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Limit theorems for higher rank actions on Heisenberg nilmanifolds

  • *Corresponding author: Minsung Kim

    *Corresponding author: Minsung Kim

The first author is supported by NSF grant DMS 1600687 and by the Centre of Excellence "Dynamics, mathematical analysis and artificial intelligence" at Nicolaus Copernicus University in Toruń

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  • 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 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.

    Mathematics Subject Classification: Primary: 37A17, 37A20, 37A44; Secondary: 60F05.


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  • Figure 1.  Illustration of the rectangles $ U({\textbf{T}_1}) $ and $ U({\textbf{T}_2}) $ on $ i,j $-th coordinate

    Figure 2.  Illustration of the standard $ d $-rectangles $ \Gamma $, $ \Gamma_Q $, $ d+1 $ dimensional current $ D(\Gamma, \Gamma_ \mathsf{Q}) $ and supports of $ r_{-t}(\Gamma) $ and $ r_{-t}(\Gamma_ \mathsf{Q}) $

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