5 problems

Fix an initial value $z(0)=z_0$. The $xy$-plane is divided into disjoint KAM and non-KAM regions, and a KAM region is one of the regions in which the associated trajectories are tr…

Consider initial conditions for the ABC flow in the non-KAM region. Principal-vortex conjecture. There exists an open set of positive measure of initial conditions such that $y$ (o…

Let a trajectory of the ABC flow cross the boundary of a cell. Infinite-crossing conjecture. Almost all trajectories that cross the boundary of a cell will cross cell boundaries in…

Consider trajectories of the ABC flow with initial condition close to $(\frac{\pi}{2},0,a_c)$, where $a_c$ is the critical value for the $z$-periodic trajectory described above. Ba…

Let $X(t)=(x(t),y(t),z(t))$ be a solution of the ABC flow with perturbation parameter $\tilde\epsilon>0$, and let $T>0$ be a period modulo a lattice translation in the $x$- and $y$…