Positive topological entropy and ${\ell }_1$

Abstract

We characterize positive topological entropy for quasi-state space homeomorphisms induced from $C^{\ast }$-algebra automorphisms in terms of dynamically generated subspaces isomorphic to ${\ell }_1$. This geometric condition is also used to give a description of the topological Pinsker algebra. In particular we obtain a geometric characterization of positive entropy for topological dynamical systems, as well as an analogue for completely positive topological entropy of Glasner and Weiss's combinatorial characterization of completely positive Kolmogorov-Sinai entropy.

9 pages