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

Abstract.

If (X,T) is a measure-preserving system, $\alpha$ 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 $\{2H(\alpha_{0}^{n-1})/f(n)\}_{n=1}^{\infty}$ 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 $\alpha_{0}^{n-1}$ and nf(log₂(n))/log₂(n). A similar statement also holds for the limit superior.