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

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

2006-05-10

arxiv.org/abs/math/0605250v1

Mathematics

Dynamical Systems

francais, 80 pages

Scientific paper

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.

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