Limit theorems for higher rank actions on Heisenberg nilmanifolds

Abstract

<p style='text-indent:20px;'>The main result of this paper is a construction of finitely additive measures for higher rank abelian actions on Heisenberg nilmanifolds. Under a full measure set of Diophantine conditions for the generators of the action, we construct <i>Bufetov functionals</i> on rectangles on <inline-formula><tex-math id="M1">\begin{document}$ (2g+1) $\end{document}</tex-math></inline-formula>-dimensional Heisenberg manifolds. We prove that deviation of the ergodic integral of higher rank actions is described by the asymptotic of Bufetov functionals for a sufficiently smooth function. As a corollary, the distribution of normalized ergodic integrals which have variance 1, converges along certain subsequences to a non-degenerate compactly supported measure on the real line.</p>