Moebius number systems represent points using sequences of Moebius
transformations. Thorough the paper, we are mainly interested in representing
the unit circle (which is equivalent to representing R\cup\{\infty\}).

The main aim of the paper is to improve already known tools for proving that
a given subshift--iterative system pair is in fact a Moebius number system. We
also study the existence problem: How to describe iterative systems resp.
subshifts for which there exists a subshift resp. iterative system such that
the resulting pair forms a Moebius number system. While we were unable to
provide a complete answer to this question, we present both positive and
negative partial results.

As Moebius number systems are also subshifts, we can ask when a given Moebius
number system is sofic. We give this problem a short treatment at the end of
our paper.