Fractal geometry of electron orbits in random systems with strong magnetic field

We study electron orbits under a random electric potential and a strong magnetic field, at the threshold energy above (and below) which all orbits are localized. In two dimensions (2D), these orbits are equipotential lines, which have the same fractal dimension as the hulls of percolation clusters, $D_h$=7/4. In 3D the electron moves on cylinders between equipotential surfaces, with approximately constant heights. The slowest motion is on transverse projections of these cylinders, which are very similar to the two-dimensional orbits. The results are relevant to the quantum Hall effect.