• The Pure and Applied Mathematics
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• v.31 no.4
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• pp.439-451
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• 2024

Motivated by an algorithm to generate all Pythagorean triples, Romik introduced a dynamical system on the unit circle, which corresponds the continued fraction algorithm on the index-2 sublattice. Cha et al. extended Romik's work to other ellipses and spheres and developed a dynamical system generating all Eisenstein triples. In this article, we review the dynamical systems by Romik and by Cha et al. and find connections to the continued fraction algorithms.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.251-274
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• 2020

We study a dynamical system that was originally defined by Romik in 2008 using an old theorem of Berggren concerning Pythagorean triples. Romik's system is closely related to the Farey map on the unit interval which generates an additive continued fraction algorithm. We explore some number theoretical properties of the Romik system. In particular, we prove an analogue of Lagrange's theorem in the case of the Romik system on the unit quarter circle, which states that a point possesses an eventually periodic digit expansion if and only if the point is defined over a real quadratic extension field of rationals.