On the relation between entropy and the average complexity of trajectories in dynamical systems

If (X,T) is a measure-preserving system, ct a nontrivial partition of X into two sets and f a positive increasing function defined on the positive real numbers, then the limit inferior of the sequence (2H(alpha (n-1)(0))/f(n))(n=1)(infinity) is greater than or equal to the limit inferior of the sequence of quotients of the average complexity of trajectories of length n generated by alpha (n-1)(0) and nf(log(2)(n))/(log(2)(n). A similar statement. also holds for the limit superior.