The Natural Extension for the Triangle Map (a Multi-dimensional Continued Fraction) with An Internal Symmetry from Young Conjugation

The natural extension of the triangle map (a type of multi-dimensional continued fraction algorithm) is completely described in all possible dimensions. The motivation and inspiration for this natural extension stems from the triangle map’s recent link to the classical study of integer partitions. Inspired by Young conjugation for integer partitions, we show that the natural extension has an internal symmetry and allows the natural extension to be subdivided into four natural subdomains. This appears to be new even for the classical case of the natural extension for continued fractions, namely for both the classical Gauss map and the classical Farey map.