On the Mahler measure of the spectrum of rank one maps

Abstract

We extend partially the Kakutani-Zygmund dichotomy theorem to a class of generalized Riesz-product type measures by proving that the generalized Riesz-product is singular if and only if its Mahler measure is zero. As a consequence, we exhibit a new subclass of rank one maps acting on a finite measure space with singular spectrum. In our proof the $H^p$ theory coming to play. Furthermore, by appealing to a deep result of Bourgain, we prove that the Mahler measure of the spectrum of rank one map with cutting parameter $p_n=O(n^{\beta })$, $\beta \le 1$ is zero, and we establish that the integral of the absolute part of any generalized Riesz-product is strictly less than 1. This answer partially a question asked by M. Nadkarni.

24 pages, 1 figure and 60 references. This work extend as some level the work initiated by J. Bourgain, the author and M. Nadkarni in arXiv:1307.6513 [math.DS], arXiv:1402.5457 [math.CV], arXiv:1508.00417 [math.DS] and the contribution of the author to the well known Banach-Rohklin problem arXiv:1508.06439 [math.DS]. In this version few misprints are corrected. Scientific comments are welcome!

References in corpus (2)

• A non-singular transformation whose spectrum has Lebesgue component of multiplicity one
• Some Notes on Flat Polynomials