[Paper Review] Seminorms for multiple averages along polynomials and applications to joint ergodicity

Sebastián Donoso, Andreas Koutsogiannis+1 more|arXiv (Cornell University)|Feb 26, 2019

Limits and Structures in Graph Theory22 references4 citations

TL;DR

This paper establishes explicit characteristic factors for multiple ergodic averages along polynomial orbits in systems with commuting transformations, using a recent concatenation theorem by Tao and Ziegler. It provides a sufficient condition for joint ergodicity of sequences of the form $(T_1^{p_{1,j}(n)} \cdots T_d^{p_{d,j}(n)})_{n \in \mathbb{Z}}$, resolving a question posed by Bergelson on joint ergodicity for polynomial sequences in commuting systems.

ABSTRACT

Exploiting the recent work of Tao and Ziegler on a concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study criteria of joint ergodicity for sequences of the form $(T^{p_{1,j}(n)}_{1}\cdot\ldots\cdot T^{p_{d,j}(n)}_{d})_{n\in\mathbb{Z}},$ $1\leq j\leq k$, where $T_{1},\dots,T_{d}$ are commuting measure preserving transformations on a probability measure space and $p_{i,j}$ are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic, giving also a characterization for sequences of the form $(T^{p(n)}_{i})_{n\in\mathbb{Z}}, 1\leq i\leq d$ to be jointly ergodic, answering a question due to Bergelson.

Motivation & Objective

• To identify explicit characteristic factors for multiple ergodic averages along polynomial orbits in systems with commuting measure-preserving transformations.
• To extend the theory of Host-Kra seminorms to multiple commuting transformations and polynomial sequences.
• To provide a sufficient condition for joint ergodicity of sequences generated by integer polynomials under commuting transformations.
• To resolve a question posed by Bergelson regarding joint ergodicity of polynomial sequences in the context of multiple commuting systems.
• To leverage the recent concatenation theorem of Tao and Ziegler to construct a hierarchy of factors controlling the $L^2$-norm of multiple averages.

Proposed method

• Utilizes the concatenation theorem of Tao and Ziegler to analyze intersections of factors in nilsystems and their behavior under polynomial sequences.
• Introduces a hierarchy of seminorms—generalizing Host-Kra seminorms—to control the $L^2$-norm of multiple averages along polynomials.
• Employs a recursive approximation argument using Følner sequences to reduce the norm of the average to a finite combination of factor averages.
• Applies the pigeonhole principle to identify a finite set of directions in the parameter space that capture the essential dynamics of the polynomial sequences.
• Uses the structure of rational span and integer lattice intersections to characterize the limiting factors as extensions of nilsystems.
• Establishes admissibility conditions on finite tuples of multi-indices to ensure convergence of the approximating averages in $L^2$.

Experimental results

Research questions

• RQ1 What are the characteristic factors for multiple ergodic averages along polynomial orbits in systems with commuting transformations?
• RQ2 Under what conditions are sequences of the form $(T_1^{p_{1,j}(n)} \cdots T_d^{p_{d,j}(n)})_{n \in \mathbb{Z}}$ jointly ergodic?
• RQ3 How can the recent concatenation theorem of Tao and Ziegler be used to analyze the structure of multiple averages in multiple commuting systems?
• RQ4 Can Host-Kra seminorms be generalized to control multiple averages involving multiple commuting transformations and polynomial sequences?
• RQ5 What is the precise role of nilsystems and their extensions in characterizing the limits of such multiple averages?

Key findings

• The paper establishes a sufficient condition for joint ergodicity of polynomial sequences in commuting systems, generalizing known results for linear sequences.
• It provides a complete characterization of joint ergodicity for sequences of the form $(T^{p(n)}_i)_{n \in \mathbb{Z}}$, $1 \leq i \leq d$, answering a question of Bergelson.
• The $L^2$-norm of the multiple average along polynomials is bounded above by a finite combination of Host-Kra seminorms associated with the system’s characteristic factors.
• The authors prove that if the $L^2$-limit of the average is zero when restricted to a certain factor, then the full average converges to zero in $L^2$-norm.
• The proof relies on a finite approximation argument using Følner sequences and the admissibility of finite tuples of multi-indices to control the error in the limit.
• The key structural result is that the intersection of certain factor algebras associated with polynomial directions contains the relevant characteristic factor, ensuring convergence.