Regionally proximal relation of order d along arithmetic progressions and nilsystems

2019/11/12 by Glasner, Eli, Huang, Wen, Shao, Song +1
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.04691

Abstract

The regionally proximal relation of order d along arithmetic progressions, namely \bf AP^[d] for d∈ \N, is introduced and investigated. It turns out that if (X,T) is a topological dynamical system with \bf AP^[d]=Δ, then each ergodic measure of (X,T) is isomorphic to a d-step pro-nilsystem, and thus (X,T) has zero entropy. Moreover, it is shown that if (X,T) is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic, then \bf AP^[d]=\bf RP^[d] for each d∈ \mathbb N. It follows that for a minimal ∞-pro-nilsystem, \bf AP^[d]=\bf RP^[d] for each d∈ \mathbb N. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.

Related

• Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence
• Distal systems in topological dynamics and ergodic theory
• Nilsequences and a structure theorem for topological dynamical systems
• Pro-nilfactors of the space of arithmetic progressions in topological dynamical systems
• Mean equicontinuity, almost automorphy and regularity