Hofstadter rules and generalized dimensions of the spectrum of Harper's equation

We consider the Harper model which describes two-dimensional Bloch electrons in a magnetic field. For irrational flux through the unit-cell the corresponding energy spectrum is known to be a Cantor set with multifractal properties. In order to relate the maximal and minimal fractal dimension of the spectrum of Harper's equation to the irrational number involved, we combine a refined version of the Hofstadter rules with results from semiclassical analysis and tunnelling in phase space. For quadratic irrationals with continued fraction expansion the maximal fractal dimension exhibits oscillatory behaviour as a function of n, which can be explained by the structure of the renormalization flow. The asymptotic behaviour of the minimal fractal dimension is given by . As the generalized dimensions can be related to the anomalous diffusion exponents of an initially localized wavepacket, our results imply that the time evolution of high order moments is sensible to the parity of n.