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 Canter 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 omega with continued fraction expansion omega = [0; (n) over bar] 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 alpha(min) similar to constant ln n/n. 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 (r(q)), q --> infinity is sensible to the parity of n.