Limit theorems for higher rank actions on Heisenberg nilmanifolds

TL;DR

This paper constructs Bufetov functionals for higher rank abelian actions on Heisenberg nilmanifolds, describing ergodic integral deviations and their limiting distributions under Diophantine conditions.

Contribution

It introduces Bufetov functionals for higher rank actions on Heisenberg nilmanifolds and links their asymptotics to ergodic integral deviations.

Findings

Deviation of ergodic integrals described by Bufetov functionals

Distribution of normalized ergodic integrals converges to a non-degenerate measure

Construction of finitely additive measures under Diophantine conditions

Abstract

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 \emph{Bufetov functionals} on rectangles on $(2g+1)$-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.