The entropy of a special overlapping dynamical system

The term special overlapping refers to a certain simple type of piecewise continuous function from the unit interval to itself and also to a simple type of iterated function system (IFS) on the unit interval. A correspondence between these two classes of objects is used (1) to find a necessary and sufficient condition for a fractal transformation from the attractor of one special overlapping IFS to the attractor of another special overlapping IFS to be a homeomorphism and (2) to find a formula for the topological entropy of the dynamical system associated with a special overlapping function.