Periodicity of general multidimensional continued fractions using repetend matrix form

TL;DR

This paper investigates the structure of periodic expansions in multidimensional continued fractions, providing criteria for periodicity and applying these to algebraic vectors using the Jacobi–Perron Algorithm.

Contribution

It characterizes the structure of repetend matrices for periodic expansions and establishes conditions for purely periodic expansions in multidimensional continued fractions.

Findings

Repetend matrices have a specific structure for eventually periodic expansions.

Vectors with certain properties cannot have purely periodic expansions.

Application to algebraic vectors via the Jacobi–Perron Algorithm.

Abstract

We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it to prove that a number of vectors has an eventually periodic expansion in the Algebraic Jacobi--Perron Algorithm. Further, we give criteria for vectors to have purely periodic expansions; in particular, the vector cannot be totally positive.