Products of exact dynamical systems and Lorentzian continued fractions

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.