  Area non-preserving transformations in the non-commutative plane are
introduced with the aim to map the $\nu=1$ integer quantum Hall effect (IQHE)
state on the fractional quantum Hall effect (FQHE) $\nu=\frac{1}{2p+1}$ FQHE
states. Using the hydrodynamical description of the quantum Hall fluid, it is
shown that these transformations are generated by vector fields satisfying the
Gauss law in the interacting non-commutative Chern-Simons gauge theory, and the
corresponding field-theory Lagrangian is reconstructed. It is demonstrated that
the geometric transformations induce quantum-mechanical non-unitary similarity
transformations, establishing the interplay between integral and fractional
QHEs.
