Polynomial orbits in totally minimal systems

Abstract

Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye [14], we prove that the maximal ∞-step pro-nilfactor X∞ of a minimal system (X,T) is the topological characteristic factor along polynomials in a certain sense. Namely, we show that by an almost one to one modification of π:X→X∞, the induced open extension π⁎:X⁎→X∞⁎ has the following property: for any d∈N, any open subsets V0,V1,…,Vd of X⁎ with ⋂i=0dπ⁎(Vi)≠∅ and any distinct non-constant integer polynomials pi with pi(0)=0 for i=1,…,d, there exists some n∈Z such that V0∩T−p1(n)V1∩…∩T−pd(n)Vd≠∅.As an application, the following result is obtained: for a totally minimal system (X,T) and integer polynomials p1,…,pd, if every non-trivial integer combination of p1,…,pd is not constant, then there is a dense Gδ subset Ω of X such that the set{(Tp1(n)x,…,Tpd(n)x):n∈Z} is dense in Xd for every x∈Ω.