Mathematics – Dynamical Systems
Scientific paper
2012-02-21
Mathematics
Dynamical Systems
16 pages, 3 figures. To appear in the Bulletin of the London Mathematical Society
Scientific paper
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.
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