Journal of Modern Dynamics (JMD)

Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

Pages: 305 - 353, Volume 9, 2015      doi:10.3934/jmd.2015.9.305

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Salvatore Cosentino - Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal (email)
Livio Flaminio - UMR CNRS 8524, UFR de Mathématiques, Université de Lille 1, F59655 Villeneuve d’Asq CEDEX, France (email)

Abstract: We prove quantitative equidistribution results for actions of Abelian subgroups of the $(2g+1)$-dimensional Heisenberg group acting on compact $(2g+1)$-dimensional homogeneous nilmanifolds. The results are based on the study of the $C^\infty$-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalization method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in $g$ variables, generalizing the classical results of Hardy and Littlewood [25,26] and the optimal result of Fiedler, Jurkat, and Körner [17] to higher dimension.

Keywords:  Equidistribution, Heisenberg group, cohomological equation.
Mathematics Subject Classification:  Primary: 37C85, 37A17, 37A45; Secondary: 11K36, 11L07.

Received: June 2015;      Revised: September 2015;      Available Online: November 2015.