  Asymptotically entropy of chaotic systems increases linearly and the
sensitivity to initial conditions is exponential with time: these two behaviors
are related. Such relationship is the analogous of and under specific
conditions has been shown to coincide with the Pesin identity. Numerical
evidences support the proposal that the statistical formalism can be extended
to the edge of chaos by using a specific generalization of the exponential and
of the Boltzmann-Gibbs entropy. We extend this picture and a Pesin-like
identity to a wide class of deformed entropies and exponentials using the
logistic map as a test case. The physical criterion of finite-entropy growth
strongly restricts the suitable entropies. The nature and characteristics of
this generalization are clarified.
