  We obtain the distribution functions for anyonic excitations classified into
equivalence classes labeled by Hausdorff dimension $h$ and as an example of
such anyonic systems, we consider the collective excitations of the Fractional
Quantum Hall Effect (FQHE). We also introduce the concept of duality between
such classes, defined by $\tilde{h}=3-h$. In this way, we confirm that the
filling factors for which the FQHE were observed just appears into these
classes and the internal duality for a given class $h$ or $\tilde{h}$ is
between quasihole and quasiparticle excitations for these FQHE systems.
Exchanges of dual pairs $(\nu,\tilde{\nu})$, suggests conformal invariance. A
connection between equivalence classes $h$ and the modular group for the
quantum phase transitions of the FQHE is also obtained. A $\beta-$function is
also defined for the complex conductivity which embodies the $h$ classes.
