Fonction complexit\'e associ\'ee \`a une application ergodique du tore

TL;DR

This paper establishes the optimal lower bound for the complexity function of planar translations inducing ergodic torus rotations and provides an explicit calculation for a specific application involving the golden mean.

Contribution

It introduces the optimal lower bound for the complexity function of ergodic planar translations and computes it explicitly for a particular application involving the golden mean.

Findings

Established the optimal lower bound for the complexity function.

Explicitly calculated the complexity for a specific ergodic translation.

Demonstrated the application of theoretical bounds to a concrete example.

Abstract

In this article we give the optimal lower bound for the complexity function of a planar translation which induces an ergodic rotation of the torus $\mathbb{R}^2 / \mathbb{Z}^2$. In addition, we give an explicit calculation of this complexity for the application $(x,y) \mapsto (x+1/\phi^2,y+x-1/ (2\phi^3))$, where $\phi$ is the golden mean. Nous proposons dans ce travail de minorer de mani\`ere optimale la fonction complexit\'e associ\'ee \`a une translation du plan, qui induit une rotation ergodique du tore $\mathbb R^2/\mathbb Z^2$. De plus, nous donnons un calcul explicite de cette complexit\'e pour l'application $(x,y)\mapsto(x+1/\phi^2,y+x-1/(2\phi^3))$, o\`u $\phi$ d\'esigne le nombre d'or.