  We study the structural stability of a cosmic acceleration (inflation) in a
class of k-essence cosmologies against changes in the shape of the potential.
Those models may be viewed as generalized tachyon cosmologies and this analysis
extends previous results on the structural stability of cosmic acceleration in
tachyon cosmologies. The study considers both phantom and non-phantom cases.
The concepts of rigidity and fragility are defined through a condition on the
functional form of the Hubble factor. Given the known result of the existence
of inflationary (non-phantom) and super-inflationary (phantom) attractors we
formulate the question of their structural stability. We find that those
attractors are rigid in the sense that they never change as long as the
conditions for inflation or super-inflation are met.
