Transitively twisted flows of 3-manifolds (Q5956902)

Let a non-singular \(C^1\) vector field \(X\) of a closed three-dimensional manifold \(M\) generate a flow \(\varphi_t.\) Further, let \(TM\) denote the tangent bundle of \(M\) and let \(NX\) stand for the quotient bundle of \(TM\) by the 1-dimensional bundle tangent to \(X.\) The flow \(\varphi_t\) induces a flow \(P\varphi_t\) of the projectivized bundle of \(NX.\) Under the assumption that the projectivized bundle is a trivial one, the author proves that the angular flow of \(\varphi_t,\) denoted by \(\angle\varphi_t\) and defined as the lift of \(P\varphi_t,\) is not minimal, and constructs an example of \(\varphi_t\) such that \(\angle\varphi_t\) has a dense orbit. Under assumptions that \(\varphi_t\) is almost periodic and minimal, the angular flow \(\angle\varphi_t\) is classified into the following three cases:NEWLINENEWLINENEWLINE(1) all the orbits of \(\angle\varphi_t\) are bounded;NEWLINENEWLINENEWLINE(2) all the orbits of \(\angle\varphi_t\) are proper; NEWLINENEWLINENEWLINE(3) \(\angle\varphi_t\) is transitive.