Topological characteristic factors and nilsystems

  • Eli Glasner

    Tel Aviv University, Israel
  • Wen Huang

    University of Science and Technology of China, Hefei, China
  • Song Shao

    University of Science and Technology of China, Hefei, China
  • Benjamin Weiss

    The Hebrew University of Jerusalem, Israel
  • Xiangdong Ye

    University of Science and Technology of China, Hefei, China
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Abstract

We prove that the maximal infinite step pro-nilfactor of a minimal dynamical system is the topological characteristic factor in a certain sense. Namely, we show that by an almost one-to-one modification of , the induced open extension has the following property: for in a dense subset of , the orbit closure is -saturated, i.e., . Using results derived from the above fact, we are able to answer several open questions: (1) if is minimal for some , then for any and any there is a sequence of with such that for in a dense subset of ; (2) if is totally minimal, then is dense in for in a dense subset of ; (3) for any and any minimal t.d.s. which is an open extension of its maximal distal factor, , where the former is the regionally proximal relation of order and the latter is the regionally proximal relation of order along arithmetic progressions.

Cite this article

Eli Glasner, Wen Huang, Song Shao, Benjamin Weiss, Xiangdong Ye, Topological characteristic factors and nilsystems. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1379