  A classification of incompressible quantum Hall fluids in terms of integral
lattices and arithmetical invariants thereof is proposed. This classification
enables us to characterize the plateau values of the Hall conductivity $\sH$ in
the interval $\,(0,1]\,$ (in units where $\,e^2/h=1$) corresponding to
``stable'' incompressible quantum Hall fluids. A bijection, called shift map,
between classes of stable incompressible quantum Hall fluids corresponding to
plateaux of $\sH$ in the intervals $\,[1/(2\mini p+1),1/(2\mini p-1)\mini)\,$
and $\,[1/(2\mini q+1),1/(2\mini q-1)\mini)$, respectively, is constructed,
with $\,p,q=1,2,3, (\ldots),\ p\neq q$.
  Our theoretical results are carefully compared to experimental data, and
various predictions and experimental implications of our theory are discussed.
