Dimension conjectures for chaotic magnetic-field directions

Let $M_Fcsubset\mathbb T^3$ be a connected smooth two-manifold homologous to zero, and let a magnetic field determine trajectories on it. For a smooth one-parameter family $M_{F,t}\subset\mathbb T^3$ of such Fermi surfaces, consider the corresponding sets of chaotic directions of the magnetic field in $S^2$. Dimension conjecture for chaotic directions. (i) For a generic connected smooth two-manifold $M_F\subset\mathbb T^3$ homologous to zero, the set of chaotic directions of the magnetic field has Hausdorff dimension less than one in $S^2$. (ii) For a generic one-parameter smooth family $M_{F,t}\subset\mathbb T^3$ of such Fermi surfaces, this set has Hausdorff dimension less than two. The conjecture concerns the exceptional, nongeneric directions supporting chaotic trajectories; it predicts that these directions remain dimensionally small both for a single generic Fermi surface and for a generic one-parameter family.