Fractal Structure of the Harper Map Phase Diagram from Topological Hierarchical Classification

It is suggested a topological hierarchical classification of the infinite many Localized phases figuring in the phase diagram of the Harper equation for anisotropy parameter $\epsilon$ versus Energy $E$ with irrational magnetic flux $\omega$. It is also proposed a rule that explain the fractal structure of the phase diagram. Among many other applications, this system is equivalent to the Semi-classical problem of Bloch electrons in a uniform magnetic field, the Azbel-Hofstadter model, where the discrete magnetic translations operators constitute the quantum algebra $U_q(sl_2)$ with $q^2=e^{i2\pi\omega}$. The magnetic flux is taken to be the golden mean $\omega^*=(\sqrt{5}-1)/2$ and is obtained by successive rational approximants $\omega_m=F_{m-1}/F_m$ with $F_m$ given by the Fibonacci sequence $F_m$.[OUTP-00-08S, \texttt{cond-mat/0011396}]