action of a $d$ to$1$ expanding transformation $T:S^1 \to S^1$ of class $C^{1+\alpha}$ (\,forexample $x \to T(x) =d\, x $ (mod $1) $\,), where $M=S^1$ is the unit circle.It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibriumprobabilities for H\"older potentials $A:M \to \mathbb{R}$ is an analyticalmanifold and carries a natural Riemannian metric associated with the asymptoticvariance. We show here that under the assumption of the existence of aFourier-like Hilbert basis for the kernel of the Ruelle operator there existsgeodesics 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 metricin $\mathcal{N}$, it describes the proximity between $\mu_1$ and $\mu_2$. Anatural problem is: for a fixed probability $\mu_1\in \mathcal{N}$ consider theprobability $\mu_2$ in a convex set of probabilities in $\mathcal{N}$ whichminimizes $D_{KL}(\mu_1,\mu_2)$. This minimization problem is a dynamicalversion of the main issues considered in information projections. We considerthis problem in $\mathcal{N}$, a case where all probabilities are dynamicallyinvariant, getting explicit equations for the solution sought. Triangle andPythagorean inequalities will be investigated.

Geodesics and dynamical information projections on the manifold of H\"older equilibrium probabilities