  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 and number
theory.
