Topological correspondence of multiple ergodic averages of nilpotent group actions

Abstract

Let $(X,\Gamma )$ be a topological system, where $\Gamma$ is a nilpotent group generated by $T_1,\dots ,T_d$ such that for each $T\in \Gamma$, $T\ne e_{\Gamma }$, $(X,T)$ 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 $i\ne j\in \{1,2,\dots ,k\}$ 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 $X^k$.

24 pages, revised version following referees' reports, to appear in Journal d'Analyse Mathematique

References in corpus (2)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• Pointwise convergence of some multiple ergodic averages