Ergodic averages of commuting transformations with distinct degree polynomial iterates
By Q. Chu, Nikos Frantzikinakis, B. Host. 2009
67 citations recorded by the source corpus.
We prove mean convergence, as N → ∞, for the multiple ergodic averages 1N∑n=1Nf1(T1p1(n)x)⋅…⋅fl(Tlplx) where p1, …, pℓ are integer polynomials with distinct degrees, and T1, …, Tℓ are commuting, invertible measure preserving transformations, acting on the same probability space. This establishes several cases of a conjecture of Bergelson and Leibman that complement the case of linear polynomials, recently established by Tao. Furthermore, we show that, unlike the case of linear polynomials, for polynomials of distinct degrees, the corresponding characteristic factors are mixtures of inverse limits of nilsystems. We use this particular structure, together with some equidistribution results on nilmanifolds, to give an application to multiple recurrence and a corresponding one to combinatorics.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/75af746b42661aebb02867183cd4cb835b1f0824
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In Discrete Mathematics and Combinatorics
In Mathematical Physics
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Citations in the corpus
- Norm convergence of nilpotent ergodic averages (2011, background): “It was shown by Chu, Frantzikinakis and Host [7] that 1 N N∑ n=1 T p1(n) 1 f1 . . .”
- Some open problems on multiple ergodic averages (2011, unknown): “33, 36, 37, 59, 60, 63, 64, 65, 68, 69, 73, 75, 76, 84, 91, 97, 98, 99, 111, 112, 113, 114, 115, 119, 123, 139, 151, 154, 155, 162, 173, 178, 181, 184, 186, 192, 193, 195, 196]. Polynomial sequences: [11, 12, 13, 22, 29, 30, 34, 35, 39, 40, 41, 43, 44, 52, 53, 54, 55, 58, 61, 62, 70, 74, 77, 79, 82,”