Convergence of multiple ergodic averages for some commuting transformations

By Nikos Frantzikinakis, Bryna Kra. Ergodic Theory and Dynamical Systems, 2004

50 citations recorded by the source corpus.

We prove the L2-convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, \dots, T_{l}$, assuming that each map Ti and each pair TiTj-1 is ergodic for $i\neq j$. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/50a718496257de43d2469fc8190d12f5eea0c6f2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Nikos Frantzikinakis
2. Q. Chu
3. B. Host
4. M. N. Walsh
5. Terence Tao
6. T. Ziegler
7. Bryna Kra
8. Tim Austin

In Geometry and Topology

1. Nikos Frantzikinakis
2. Q. Chu
3. M. N. Walsh
4. B. Host
5. Terence Tao
6. T. Ziegler
7. Bryna Kra
8. Ben Joseph Green

In Mathematical Physics

1. Nikos Frantzikinakis
2. Q. Chu
3. B. Host
4. M. N. Walsh
5. Terence Tao
6. T. Ziegler
7. Bryna Kra
8. Ben Joseph Green

Citations in the corpus

• Some open problems on multiple ergodic averages (2011, background): “The lower bounds are known when all transformations are equal [23] and they are also known for general commuting transformations when the polynomials are monomials with distinct degrees [15].”
• Norm convergence of nilpotent ergodic averages (2011, background): “Here, the case d = 2 was proven by Conze and Lesigne [8] and assuming extra ergodicity hypothesis on the transformations Zhang [21] gave a proof for d = 3 and Frantzikinakis and Kra [9] for general d.”
• Ergodic averages of commuting transformations with distinct degree polynomial iterates (2009, background): “When all the polynomials in (1) are linear, after a series of partial results [11, 16, 20, 29, 33] that were obtained using ergodic theory, convergence was established in [31] using a finitary argument.”
• Ergodic seminorms for commuting transformations and applications (2008, background): “1,...,Sd) be a system where S1,...,Sd are commuting measure preserving transformations. Then, for every f1,...,fd ∈ L∞(µ), the averages (2) 1 |Ij| X n∈Ij Sn 1 f1 ·...·S n d fd 1The case considered in [FK1] is very particular. ERGODIC SEMINORMS FOR COMMUTING TRANSFORMATIONS 3 converge in L2(µ) for an”