Eigenvalues of transformations arising from irrational rotations and step functions. (Valeurs propres de transformations liées aux rotations irrationnelles et aux fonctions en escalier)
arXiv:math/0605250
Abstract
Given an irrational rotation on $\M T$ we settle necessary and sufficient conditions on a step function and $t\in \M T$ for the existence of measurable solutions to the cohomogical equation $$\exp{(2iπϕ)}=\e{2iπt}f/f\rond T.$$ 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.
francais, 80 pages