Minimal rates of entropy convergence for rank one systems. (Q1864062)

In [\textit{F. Blume}, Ergodic Theory Dyn. Syst. 17, No.1, 45--70 (1997; Zbl 0891.28012)] the author gives a universal lower bound for entropy convergence rates of measure-preserving systems in the following sense: Let \((X,\mathcal{B},\mu,T)\) be an aperiodic measure-preserving system and \(g:[0,\infty)\longrightarrow\mathbb{R}_+\) be a monotone increasing function with \(\frac{g(x)}{x^2}\in L^1\). Then \[ \limsup_{n\rightarrow\infty}\frac{H(\alpha_0^{n-1})}{g(\log_2 n)}=\infty, \] for any partition \(\alpha\) of \(X\) into two sets such that \(\lim_{n\rightarrow\infty}\,\max\{\mu( A)| \; A\in\alpha_0^{n-1}\}=0\). In [\textit{F. Blume}, ``The rate of entropy convergence'', Doctoral Dissertation, University of North Carolina at Chapel Hill (1995)] it is shown that the above cannot be improved in general and the following result is given: Let \((X,\mathcal{B},\mu,T)\) be a rank one mixing measure-preserving system. Then \[ \limsup_{n\rightarrow\infty}\frac{H(\alpha_0^{n-1})}{\log_2n}>0, \] for any non-trivial partition \(\alpha\) of \(X\) into two sets. In the present paper the author tries to get rid of the assumption of mixing. He partially succeeds but for an arbitrary rank one system obtains the weaker convergence rates than \(\log_2n\). Namely, the main result of the paper is the following. Let \((X,\mathcal{B},\mu,T)\) be a rank one measure-preserving system and \(g:[0,\infty)\longrightarrow\mathbb{R}_+\) be a concave function with \(\frac{g(x)^2}{x^3}\in L^1\). Then \[ \limsup_{n\rightarrow\infty}\frac{H(\alpha_0^{n-1})}{g(\log_2 n)}=\infty, \] for any non-trivial partition of \(X\) into two sets such that \(\lim_{n\rightarrow\infty}\,\max\{\mu( A)| \; A\in\alpha_0^{n-1}\}=0\).