On the maximal spectral type of a class of rank one transformations

Abstract

Rank one transformations are transformations which can be obtained by cutting and stacking, using a single column in each step. Such a transformation is defined by a sequence of cutting parameters (p _k ) _k≥1 and a sequence of parameters of spacers . Rank one transformations are ergodic and have simple spectrum. By a result of Klemes and Reinhold, a rank one transformation is of singular maximal spectral type if . El Abdalaoui showed that for arbitrary (p _k ) _k≥1 the transformation has singular maximal spectral type if for each k all the numbers are of different order of magnitude. In this article we prove a counterpart of El Abdalaoui's result: if for infinitely many indices k a certain (relatively small) proportion of the coefficients are all equal, then the transformation is of singular maximal spectral type.

Keywords:

• rank one transformations
• maximal spectral type
• exponential sums
• martingale approximation

AMS Subject Classifications::

• 37A30
• 28D05
• 37A05

Acknowledgements

Research supported by the Austrian Research Foundation (FWF), Project S9603-N23.