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Erratum: On Omri Sarig's work on the dynamics of surfaces
2015, 9: 305-353. doi: 10.3934/jmd.2015.9.305
Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds
Salvatore Cosentino 1, and Livio Flaminio 2,
Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal
UMR CNRS 8524, UFR de Mathématiques, Université de Lille 1, F59655 Villeneuve d'Asq CEDEX, France
Received June 2015 Revised September 2015 Published November 2015
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: cohomological equation., Equidistribution, Heisenberg group.
Mathematics Subject Classification: Primary: 37C85, 37A17, 37A45; Secondary: 11K36, 11L0.
Citation: Salvatore Cosentino, Livio Flaminio. Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds. Journal of Modern Dynamics, 2015, 9: 305-353. doi: 10.3934/jmd.2015.9.305
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Salvatore Cosentino Livio Flaminio