Polynomial decay of correlations for nonpositively curved surfaces (Q6597528)

The authors provide examples of geodesic flows on non-positively curved manifolds with decay of correlations that is polynomial.\N\NTheir primary theorem is as follows: Take \(S\) to be a closed Riemannian surface with non-positive curvature constructed by isometrically gluing two negatively curved surfaces with boundaries to the boundaries of the surface of revolution with profile \(1 + |s|^r\) where \(|s| \leq 1\) and \(r \in [4, \infty)\). \(S\) is assumed to be of class \(C^\infty\) when \(r\) is an even integer, and otherwise that \(S\) is \(C^\infty\) away from the circle \(s = 0\). Let \(g_t: M \rightarrow M\) be the geodesic flow on \(M = T^1S\).\N\NThen, if \(a = \frac{r+2}{r-2} \in (1,3]\), the geodesic flow \(g_t\) on \(M\) has a polynomial decay of correlations with respect to the normalized Riemannian volume bounded by a constant times \( \frac{1}{t ^{a- \epsilon}}\) for all \(t > 0\) and \(\epsilon > 0\).\N\NTo prove their main theorem, the authors use an axiomatic approach using the \textit{N. Chernov}'s axioms as presented in [J. Statist. Phys. 94, No. 3--4, 513--556 (1999; Zbl 1047.37503].\N\NThe paper also includes a proof of a version of the central limit theorem for this class of flows, as well as other related statistical limit laws.