Eigenvalues of transformations arising from irrational rotations and step functions. (Valeurs propres de transformations liées aux rotations irrationnelles et aux fonctions en escalier)

Melanie Guenais (1) , Francois Parreau (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LAGA - Laboratoire Analyse, Géométrie et Applications

Given an irrational rotation $T$ on $\M T$ we settle necessary and sufficient conditions on a step function $\phi$ and $t\in \M T$ for the existence of measurable solutions to the cohomogical equation $$\exp{(2i\pi\phi)}=\e{2i\pi t}f/f\rond T.$$ This yields a characterization of eigenvalues and eigenfunctions for several transformations arising from irrational rotations and step functions~: cylinder flows, special flows, induced maps... From there we give constructions of special flows and three-interval exchange transformations with unusual spectral properties. In both cases we exhibit examples with Kronecker factors of infinite rank. We also construct three-interval exchange transformations which are non-trivially conjugate to irrational rotations or to odometers. Similarly there exist special flows over irrational rotations which are non-trivially conjugate to translations flows on $\M T^2$ or on solenoids. Finally, we prove a regularization property which allows us to give similar examples of special flows with smooth ceiling functions, under natural Diophantine conditions for the rotation.

Domaines: Systèmes dynamiques [math.DS]

Melanie Guenais, Francois Parreau. Eigenvalues of transformations arising from irrational rotations and step functions. (Valeurs propres de transformations liées aux rotations irrationnelles et aux fonctions en escalier). 2006. ⟨hal-00067970⟩