Topological characteristic factors and nilsystems
Eli Glasner, Wen Huang, Song Shao, Benjamin Weiss, Xiangdong Ye · Journal of the European Mathematical Society · 2023
We prove that the maximal infinite step pro-nilfactor X_\infty of a minimal dynamical system (X,T) is the topological characteristic factor in a certain sense. Namely, we show that by an almost one-to-one modification of \pi:X \rightarrow X_\infty , the induced open extension \pi^{*}:X^{*} \rightarrow X^*_\infty has the following property: for x in a dense G_\delta subset of X^* , the orbit closure L_x=\overline{\mathcal{O}}((x,\ldots,x),\allowbreak T\times obreak T^2\times obreak \cdots \times T^d) is (\pi^*)^{(d)} -saturated, i.e., L_x=((\pi^*)^{(d)})^{-1}(\pi^*)^{(d)}(L_x) . Using results derived from the above fact, we are able to answer several open questions: (1) if (X,T^k) is minimal for some k\ge 2 , then for any d\in \mathbb{N} and any 0\le j there is a sequence \{n_i\} of \mathbb{Z} with n_i\equiv j\ (\text{mod}\ k) such that T^{n_i}x\rightarrow x, T^{2n_i}x\rightarrow x, \ldots, T^{dn_i}x\rightarrow x for x in a dense G_\delta subset of X ; (2) if (X,T) is totally minimal, then \{T^{n^2}x:n\in\mathbb{Z}\} is dense in X for x in a dense G_\delta subset of X ; (3) for any d\in\mathbb{N} and any minimal t.d.s. which is an open extension of its maximal distal factor, \mathbf{RP}^{[d]}=\mathbf{AP}^{[d]} , where the former is the regionally proximal relation of order d and the latter is the regionally proximal relation of order d along arithmetic progressions.