  We analytically link three properties of nonlinear dynamical systems, namely
sensitivity to initial conditions, entropy production, and escape rate, in
$z$-logistic maps for both positive and zero Lyapunov exponents. We unify these
relations at chaos, where the Lyapunov exponent is positive, and at its onset,
where it vanishes. Our result unifies, in particular, two already known cases,
namely (i) the standard entropy rate in the presence of escape, valid for
exponential functionality rates with strong chaos, and (ii) the Pesin-like
identity with no escape, valid for the power-law behavior present at points
such as the Feigenbaum one.
