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

If (X,T) is a measure-preserving system, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \alpha $\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \{2H(\alpha_{0}^{n-1})/f(n)\}_{n=1}^{\infty} $\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \alpha_{0}^{n-1} $\end{document} and nf(log2(n))/log2(n). A similar statement also holds for the limit superior.