  Using tools from computable analysis we develop a notion of effectiveness for
general dynamical systems as those group actions on arbitrary spaces that
contain a computable representative in their topological conjugacy class. Most
natural systems one can think of are effective in this sense, including some
group rotations, affine actions on the torus and finitely presented algebraic
actions. We show that for finitely generated and recursively presented groups,
every effective dynamical system is the topological factor of a computable
action on an effectively closed subset of the Cantor space. We then apply this
result to extend the simulation results available in the literature beyond
zero-dimensional spaces. In particular, we show that for a large class of
groups, many of these natural actions are topological factors of subshifts of
finite type.
