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PreprintCC BY 4.0arXiv

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

  • Joe Fox
  • Thomas Garrity
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{fox2026natural,
  title = {{The Natural Extension for the Triangle Map (a Multi-dimensional Continued Fraction) with An Internal Symmetry from Young Conjugation}},
  author = {Joe Fox and Thomas Garrity},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.28014},
  eprint = {2606.28014},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.28014},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Natural Extension for the Triangle Map (a Multi-dimensional Continued Fraction) with An Internal Symmetry from Young Conjugation
AU  - Joe Fox
AU  - Thomas Garrity
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2606.28014
UR  - https://arxiv.org/abs/2606.28014
AB  - 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.
ER  -