Classification of rotations on the torus $\mathbb{T}^2$

TL;DR

This paper classifies rotations on the two-dimensional torus based on their complexity functions, extending the concept of Sturmian words from one dimension to two dimensions.

Contribution

It generalizes the classification of minimal rotations from one dimension to two dimensions using complexity functions.

Findings

Classified rotations on $\

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Abstract

We consider rotations on the torus $\mathbb{T}^2$, and we classify them with respect to the complexity functions. In dimension one, a minimal rotation can be coded by a sturmian word. A sturmian word has complexity $n+1$ by the Morse-Hedlund theorem. Here we make a generalization in dimension two.