  We consider the fractal characteristic of the quantum mechanical paths and we
obtain for any universal class of fractons labeled by the Hausdorff dimension
defined within the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$, a fractal
distribution function associated with a fractal von Neumann entropy. Fractons
are charge-flux systems defined in two-dimensional multiply connected space and
they carry rational or irrational values of spin.
  This formulation can be considered in the context of the fractional quantum
Hall effect-FQHE, where we discovered that the quantization of the Hall
resistance occurs in pairs of dual topological quantum numbers, the filling
factors. In this way, these quantum numbers get their topological character
from the Hausdorff dimension associated with the fractal quantum path of such
particles termed fractons. On the other hand, the universality classes of the
quantum Hall transitions can be classified in terms of $h$. Another consequence
of our approach, which is supported by symmetry principles, is the prediction
of the FQHE. The connection between Physics and Number Theory appears naturally
in this context.
