The spectral multiplicity function and geometric representations of interval exchange transformations

The article solves the problem of which sets can be the set $\mathscr M$ of values of the spectral multiplicity function of an ergodic (strictly ergodic) interval exchange transformation and also of a simply ergodic dynamical system. By means of the method of geometric representations subsets of transformations are constructed that are generic in the metric and topological sense in the respective subclasses, and whose spectral multiplicity function has a prescribed set of values (in $\mathbb N\cup\{\infty\}$ naturally). These classes of transformations exhibit a new spectral effect: the component of multiplicity 1 in the spectrum is not the same as the spectrum of any factor of these transformations. Specific examples of strictly ergodic interval exchange transformations are constructed whose spectral multiplicity functions have all possible sets of values.