Superdiffusive central limit theorems for geodesic flows on nonpositively curved surfaces

TL;DR

This paper establishes superdiffusive central limit theorems and decay of correlations for geodesic flows on certain nonpositively curved surfaces, advancing understanding of their statistical properties.

Contribution

It introduces a nonstandard CLT with superdiffusive normalization and improves regularity results for stable/unstable foliations in this context.

Findings

Proves a superdiffusive CLT with normalization (t log t)^{1/2}

Shows correlations decay at rate t^{-1}

Provides improved regularity results for foliations

Abstract

We prove a nonstandard central limit theorem and weak invariance principle, with superdiffusive normalisation $(t\log t)^{1/2}$, for geodesic flows on a class of nonpositively curved surfaces with flat cylinder. We also prove that correlations decay at rate $t^{-1}$. An important ingredient of the proof, which is of independent interest, is an improved results on the regularity of the stable/unstable foliations induced by the Green bundles.