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 Kq≡limt→∞{Sq(t)−Sq(0)}/t of increase in Sq(t) remains finite whereas Kq vanishes (diverges) for q>q∗(q<q∗). We show that, in spite of the fact that the associated whole time evolution cannot be factorized into a product of independent sub-interval time evolutions, the pseudo-additive conditional entropy Sq(t|0)≡{Sq(t)−Sq(0)}/{1+(1−q)Sq(0)} becomes additive when q=q∗. The connection between Kq∗ and the rate Kq∗′≡Sq∗(t|0)/t of increase in the conditional entropy is discussed.