A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs

Autor: Claudio Bonanno, Sara Munday, Alessio Del Vigna
Rok vydání: 2021
Předmět:
Zdroj: Monatshefte für Mathematik. 194:1-40
ISSN: 1436-5081
0026-9255
DOI: 10.1007/s00605-020-01500-w
Popis: We study the two-dimensional continued fraction algorithm introduced in \cite{garr} and the associated \emph{triangle map} $T$, defined on a triangle $\triangle\subset \R^2$. We introduce a slow version of the triangle map, the map $S$, which is ergodic with respect to the Lebesgue measure and preserves an infinite Lebesgue-absolutely continuous invariant measure. We discuss the properties that the two maps $T$ and $S$ share with the classical Gauss and Farey maps on the interval, including an analogue of the weak law of large numbers and of Khinchin's weak law for the digits of the triangle sequence, the expansion associated to $T$. Finally, we confirm the role of the map $S$ as a two-dimensional version of the Farey map by introducing a complete tree of rational pairs, constructed using the inverse branches of $S$, in the same way as the Farey tree is generated by the Farey map, and then, equivalently, generated by a generalised mediant operation.
32 pages. The main results have slightly changed due to a mistake in the previous version
Databáze: OpenAIRE
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