Admissible digit sets

We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “digit set” yields an admissible representation of [0,+∞]. Furthermore, we establish the productivity and correctness of the homographic algorithm for such “admissible” digit sets. We present the Stern–Brocot representation and a modification of same as a working example throughout.