  We study the dynamics of the attractor of the doubling map with an
asymmetrical hole by associating to each hole an element of the lexicographic
world. A description of the topological entropy function is given. We show that
the set of parameters $(a,b)$ such that the dynamics of the mentioned attractor
corresponds to a subshift of finite type is open and dense. Using the
connections between this family of open dynamical systems, intermediate
$\beta$-expansions and Lorenz maps we study the topological transitivity and
the specification property for such maps.
