Large-scale b.a.u. convergence for polynomial ergodic averages

Prove that, for every 1<p<\infty, every polynomial map P:\mathbb{R}^k\to\mathbb{R}^n of finite type at the origin, and every x\in L^p(\mathcal{M}), the ergodic means N_r^P(x) converge bilaterally almost uniformly to the projection F(x) onto the fixed-point subspace as r\to\infty.

Background

The paper establishes maximal inequalities for normalized polynomial ergodic averages and proves bilateral almost uniform convergence at small scales. At large scales, it proves convergence only along the dyadic subsequence r=2m, under the finite-type assumption on the polynomial map P at the origin.

The unresolved issue is convergence over the full continuous parameter r\to\infty. Here F denotes the projection from Lp(\mathcal{M}) onto the fixed-point subspace of the trace-preserving \mathbb{R}n-action. The authors formulate the desired full-parameter convergence as a conjecture because the telescoping methods used for positive contractive semigroups do not directly apply to averages over lower-dimensional polynomial sets.

References

This raises the question of whether the same conclusion holds for ergodic averages over lower-dimensional submanifolds in $n$. That is, does $N_rP(x)$ converge b.a.u. to $F(x)$ for $x\in Lp()$?

\begin{conj} Let $1<p<\infty$, and let $P:k\ton$ be a polynomial map of finite type at $t=0$. Then for all $x\in Lp()$, the ergodic mean $N_rP(x)$ converges to $F(x)$ bilaterally almost uniformly as $r\to\infty$. \end{conj}

— Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces  (2609.34906 - Lai et al., 28 Sep 2026) in Section 8, subsection “Large-scale ergodic theorems”