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) =d\, x $ (mod $1) $\,), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibrium probabilities for H\"older 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.

翻译：本文考虑由变换$T:M \to M$描述的离散时间动力学，其中$T$要么是符号空间$M=\{1,2,...,d\}^\mathbb{N}$上的移位算子$T=\sigma$，要么是$C^{1+\alpha}$类$d$到$1$扩张变换$T:S^1 \to S^1$（例如$x \to T(x)=d\, x$ (mod $1)$），其中$M=S^1$为单位圆周。已知Hölder势$A:M \to \mathbb{R}$的平衡概率构成的无限维流形$\mathcal{N}$是一个解析流形，并承载了与渐近方差相关的自然黎曼度量。本文证明：在假设Ruelle算子核存在类似傅里叶的Hilbert基的条件下，存在测地线路径。当$T=\sigma$且$M=\{0,1\}^\mathbb{N}$时，该基存在。另一方面，我们考虑一对平衡概率的KL散度$D_{KL}(\mu_1,\mu_2)$。若$D_{KL}(\mu_1,\mu_2)=0$，则$\mu_1=\mu_2$。尽管$D_{KL}$不是$\mathcal{N}$上的度量，但它描述了$\mu_1$与$\mu_2$之间的接近程度。一个自然的问题是：对于固定的概率$\mu_1\in \mathcal{N}$，考虑$\mathcal{N}$的凸概率集中使$D_{KL}(\mu_1,\mu_2)$最小的概率$\mu_2$。这一最小化问题是信息投影中主要问题的动力学版本。我们在所有概率均具有动力学不变性的$\mathcal{N}$中考虑该问题，给出了所求解的显式方程，并探讨了三角不等式与毕达哥拉斯不等式。