The infinite dimensional manifold of Hölder equilibrium probabilities has non-negative curvature

Artur O. Lopes, Rafael O. Ruggiero

Published 2018-11-19, updated 2019-01-22Version 2

Here we consider 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 unitary circle. 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 certain normalized H\"older potential $A$ denote by $\mu_A \in \mathcal{N}$ the associated equilibrium probability. The set of tangent vectors $X$ (a function $X: M \to \mathbb{R}$) to the manifold $\mathcal{N}$ at the point $\mu_A$ coincides with the kernel of the Ruelle operator for the normalized potential $A$. 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 \smallskip $\,\,\,\,\,\,\,\,\,\,\,\,\,|X|^2\,\,= \,\,<X,X>\,\,=\,\,\lim_{n \to \infty} \frac{1}{n} \int (\sum_{i=0}^{n-1} X\circ T^i )^2 \,d \mu_A$. Given two unitary tangent vectors to the manifold $\mathcal{N}$ at $\mu_A$, denoted by $X$ and $Y$, we will show that the sectional curvature $K(X,Y)$ equals to $\int X^{2}Y^{2}d\mu_{A}$, so it is always non-negative. The curvature vanishes, if and only if, the supports of the functions $X$ and $Y$ are disjoint. In our proof for the above expression for the curvature it is necessary in some moment to show the existence of geodesics for such Riemannian metric.