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

## TL;DR

This paper introduces a slow version of a 2D continued fraction algorithm's triangle map, demonstrating its ergodic properties, invariant measures, and its role as a 2D Farey map with a complete rational pair tree.

## Contribution

It presents a new slow triangle map with ergodic properties and constructs a complete tree of rational pairs, extending classical 1D continued fraction concepts to two dimensions.

## Key findings

- The slow triangle map is ergodic with respect to Lebesgue measure.
- It preserves an infinite Lebesgue-absolutely continuous invariant measure.
- A complete tree of rational pairs is constructed, analogous to the Farey tree.

## Abstract

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.