On the relation between entropy convergence rates and Baire category
Let \(G\) denote the group of all invertible Lebesgue measure-preserving transformations on the unit interval \([0,1]\). For \(T\in G\) and \(\alpha\) a finite measurable partition of \([0,1]\), let \(\alpha_T^n\) denote the \(n\)th refinement of \(\alpha\) under \(T\). In this paper the author shows that if \(\{a_n\}\) is a sequence of positive integers whose limit superior is \(\infty\) then \[ \{T\in G: \liminf_{n\longrightarrow \infty}\frac{H(\alpha_T^n)}{a_n}>0 \text{ for all finite nontrivial partitions }\alpha\} \] is of first category in \(G\), and if \(\{a_n\}\) is a sequence of positive integers such that \(\displaystyle\lim_{n\longrightarrow \infty}\frac{a_n}{n}=0\) then \[ \{T\in G: \limsup_{n\longrightarrow \infty}\frac{H(\alpha_T^n)}{a_n}>0 \text{ for all finite nontrivial partitions }\alpha\} \] is residual in \(G\).
scientific article; zbMATH DE number 6065378