Self-duality of multidimensional continued fractions

TL;DR

This paper investigates the self-duality property of fibred systems related to multidimensional continued fractions, revealing algebraic self-duality in many cases and exploring systems with partial self-duality, thus advancing understanding of their symmetries.

Contribution

It introduces the concept of self-duality for fibred systems and demonstrates its algebraic nature in various multidimensional continued fraction algorithms.

Findings

Explicit algebraic self-duality in many systems

Existence of systems with partial self-duality

Enhanced understanding of symmetries in continued fraction algorithms

Abstract

F.~Schweiger introduced the fibred system in \cite{Schweiger-MCF}, to unify and generalize many known continued fraction algorithms. An advantage of a fibred system is that it often provides a systematic construction of absolutely continuous invariant density. In this paper, we define and study the self-duality of fibred systems, a strong symmetry of a given system. We show that explicit algebraic self-duality holds in many systems and presents a curious system with "partial" self-duality.