The sectional curvature of the infinite dimensional manifold of Hölder 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 Hölder equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a normalized Hölder 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 $|X{|}^2=⟨X,X⟩={\lim }_{n\to \infty }\frac{1}{n}\int ({\sum }_{i=0}^{n-1}X\circ T^i{)}^{2}d{\mu }_A$. 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 $K(X,Y)=\frac{1}{4}[{\sum }_{i=1}^{\infty }(\int XY{X}_{i}d{\mu }_A{)}^2-{\sum }_{i=1}^{\infty }\int X^2{X}_{i}d{\mu }_A\int Y^2{X}_{i}d{\mu }_A].$ 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 $K({\hat{a}}_x,{\hat{a}}_z)=0$.

We fixed the expressions of the sectional curvature

References in corpus (2)

• Spectral estimates for Ruelle operators with two parameters and sharp large deviations
• Spectral estimates for Ruelle transfer operators with two parameters and applications