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 $M=\{1,2,...,d\}^\mathbb{N}$ 上的移位作用 $T=\sigma$，要么描述 $d$ 对 $1$ 的 $C^{1+\alpha}$ 类扩张变换 $T:S^1 \to S^1$ 的作用（例如 $x \to T(x) =d\, x $ (模 $1) $\,），此时 $M=S^1$ 为单位圆。已知 Hölder 势函数 $A:M \to \mathbb{R}$ 对应的平衡概率构成的无穷维流形 $\mathcal{N}$ 是一个解析流形，并具有与渐近方差相关的自然黎曼度量。本文证明，在假设 Ruelle 算子核存在傅里叶类希尔伯特基的条件下，存在测地线路径。当 $T=\sigma$ 且 $M=\{0,1\}^\mathbb{N}$ 时，此类基存在。另一方面，我们也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 $\mu_2$ 之间的邻近性。一个自然的问题是：对于固定的概率 $\mu_1\in \mathcal{N}$，考虑 $\mathcal{N}$ 中凸概率集合内使 $D_{KL}(\mu_1,\mu_2)$ 最小化的概率 $\mu_2$。该最小化问题是信息投影中核心问题的动态版本。我们在 $\mathcal{N}$ 中研究此问题（此时所有概率均为动态不变），得到了所求解的显式方程。三角形不等式与勾股不等式也将被探讨。