The additivity of the pseudo-additive conditional entropy for a proper Tsallis' entropic index

For Tsallis' entropic analysis to the time evolutions of standard logistic map at the Feigenbaum critical point, it is known that there exists a unique value $q^*$ of the entropic index such that the asymptotic rate $K_q \equiv \lim_{t \to \infty} \{S_q(t)-S_q(0)\} / t$ of increase in $S_q(t)$ remains finite whereas $K_q$ vanishes (diverges) for $q > q^* (q < q^*)$. We show that in spite of the associated whole time evolution cannot be factorized into a product of independent sub-interval time evolutions, the pseudo-additive conditional entropy $S_q(t|0) \equiv \{S_q(t)-S_q(0)\}/ \{1+(1-q)S_q(0)\}$ becomes additive when $q=q^*$. The connection between $K_{q^*}$ and the rate $K'_{q^*} \equiv S_{q^*}(t | 0) / t$ of increase in the conditional entropy is discussed.