Mobius number systems based on interval covers

Given a finite alphabet A, a system of real orientation-preserving Mobius transformations (F-a : (R) over bar --> (R) over bar)(a is an element of A), a subshift Sigma subset of A(N) and an interval cover W = {W-a : a is an element of A} of (R) over bar, we consider the expansion subshift Sigma(W) subset of Sigma of all expansions of real numbers with respect to W. If the expansion quotient Q(Sigma, W) is greater than 1 then there exists a continuous and surjective symbolic mapping Phi : Sigma(W) --> (R) over bar and we say that (F, Sigma(W)) is a Mobius number system. We apply our theory to the system of binary continued fractions which is a combination of the binary signed system with the continued fractions, and to the binary square system whose transformations have stable fixed points -1, 0, 1 and infinity.