  In this paper we develop unifying graph theoretic techniques to study the
dynamics and the structure of the space of homeomorphisms and the space of
self-maps of the Cantor space. Using our methods, we give characterizations
which determine when two homeomorphisms of the Cantor space are conjugate to
each other. We also give a new characterization of the comeager conjugacy class
of the space of homeomorphisms of the Cantor space. The existence of this class
was established by Kechris and Rosendal and a specific element of this class
was described concretely by Akin, Glasner and Weiss. Our characterization
readily implies many old and new dynamical properties of elements of this
class. For example, we show that no element of this class has a Li-Yorke pair,
implying the well known Glasner-Weiss result that there is a comeager subset of
homeomorphism space of the Cantor space each element of which has topological
entropy zero. Our analogous investigation in the space of continuous self-maps
of the Cantor space yields a surprising result: there is a comeager subset of
the space of self-maps of the Cantor space such that any two elements of this
set are conjugate to each other by an homeomorphism. Our description of this
class also yields many old and new results concerning dynamics of a comeager
subset of the space of continuous self-maps of the Cantor space.
