Higher order Wiener–Wintner systems: examples and applications

Abstract

We will construct “higher-dimensional" versions of the Wiener–Wintner dynamical system that was originally studied by I. Assani in 2003. We will show that on these systems we can provide very simple proofs of the a.e. convergence of the multiple recurrence averages, as well as the multiple recurrence return times averages. We will do so by obtaining a quantitative control of the multiple ergodic averages by extending the estimate for the double recurrence that was attained by J. Bourgain. We will also observe that this class of dynamical systems contains numerous examples that are not bounded by the standard classifications (e.g. entropy, mixing), such as Kolmogorov systems, classical skew products, as well as systems for which the a.e. convergence of multiple recurrence is not currently known. Along our way, we will also provide alternative characteristics of the Host–Kra–Ziegler factors from the point of view of the uniform Wiener–Wintner theorem.