Bianchi IX Chaoticity: BKL Map and Continuous Flow

TL;DR

This paper investigates the chaotic behavior of Bianchi IX cosmological models by comparing the BKL map with a Hamiltonian flow approach, revealing their relationship and underlying geometric structures.

Contribution

It provides a comparative analysis of the BKL and ADM frameworks, elucidating how the BKL map relates to the geodesic flow in the ADM description of Bianchi IX dynamics.

Findings

The BKL map is related to the geodesic flow of the ADM Hamiltonian.

A connection between anisotropy parameters and Kasner indices is established.

The study clarifies the stochastic properties of Bianchi IX models.

Abstract

We analyze the Bianchi IX dynamics (Mixmaster) in view of its stochastic properties; in the present paper we address either the original approach due to Belinski, Khalatnikov and Lifshitz (BKL) as well as a Hamiltonian one relying on the Arnowitt--Deser--Misner (ADM) reduction. We compare these two frameworks and show how the BKL map is related to the geodesic flow associated with the ADM dynamics. In particular, the link existing between the \textit{anisotropy parameters} and the \textit{Kasner indices} is outlined.