Products of exact dynamical systems and Lorentzian continued fractions

TL;DR

This paper introduces a new continued fraction system in Minkowski space, proves its convergence and ergodicity, and explores the exactness of product systems under certain conditions, with new systems conjectured to be convergent and ergodic.

Contribution

It develops a novel continued fraction framework in Minkowski space, establishes key properties, and extends ergodic theory results to product systems under Renyi's condition.

Findings

Proved convergence and ergodicity of the new continued fraction system.

Established that products of exact systems are again exact under Renyi's condition.

Conjectured convergence and ergodicity of new CF systems based on experimental evidence.

Abstract

We describe a new continued fraction system in Minkowski space $\mathbb R^{1,1}$, proving convergence, ergodicity with respect to an explicit invariant measure, and Lagrange's theorem. The proof of ergodicity leads us to the question of exactness for products of dynamical systems. Under technical assumptions, namely Renyi's condition, we show that products of exact dynamical systems are again exact, allowing us to study $\alpha$-type perturbations of the system. In addition, we describe new CF systems in $\mathbb R^{1,1}$ and $\mathbb R^{2,1}\cong \mathrm{Sym}_2(\mathbb R)$ that, based on experimental evidence, we conjecture to be convergent and ergodic with respect to a finite invariant measure.