Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps -: art. no. 020601

Ensemble averages of the sensitivity to initial conditions xi(t) and the entropy production per unit of time of a new family of one-dimensional dissipative maps, x(t+1)=1-ae(-1/\x t\ z)(z>0), and of the known logisticlike maps, x(t+1)=1-a\x(t)\(z)(z>1), are numerically studied, both for strong (Lyapunov exponent lambda(1)>0) and weak (chaos threshold, i.e., lambda(1)=0) chaotic cases. In all cases we verify the following: (i) both <ln(q)xi> [ln(q)xequivalent to(x(1-q)-1)/(1-q); ln(1)x=lnx] and <S-q> [S(q)equivalent to(1-Sigma(i)p(i)(q))/(q-1); S-1=-Sigma(i)p(i)lnp(i)] linearly increase with time for (and only for) a special value of q, q(sen)(av), and (ii) the slope of <ln(q)xi> and that of <S-q> coincide, thus interestingly extending the well known Pesin theorem. For strong chaos, q(sen)(av)=1, whereas at the edge of chaos q(sen)(av)(z)<1.