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arXiv · 2307.00898

Periodicity of general multidimensional continued fractions using repetend matrix form

Hanka Řada·Štěpán Starosta·Vítězslav Kala
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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.

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Hanka Řada, Štěpán Starosta, Vítězslav Kala. 2023-07-03. Periodicity of general multidimensional continued fractions using repetend matrix form. https://doi.org/10.1016/j.exmath.2024.125571

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