2019/04/15 by Claudio Bonanno, Bonanno, Claudio, Alessio Del Vigna +3 · 1 citation
Computer Science · Mathematics · #37A40 #37A45 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Numerical Methods and Algorithms
paper · pdf · doi:10.48550/arxiv.1904.07095
openalex publication_date 2019/04/15 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We study the two-dimensional continued fraction algorithm introduced in\n citegarr and the associated \triangle map T, defined on a triangle\n triangle\⊂ R2. We introduce a slow version of the triangle map, the\nmap S, which is ergodic with respect to the Lebesgue measure and preserves an\ninfinite Lebesgue-absolutely continuous invariant measure. We discuss the\nproperties that the two maps T and S share with the classical Gauss and\nFarey maps on the interval, including an analogue of the weak law of large\nnumbers and of Khinchin's weak law for the digits of the triangle sequence, the\nexpansion associated to T. Finally, we confirm the role of the map S as a\ntwo-dimensional version of the Farey map by introducing a complete tree of\nrational pairs, constructed using the inverse branches of S, in the same way\nas the Farey tree is generated by the Farey map, and then, equivalently,\ngenerated by a generalised mediant operation.\n