Ensemble Averages and Nonextensivity at the Edge of Chaos of One-Dimensional Maps

Ensemble averages of the sensitivity to initial conditions ξ(t) and the entropy production per unit of time of a new family of one-dimensional dissipative maps, xₜ₊₁=1-ae^-1/|xₜ|ᶻ(z>0), and of the known logisticlike maps, xₜ₊₁=1-a|xₜ|ᶻ(z>1), are numerically studied, both for strong (Lyapunov exponent λ₁>0) and weak (chaos threshold, i.e., λ₁=0) chaotic cases. In all cases we verify the following: (i) both lnqξ [lnqx≡(x^1-q-1)/(1-q); ln₁x=lnx] and ${}{S}q{} [{S}q{≡}{{0ex}{0ex}}(1{-}{∑}ᵢ{p}ᵢq)/(q{-}1); {S}₁={-}{∑}ᵢ{p}ᵢln{p}ᵢ]$ linearly increase with time for (and only for) a special value of $q$, ${q}ₛₑₙᵃᵛ$, and (ii) the slope of ${}{ln}q{ξ}{}$ and that of ${}{S}q{}$ coincide, thus interestingly extending the well known Pesin theorem. For strong chaos, ${q}ₛₑₙᵃᵛ=1$, whereas at the edge of chaos ${q}ₛₑₙᵃᵛ(z)<1$.