Complete special affine flows without closed orbits on 3-dimensional torus bundle over circle (Q1384462)

Let \(SA(n)\) be the group of volume preserving affine transformations of \(\mathbb{R}^n\) and \(M(P)\) denote a \(T^3\)-bundle over \(S^1\) with monodromy \(P\in\text{SL}(3;\mathbb{Z})\). Starting with a work of \textit{A. Sato} [Note on complete affine flows without closed orbits on 3-manifolds, Preprint] the author determines monodromies of 3-dimensional torus bundles over the circle such that the corresponding manifolds admit complete special affine flows without closed orbits. Main results: (1) There exists a complete \((SA(3),\mathbb{R}^3)\)-flow without closed orbits on \(M(P)\) if and only if \(\text{det}(P-I)=0\); (2) There exists a minimal complete \((SA(3),\mathbb{R}^3)\)-flow on \(M(P)\) if and only if all eigenvalues of \(P\) are 1, (3) the torus \(T^4=M(I)\) has an exotic minimal flow. Here \(SA(3)\) acts real analytically on \(\mathbb{R}^3\), and \(I\) denotes the identity matrix. The author constructs exotic (nonlinear) minimal flows on \(T^4\). Some important particular cases are studied, too.