Ergodicity of Iwasawa continued fractions via markable hyperbolic geodesics

TL;DR

This paper establishes the convergence and ergodicity of various advanced continued fraction algorithms, linking hyperbolic geometry and geodesic flow to extend classical results to complex, quaternionic, octonionic, and Heisenberg cases.

Contribution

It introduces a unified framework for Iwasawa continued fractions and proves ergodicity using geodesic marking, extending classical theorems to higher-dimensional and non-commutative settings.

Findings

Proves ergodicity for a broad class of continued fractions.

Generalizes Serret's tail-equivalence theorem.

Connects continued fractions with hyperbolic geodesic flow.

Abstract

We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and $\alpha$-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a generalization of Serret's tail-equivalence theorem for almost all points. The results are new even in the case of complex continued fractions.