Topological multiple recurrence for polynomial configurations in nilpotent groups
Let \(G\) be a nilpotent group of self-homeomorphisms of a compact metric space \((X,\rho)\) and let \(P_1, \dots, P_k\) be polynomial mappings from the collection of finite sets of natural numbers to \(G\) satisfying \(P_1(\emptyset) = \cdots = P_k(\emptyset) = 1_G\). The main theorem of this paper shows that, for any \(\varepsilon > 0\), there exist \(x \in X\) and a nonempty finite set of natural numbers \(\alpha\) such that \(\rho(P_i(\alpha)x,x) < \varepsilon\) for all \(i = 1, \dots, k\). This theorem can be viewed as a nilpotent version of the authors' polynomial Hales-Jewett theorem [cf. Ann. Math. 150, 33-75 (1999; Zbl 0969.05061)] and contains nilpotent extensions of many known ``abelian'' results as special cases.
scientific article; zbMATH DE number 1915638