Ergodicity and Algebraticity of the Fast and Slow Triangle Maps

TL;DR

This paper proves ergodicity of both fast and slow triangle maps in higher dimensions, confirming a conjecture and linking their dynamics to combinatorial properties related to partition numbers.

Contribution

It establishes the ergodic nature of higher-dimensional triangle maps, resolving a conjecture and connecting their dynamics to combinatorial implications.

Findings

Both fast and slow triangle maps are ergodic in dimension n.

The results confirm a conjecture by Messaoudi, Noguiera, and Schweiger.

The dynamics of these maps have implications for partition number studies.

Abstract

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension $n$ are ergodic, resolving a conjecture of Messaoudi, Noguiera and Schweiger. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.