The "flux-energy" diagram for the Hamiltonian of the Bloch electron in
a uniform magnetic field is known as the "Hofstadter butterfly" (or
"Azbel-Hofstadter butterfly"); it is among the best knownfractal structures of condensed matter physics. Originally this diagram
was constructed numerically by Hofstadter for tight-binding Hamiltonian;
now these diagrams are known for a few periodic potentials.
In the case of three-dimensional Bloch elecrons, the electron motion
along the field is not quantized by the field, and the most gaps of the
transversal motion are smeared; therefore, the fractal structure of the
"flux-energy" diagram disappears . Recently it was shown
that the fractal structure appears in the "angle-energy" diagram
for the tight-binding Hamiltonian (the strenghth of the field is fixed but
the direction is varied); the result is published in
 M.Koshino, H.Aoki, K.Kuroki, S.Kagoshima, T.Osada: Hofstadter butterfly
and integer quantum Hall effect in three dimensions, Phys. Rev. Lett.
86 (2001), 1062-1065.
 M..Koshino, H.Aoki, T.Osada, K.Kuroki, S.Kagoshima: Phase diagramm for
the Hofstadter butterfly and integer quantum Hall effect in three dimensions,
Phys. Rev. B65 (2002), 045310.
In the talk, the results concerning the fractal structure of the
"angle-energy" diagram for the Hamiltonian of an elecron in the
presence of a uniform magnetic field and a periodic array of point scatterers
will be reported. The results are obtained in collaboration with V.Demidov.