Convergence of improper Iwasawa Continued Fractions

TL;DR

This paper proves the convergence of a broad class of continued fractions, including those over quaternions and octonions, by analyzing digit sequences for points approaching the unit sphere, extending previous methods.

Contribution

It introduces new convergence proofs for generalized continued fractions over non-commutative algebras, overcoming challenges posed by non-uniform expansion.

Findings

Proves convergence for continued fractions over quaternions and octonions.

Develops methods to analyze digit sequences near the unit sphere.

Extends Dani-Nogueira's techniques to broader classes of continued fractions.

Abstract

We prove the convergence of a wide class of continued fractions, including generalized continued fractions over quaternions and octonions. Fractional points in these systems are not bounded away from the unit sphere, so that the iteration map is not uniformly expanding. We bypass this problem by analyzing digit sequences for points that converge to the unit sphere under iteration, expanding on previous methods of Dani-Nogueira.