The sectional curvature of the infinite dimensional manifold of H\"older equilibrium prababilities

Abstract

Here we consider the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is the shift and $M=\{1,2,...,d\}^\mathbb{N}$. It is known that the infinite-dimensional manifold $\mathcal{N}$ of H\"older equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a normalized H\"older potential $A$ denote by $\mu_A \in \mathcal{N}$ the associated equilibrium probability. The set of tangent vectors $X$ to the manifold $\mathcal{N}$ at the point $\mu_A$ coincides with the kernel of the Ruelle operator for $A$. The Riemannian norm $|X|=|X|_A$ of the vector $X$, which is tangent to $\mathcal{N}$ at the point $\mu_A$, is described via the asymptotic variance, that is, satisfies $|X|^2\,\,= \langle X, X \rangle =\lim_{n \to \infty} \frac{1}{n} \int (\sum_{i=0}^{n-1} X\circ T^i )^2 \,d \mu_A$. Consider an orthonormal basis $X_i$, $i \in \mathbb{N}$, for the tangent space at $\mu_A$. Given two unit tangent vectors $X$ and $Y$ the curvature $K(X,Y)$ satisfies $\,\,\,\,K(X,Y) = \frac{1}{4}[\, \sum_{i=1}^\infty ( \int X \,Y\, X_i \,d \mu_A)^2 - \sum_{i=1}^\infty \int X^2 X_i \,d \mu_A\, \,\int Y^2 X_i \,d \mu_A \,].$ When the equilibrium probabilities $\mu_A$ is the set of invariant Markov probabilities on $\{0,1\}^\mathbb{N}\subset \mathcal{N}$, introducing an orthonormal basis $\hat{a}_y$, indexed by finite words $y$, we show explicit expressions for $K(\hat{a}_x,\hat{a}_z)$, which is a finite sum. These values can be positive or negative depending on $A$ and the words $x$ and $z$. Words $x,z$ with large length can eventually produce large negative curvature $K(\hat{a}_x,\hat{a}_z)$. If $x, z$ do not begin with the same letter, then $K(\hat{a}_x,\hat{a}_z)=0$.