On mixing and the local central limit theorem for hyperbolic flows

TL;DR

This paper establishes abstract conditions ensuring that certain hyperbolic flows satisfy the local central limit theorem, and verifies these conditions for various dynamical systems including Sinai billiards and Lorenz attractors.

Contribution

It formulates general conditions for the local central limit theorem in hyperbolic flows and applies them to multiple complex systems, expanding understanding of their statistical properties.

Findings

Conditions verified for Sinai billiards with finite horizon

Conditions verified for systems with Young towers

Local central limit theorem holds for several hyperbolic flows

Abstract

We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young tower, such as Sinai billiard with finite horizon, suspensions over Pomeau-Manneville maps, and geometric Lorenz attractors.