$3$-dimensional Continued Fraction Algorithms Cheat Sheets

Sébastien Labbé

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arXiv · 1511.08399

$3$-dimensional Continued Fraction Algorithms Cheat Sheets

Sébastien Labbé

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Abstract

Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of $\mathbb{R}^d$. We consider multidimensional continued fraction algorithms that acts symmetrically on the positive cone $\mathbb{R}^d_+$ for $d=3$. We include well-known and old ones (Poincaré, Brun, Selmer, Fully Subtractive) and new ones (Arnoux-Rauzy-Poincaré, Reverse, Cassaigne). For each algorithm, one page (called cheat sheet) gathers a handful of informations most of them generated with the open source software Sage with the optional Sage package \texttt{slabbe-0.2.spkg}. The information includes the $n$-cylinders, density function of an absolutely continuous invariant measure, domain of the natural extension, lyapunov exponents as well as data regarding combinatorics on words, symbolic dynamics and digital geometry, that is, associated substitutions, generated $S$-adic systems, factor complexity, discrepancy, dual substitutions and generation of digital planes. The document ends with a table of comparison of Lyapunov exponents and gives the code allowing to reproduce any of the results or figures appearing in these cheat sheets.

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Sébastien Labbé. 2015-11-26. $3$-dimensional Continued Fraction Algorithms Cheat Sheets. https://arxiv.org/abs/1511.08399

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