Products of exact dynamical systems and Lorentzian continued fractions

TL;DR

This work broadens continued fractions beyond the Euclidean setting by introducing Lorentzian CFs in Minkowski spaces and product-type CFs, proving convergence, ergodicity, and exactness under Renyi’s condition. It furnishes a general framework for products of exact dynamical systems, showing that exactness passes to finite products and enabling analysis of multi-dimensional and α-perturbed CFs. A key contribution is the exactness and ergodicity results for product CFs and Lorentzian CFs, including explicit invariant measures and Lagrange-type quadratic characterizations. The findings illuminate how product and Lorentzian structures interact in CF dynamics and lay groundwork for further exploration of higher-dimensional CFs with experimental guidance guiding future proofs and conjectures.

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 $α$-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.

Figures (3)

• Figure 1: Left: depth-1 cylinders in $[0,1)^2$ for the product CFs in $\mathbb{R}^2$. Right: depth-1 cylinders for the little diamond CFs in $\mathbb{R}[\mathbbm{j}]$.
• Figure 2: Left: Experimental evidence suggests that square CFs in $\mathbb{R}^{1,1}$ have a finite invariant measure. Right: Image of the boundary of $[-1/2,1/2]^2$ under the generalized Gauss map.
• Figure 3: The fundamental domain in $\mathbb{R}^{2,1}$ of the Lorentzian continued fractions is shown in red. The light cone $\mathcal{N}$ at the origin is shown in blue, while the light cones at $(0,0,\pm 1)$ are shown in yellow. The region between these two light cones is the Euclidean expanding region for the inversion.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• Theorem 5: Rokhlin's Exactness Theorem Rokhlin1961ExactEO
• proof
• Remark 1
• proof : Proof of Theorem \ref{['thm:RokhlinProductSystem']}
• proof : Proof of Theorem \ref{['thm:MainProductAlgebras']}