Sequence entropy and rigid $\sigma$-algebras

Alvaro Coronel ¹   ; Alejandro Maass ¹   ; Song Shao ²

¹ Departamento de Ingeniería Matemática Universidad de Chile Av. Blanco Encalada 2120 Santiago, Chile
² Department of Mathematics University of Science and Technology of China Hefei, Anhui, 230026, P.R. China

Studia Mathematica, Tome 194 (2009) no. 3, pp. 207-230

Résumé

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.

Détail

DOI : 10.4064/sm194-3-1

Keywords: study relationships between sequence entropy kronecker rigid algebras mathcal factor measure theoretical dynamical system mathcal sequence positive integers positive upper density prove there exists subsequence subseteq mathcal mathcal finite partitions where mathcal kronecker algebra mathcal similar result holds rigid algebras cal application characterize compact rigid mixing extensions via relative sequence entropy