Ergodic seminorms for commuting transformations and applications

By B. Host. 2008

86 citations recorded by the source corpus.

Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although we give here one more proof of the same theorem, this is not the main goal of this paper. Our main concern is to provide some tools for the case of several commuting transformations, similar to the tools successfully used in the case of a single transformation, with the idea that they will be useful in the solution of other problems.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4de9ef92be7cb48ca8e1d991493ec6bf243b6818

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In Discrete Mathematics and Combinatorics

1. Nikos Frantzikinakis
2. Q. Chu
3. B. Host
4. Xiangdong Ye
5. S. Shao
6. M. N. Walsh
7. Tim Austin
8. Terence Tao

In Mathematical Physics

1. Nikos Frantzikinakis
2. Q. Chu
3. B. Host
4. S. Shao
5. Xiangdong Ye
6. M. N. Walsh
7. Tim Austin
8. Terence Tao

In Statistics and Probability

1. Q. Chu
2. Nikos Frantzikinakis
3. M. N. Walsh
4. Xiangdong Ye
5. S. Shao
6. Tim Austin
7. B. Host

Citations in the corpus

• Ergodic averages of commuting transformations with distinct degree polynomial iterates (2009, background): “† It is in cases like this that working with some kind of ‘pleasant’ extension (using terminology from [4]) or ‘magic’ extension (using terminology form [22]) of the system may offer an essential advantage (this is indeed the case when all the polynomials are linear).”
• Multiple recurrence for two commuting transformations (2009, methodology): “We will use more elaborate tools such as the machinery of ‘magic systems’ introduced recently by Host [12].”
• Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence (2010, methodology): “Note that the multiple ergodic average for commuting transformations was obtained by Tao [26] using finitary ergodic method, see [3, 18] for more traditional ergodic proofs.”
• Some open problems on multiple ergodic averages (2011, background): “, T` are powers of the same transformation ([32, 35]), when the polynomials are linear [42] (with alternate proofs given in [43, 3, 29]), and when the polynomials have distinct degrees [15].”
• On the norm convergence of non-conventional ergodic averages (2008, background): “Since a preprint of this paper first appeared, Bernard Host has shown in [11] that by using the above Host–Kra self-joining, one iteration of the above construction suffices to produce a pleasant system: the passage to the inverse limit is already superfluous! His proof of this requires a slightly m”
• Norm convergence of nilpotent ergodic averages (2011, result): “Without these assumptions, this result was established by Tao [19] and by now possesses several different proofs [2, 13, 20].”
• Multiple recurrence and convergence for hardy sequences of polynomial growth (2009, background): “Nevertheless, we now have several different proofs of convergence in the case where the sequences are linear ([26], [38], [33], [35],[1], [25]), and the case where all the sequences are integer polynomials was treated in [27] and [29].”