  We show that the dynamical system associated by Putnam to a pair of graph
embeddings is identical to the shift map on the limit space of a self-similar
groupoid action on a graph. Moreover, performing a certain out-split on said
graph gives rise to a Katsura groupoid action on the out-split graph whose
associated limit space dynamical system is conjugate to the previous one. We
characterise the self-similar properties of these groupoids in terms of
properties of their defining data, two matrices $A$, $B$. We prove a large
class of the associated limit spaces are bundles of circles and points which
fibre over a totally disconnected space, and the dynamics restricted to each
circle is of the form $z\to z^{n}$. Moreover, we find a planar embedding of
these spaces, thereby answering a question Putnam posed in his paper.
