Sequence entropy and rigid $\sigma$-algebras

Tom 194 / 2009

Alvaro Coronel, Alejandro Maass, Song Shao Studia Mathematica 194 (2009), 207-230 MSC: Primary 37A35, 37A25. DOI: 10.4064/sm194-3-1

Streszczenie

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.

Autorzy

  • Alvaro CoronelDepartamento de Ingeniería Matemática
    Universidad de Chile
    Av. Blanco Encalada 2120
    Santiago, Chile
    e-mail
  • Alejandro MaassDepartamento de Ingeniería Matemática
    Universidad de Chile
    Av. Blanco Encalada 2120
    Santiago, Chile
    e-mail
  • Song ShaoDepartment of Mathematics
    University of Science and Technology of China
    Hefei, Anhui, 230026, P.R. China
    e-mail

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