Sequence entropy and rigid $\sigma$-algebras
Volume 194 / 2009
Studia Mathematica 194 (2009), 207-230
MSC: Primary 37A35, 37A25.
DOI: 10.4064/sm194-3-1
Abstract
We study relationships between sequence entropy and the Kronecker and rigid algebras. Let $(Y,\mathcal Y,\nu, T)$ be a factor of a measure-theoretical dynamical system $(X,\mathcal X,\mu,T)$ and $S$ be a sequence of positive integers with positive upper density. We prove there exists a subsequence $A\subseteq S$ such that $h^A_\mu(T,\xi\,|\,\mathcal Y)= H_\mu(\xi\, |\,\mathcal K(X\,|\,Y))$ for all finite partitions $\xi$, where $\mathcal K(X\,|\,Y)$ is the Kronecker algebra over $\mathcal Y$. A similar result holds for rigid algebras over ${\cal Y}$. As an application, we characterize compact, rigid and mixing extensions via relative sequence entropy.
