  We construct two new classes of topological dynamical systems; one is a
factor of a one-sided shift of finite type while the second is a factor of the
two-sided shift. The data is a finite graph which presents the shift of finite
type, a second finite directed graph and a pair of embeddings of it into the
first, satisfying certain conditions. The factor is then obtained from a simple
idea based on binary expansion of real numbers. In both cases, we construct
natural metrics on the factors and, in the second case, this makes the system a
Smale space, in the sense of Ruelle. We compute various algebraic invariants
for these systems, including the homology for Smale space developed by the
author and the K-theory of various $C^{*}$-algebras associated to them, in
terms of the pair of original graphs.
