[Paper Review] Topological correspondence of multiple ergodic averages of nilpotent group actions

Wen Huang, Song Shao+1 more|arXiv (Cornell University)|Apr 25, 2016

Mathematical Dynamics and Fractals15 references3 citations

TL;DR

This paper establishes a topological counterpart to multiple ergodic averages for nilpotent group actions, proving that under weak mixing and minimality conditions, the orbit of a residual set of points under polynomial group actions is dense in the product space. The key result extends Glasner's theorem to non-abelian nilpotent groups using polynomial sequences with rational coefficients.

ABSTRACT

Let $(X,Γ)$ be a topological system, where $Γ$ is a nilpotent group generated by $T_1,\ldots, T_d$ such that for each $T\in Γ$, $T eq e_Γ$, $(X,T)$ is weakly mixing and minimal. For $d,k\in \mathbb{N}$, let $p_{i,j}(n), 1\le i\le k, 1\le j\le d$ be polynomials with rational coefficients taking integer values on the integers and $p_{i,j}(0)=0$. We show that if the expressions $g_i(n)=T_1^{p_{i,1}(n)}\cdots T_d^{p_{i,d}(n)}$ depends nontrivially on $n$ for $i=1,2,\cdots,k$, and for all $i eq j\in \{1,2,\ldots,k\}$ the expressions $g_i(n)g_j(n)^{-1}$ depend nontrivially on $n$, then there is a residual set $X_0$ of $X$ such that for all $x\in X_0$ \begin{equation*} \{(g_1(n)x, g_2(n)x,\ldots, g_k(n)x)\in X^k:n\in \mathbb{Z}\} \end{equation*} is dense in $X^k$.

Motivation & Objective

• To extend Glasner's topological multiple recurrence result from single transformations to nilpotent group actions.
• To establish a topological analogue of multiple ergodic averages in the context of weakly mixing, minimal systems under nilpotent group actions.
• To characterize the conditions under which polynomial sequences in nilpotent groups yield dense orbits in product spaces.
• To prove that the set of points with dense orbits is residual (Gδ), generalizing results from abelian to non-abelian nilpotent groups.
• To verify the necessity of non-degeneracy conditions on polynomial sequences for the density result.

Proposed method

• Utilizes the structure of nilpotent groups and their Malcev bases to define polynomial group elements $ g_i(n) = T_1^{p_{i,1}(n)} \cdots T_d^{p_{i,d}(n)} $ with rational coefficient polynomials vanishing at zero.
• Imposes non-degeneracy conditions: each $ g_i(n) $ is non-constant, and $ g_i(n)g_j(n)^{-1} $ is non-constant for $ i \neq j $, ensuring distinct dynamics.
• Applies topological dynamics tools: residual sets, syndetic sets, and piecewise syndeticity to analyze orbit distribution.
• Employs inductive arguments on the weight of $ \Gamma $-polynomials and uses the structure of $ \mathbf{P}\Gamma_0^* $ to control orbit recurrence.
• Leverages the fact that for minimal and weakly mixing systems, return times to open sets are syndetic, and uses this to construct dense orbit sets.
• Constructs a residual set $ X_0 \subset X $ such that for all $ x \in X_0 $, the set $ \{(g_1(n)x, \dots, g_k(n)x)\}_{n \in \mathbb{Z}} $ is dense in $ X^k $.

Experimental results

Research questions

• RQ1 Under what conditions do polynomial group actions of a nilpotent group on a minimal, weakly mixing system yield dense orbits in the product space?
• RQ2 How does the topological multiple recurrence result generalize from abelian to non-abelian nilpotent groups?
• RQ3 What is the role of non-degeneracy conditions on polynomial sequences in ensuring orbit density?
• RQ4 Can the topological counterpart of multiple ergodic averages be established for nilpotent group actions?
• RQ5 What is the structure of the set of points with dense orbits under such actions?

Key findings

• For a nilpotent group $ \Gamma $ acting minimally and weakly mixing on $ X $, and for polynomial sequences $ g_i(n) $ satisfying non-degeneracy conditions, there exists a residual set $ X_0 \subset X $ such that $ \{(g_1(n)x, \dots, g_k(n)x)\}_{n \in \mathbb{Z}} $ is dense in $ X^k $ for all $ x \in X_0 $.
• The non-degeneracy condition—that $ g_i(n) $ and $ g_i(n)g_j(n)^{-1} $ are non-constant for $ i \neq j $—is necessary for the density result.
• The result generalizes Glasner’s theorem from single transformations to multiple polynomial actions in nilpotent group settings.
• The set of points with dense orbits is residual, hence comeager, indicating that such points are generic in the topological sense.
• The proof relies on syndeticity and piecewise syndeticity of return times, showing that orbit recurrence is sufficiently rich to ensure density.
• The construction uses inductive arguments on the weight of $ \Gamma $-polynomials and properties of $ \mathbf{P}\Gamma_0^* $, the set of non-constant $ \Gamma $-polynomials.