DOI: 10.1017/s1446788726101657 ISSN: 1446-7887
RELATING INSPLITTINGS OF 2-GRAPHS AND OF TEXTILE SYSTEMS
SAMANTHA BROOKER, PRIYANGA GANESAN, ELIZABETH GILLASPY, YING-FEN LIN, DAVID PASK, JULIA PLAVNIK Abstract
The graphical operation of insplitting is key to understanding conjugacy of shifts of finite type (SFTs) in both one and two dimensions. In this paper, we consider two approaches to studying two-dimensional (2D) SFTs: textile systems and rank-2 graphs. Nasu’s textile systems describe all two-sided 2D SFTs up to conjugacy [Johnson and Madden, ‘The decomposition theorem for two-dimensional shifts of finite type’,
Proc. Amer. Math. Soc.
127
(5) (1999), 1533–1543; Aso, ‘Conjugacy of
Z
2
$Z^2$
upper Z squared
-subshifts and textile systems’,
Publ. RIMS. Kyoto Univ
36
(2000), 1–18], whereas the 2-graphs (higher-rank graphs of rank 2) introduced by Kumjian and Pask yield associated
C
∗
$C^*$
upper C Superscript asterisk
-algebras [Tang, ‘Tiling systems and 2-graphs associated to textile systems’, PhD Thesis, University of Wollongong, 2013, http://ro.uow.edu.au/theses/4003]. Both models have a naturally associated notion of insplitting (introduced for textile systems by Johnson and Madden [‘The decomposition theorem for two-dimensional shifts of finite type’,
Proc. Amer. Math. Soc.
127
(5) (1999), 1533–1543] and for 2-graphs by Eckhardt
et al.
[‘Moves on
k
-graphs preserving Morita equivalence’,
Canad. J. Math.
74
(2022), 655–685]). We show that these notions do not coincide, raising the question of whether insplitting a 2-graph induces a conjugacy of the associated one-sided 2D SFTs. Our first main result shows how to reconstruct 2-graph insplitting using textile-system insplits and inversions, and consequently proves that 2-graph insplitting induces a conjugacy of dynamical systems. We also present several other facets of the relationship between 2-graph insplitting and textile-system insplitting. Incorporating an insplit of the ‘bottom’ graph of the textile system turns out to be key to this relationship. By articulating the connection between operator-algebraic and dynamical notions of insplitting in two dimensions, this article lays the groundwork for a
C
∗
$C^*$
upper C Superscript asterisk
-algebraic framework for classifying one-sided conjugacy in higher-dimensional SFTs.