A non-conventional ergodic theorem for a nilsystem

By T. Ziegler. Ergodic Theory and Dynamical Systems, 2002

69 citations recorded by the source corpus.

We prove a non-conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0f5891bc585ee1841ae1246fb21fa0a2b1cd0e89

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Some open problems on multiple ergodic averages (2011, background): “Furthermore, explicit formulas for the limit can be given for special families of polynomials [44, 22, 24, 21, 36], but no such formula is known for general families of polynomials.”
The logarithmic Sarnak conjecture for ergodic weights (2017, background): “By [2, §5] (see also [69]), G′ is a closed subgroup of Gs+1, hence a nilpotent Lie group, and the discrete subgroup Γ′ of G′ is cocompact.”
Multiple recurrence and convergence for hardy sequences of polynomial growth (2009, background): “Furthermore, a rather explicit formula for the limit of the averages (1) can be given in the linear case (combining results from [26] and [37]), and for some special collections of integer polynomial sequences ([16], [13], [30]).”
Universal characteristic factors and Furstenberg averages (2004, methodology): “An explicit description of the limit is given in [Le89] for the case k = 3, and in general in [Z05].”
Multiple ergodic averages for three polynomials and applications (2006, unknown): “The following result of Leibman ([23],[ 24 ]) gives information about the orbit closure of polynomial sequences on nilmanifolds and helps us handle their uniform distribution properties8 by reducing them to uniform distribution”
An inverse theorem for the Gowers U^3 norm (2005, background): “ence expressions appearing for instance in the Furstenberg recurrence theorem; this fact is crucial in establishing the convergence of these recurrence expressions. We remark that the work of Ziegler [60, 61] achieves a very similar result, but avoids use of the Uk−1(T) norm and obtains a characteri”