An example on movable approximations of a minimal set in a continuous flow (Q870239)

The author presents an example of a set \(\Omega \) in \(\mathbb{R }^{3}\) that is not stable, is not a solenoid and, instead of periodic orbits, is approximated by invariant Denjoy-like sets \(\mathcal{D}_{n}\), \( n\in \mathbb{N}.\) Moreover, \(\Omega \) is not movable but the union of \( \Omega \) and the \(\mathcal{D}_{n}\)'s is (for the definition of movability, see \textit{K. Borsuk} [Fundam. Math. 66, 137--146 (1969; Zbl 0189.53802)]). This set \(\Omega\) is constructed in terms of an embedding in \(\mathbb{R}^{3}\) of the mapping torus of a suspension obtained from the direct product of an adding machine \(h_{\alpha}\) defined on a Cantor set \(S,\) and a circle homeomorphism \(h_{r_{\theta }}\) which is topologically semi-conjugated to a rotation \(r_{\theta }\) of angle \(2\pi \theta \) (\(\theta \) is an irrational number) defined on a minimal set \(D\) of the circle (\(D\) is obtained from \(r_{\theta }\) under the ``blowing up orbits'' technique).