  We survey the connections between entropy, chaos, and independence in
topological dynamics. We present extensions of two classical results placing
the following notions in the context of symbolic dynamics:
  1. Equivalence of positive entropy and the existence of a large (in terms of
asymptotic and Shnirelman densities) set of combinatorial independence for
shift spaces.
  2. Existence of a mixing shift space with a dense set of periodic points with
topological entropy zero and without ergodic measure with full support, nor any
distributionally chaotic pair.
  Our proofs are new and yield conclusions stronger than what was known before.
