On the long-run average growth rate of chaotic systems

For a discrete dynamical process defined by Xt=F(X(t-1),X(t-2),...,X(t-k)), where Xt subset R+n, the growth rate for the ith endogenous variable is conventionally defined as g(it) identical with (x(it+1)-x(it))/x(it). It is shown that, when the trajectories of x(it) exhibit periodic or aperiodic (chaotic) fluctuations within a bounded range, the long-run average growth rates are always positive, that is, g(it) identical with lim(T--> infinity )1/TSigmaTt=1 g(it) > 0, for all i, so as to induce an illusion of growth. The fact is generic in the sense of being irrespective of the dynamic structure of F. Moreover, the same characteristic exists even if the process is reversed. Economic implications and applications are discussed. It is concluded that the statistical inequalities provide additional measures to check the validity of different discrete models or to test different specifications from empirical time series.