  This work describes the way that topological mixing and chaos in continua, as
induced by discrete dynamical systems, can or can't be understood through
topological conjugacy with symbolic dynamical systems. For example, there is no
symbolic dynamical system that is topologically conjugate to any discrete
dynamical system on an entire continuum, and there is no finer topology that
can be given to a continuum which simultaneously makes the continuum
homeomorphic to a symbolic dynamical system and contains its original topology.
However, this paper demonstrates an analytical mechanism by which the existence
of topological mixing and/or chaos can be shown through conjugacy with
qualitative dynamical systems outside the usual purview of symbolic dynamics.
Two examples of this mechanism are demonstrated on classic textbook models of
chaotic dynamics; the first proving the existence of topological mixing
everywhere in the dyadic map on the interval by showing that there exists a
qualitative system that is topologically conjugate to the dyadic map on the
interval with a finer topology than the usual Euclidean topology, and the other
following a similar approach to demonstrate the existence of Devaney chaos
everywhere in the $2$-tent map on the interval. The content is presented in a
somewhat self-contained fashion, reiterating some standard results in the
field, to aid new learners of topological mixing/chaos.
