  In this paper we investigate aspects of rigidity and flexibility for
conformal iterated function systems. For the case in which the systems are not
essentially affine we show that two such systems are conformal equivalent if
and only if in each of their Lyapunov spectra there exists at least one level
set such that the corresponding Gibbs measures coincide. We then proceed by
comparing this result with the essentially affine situation. We show that
essentially affine systems are far less rigid than non--essentially affine
systems, and subsequently we then investigate the extent of their flexibility.
