On weakly mixing homeomorphisms of the two-torus that are uniformly rigid (Q1933615)

Does there exist a measure-preserving homeomorphism which is both weakly mixing and uniformly rigid? \textit{S. Glasner} and \textit{B. Weiss} proved that there is a large family of weakly mixing homeomorphisms on the infinite torus that are strictly ergodic [Ergodic Theory Dyn. Syst. 1, 151--157 (1981; Zbl 0474.54033)], this implies uniform rigidity by using a result of \textit{S. Glasner} and \textit{D. Maon} [Ergodic Theory Dyn. Syst. 9, No. 2, 309--320 (1989; Zbl 0661.58027)]. This paper considers this question on the two torus \(\mathbb{T}^2\). Let \(\mathcal{O}\) denote the closure of the set of conjugations of an aperiodic rotation by measure-preserving homeomorphism of \(\mathbb{T}^2\). It is showed that there exists a dense \(G_{\delta}\) subset \(\mathcal{R}\) of \(\mathcal{O}\) such that for every \(T\in \mathcal{R}\), the measure-theoretic system \((\mathbb{T}^2, T,\mu)\) is weakly mixing, uniformly rigid and strictly ergodic. Furthermore, given an increasing sequence of natural numbers \(\{n_m\}\) which satisfies \(\lim_{m\rightarrow\infty}\frac{n_{m+1}}{n_m}=\infty\), there exists an ergodic homeomorphism of \(\mathbb{T}^2\) that is uniformly rigid with respect to \(\{n_m\}\); and if the sequence \(\{n_m\}\) satisfies \(\frac{n_{m+1}}{n_m}\geq \psi(n_m)\), where \(\psi(x)=x^{x^3}\), there exists a weakly mixing homeomorphism of \(\mathbb{T}^2\) that is uniformly rigid with respect to \(\{n_m\}\).