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KATSURA–EXEL–PARDO SELF-SIMILAR ACTIONS, PUTNAM’S BINARY FACTORS AND THEIR LIMIT SPACES
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- Journal:
- Journal of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 31 October 2025, pp. 1-36
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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–Exel–Pardo 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 that fibre over a totally disconnected space, and the dynamics restricted to each circle are 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.
