A minimal non-distal system with trivial order-two regional proximal relation

For a topological dynamical system (X,T)(X,T), let AP[2](X,T)\textbf{AP}^{[2]}(X,T) denote its order-two approximate proximal relation and let ΔX=(x,x):xX\Delta_X=\\{(x,x):x\in X\\}. Approximate-proximality conjecture. There is a minimal system (X,T)(X,T) such that

AP[2](X,T)=ΔX\textbf{AP}^{[2]}(X,T)=\Delta_X

and (X,T)(X,T) is not distal. This asks whether trivial order-two approximate proximality can coexist with failure of distality. The source provides no resolution.

Sources & referencesView supporting material

Primary source

Eli Glasner, Wen Huang, Song Shao, Benjamin Weiss and Xiangdong Ye, “Topological characteristic factors and nilsystems”, arXiv:2006.12385 (2020).

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