On the relation between entropy and the average complexity of trajectoriesin dynamical systems

If (X,T) is a measure-preserving system, ct a nontrivial partition of X int o two sets and f a positive increasing function defined on the positive rea l 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 sequ ence of quotients of the average complexity of trajectories of length n gen erated by alpha (n-1)(0) and nf(log(2)(n))/(log(2)(n). A similar statement. also holds for the limit superior.