  We characterize positive topological entropy for quasi-state space
homeomorphisms induced from $C^*$-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.
