49 citations · 210 across the 23 of their papers we have counts for
27 papers
Katsura-Exel-Pardo self-similar actions, Putnam's binary factors and their limit spaces
Jeremy B. Hume, Michael F. Whittaker
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.…
Planar aperiodic tile sets: from Wang tiles to the Hat and Spectre monotiles
Tinka Bruneau, Michael F. Whittaker
A brief history of planar aperiodic tile sets is presented, starting from the Domino Problem proposed by Hao Wang in 1961. We provide highlights that led to the discovery of the Ta…
KK-duality for self-similar groupoid actions on graphs
Nathan Brownlowe, Alcides Buss, Daniel Gonçalves +3
We extend Nekrashevych's -duality for -algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a gr…
A geometric representative for the fundamental class in KK-duality of Smale spaces
D. M. Gerontogiannis, Michael F. Whittaker, Joachim Zacharias
A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality.…
Self-similarity and limit spaces of substitution tiling semigroups
James J. Walton, Michael F. Whittaker
We show that Kellendonk's tiling semigroup of an FLC substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend the notion of the limit…
The Cuntz-Toeplitz algebras have nuclear dimension one
Philip Easo, Esperanza Garijo, Sarunas Kaubrys +12
We prove that unital extensions of Kirchberg algebras by separable stable AF algebras have nuclear dimension one. The title follows.
C*-algebras of self-similar actions of groupoids on higher-rank graphs and their equilibrium states
Zahra Afsar, Nathan Brownlowe, Jacqui Ramagge +1
We introduce the notion of a self-similar action of a groupoid on a finite higher-rank graph. To these actions we associate a compactly aligned product system of Hilbert bimodu…
Classification of tiling -algebras
Luke J. Ito, Michael F. Whittaker, Joachim Zacharias
We prove that Kellendonk's -algebra of an aperiodic and repetitive tiling with finite local complexity is classifiable by the Elliott invariant. Our result follows from showin…
An aperiodic tile with edge-to-edge orientational matching rules
James J. Walton, Michael F. Whittaker
We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles…
An aperiodic monotile that forces nonperiodicity through dendrites
Michael Mampusti, Michael F. Whittaker
We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding…
Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs
Marcelo Laca, Iain Raeburn, Jacqui Ramagge +1
We consider self-similar actions of groupoids on the path spaces of finite directed graphs, and construct examples of such self-similar actions using a suitable notion of graph aut…
Wieler solenoids, Cuntz-Pimsner algebras and K-theory
Robin J. Deeley, Magnus Goffeng, Bram Mesland +1
We study irreducible Smale spaces with totally disconnected stable sets and their associated -theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict…
Zappa-Szép product groupoids and C*-blends
Nathan Brownlowe, David Pask, Jacqui Ramagge +2
We study the external and internal Zappa-Szép product of topological groupoids. We show that under natural continuity assumptions the Zappa-Szép product groupoid is étale if and on…
Dynamical correspondences for Smale spaces
Robin J. Deeley, D. Brady Killough, Michael F. Whittaker
We initiate the study of correspondences for Smale spaces. Correspondences are shown to provide a notion of a generalized morphism between Smale spaces and are a special case of fi…
Fractal spectral triples on Kellendonk's -algebra of a substitution tiling
Michael Mampusti, Michael F. Whittaker
We introduce a new class of noncommutative spectral triples on Kellendonk's -algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed…
Fractal dual substitution tilings
Natalie Priebe Frank, Samuel B. G. Webster, Michael F. Whittaker
Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated f…
Graph algebras and orbit equivalence
Nathan Brownlowe, Toke Meier Carlsen, Michael F. Whittaker
We introduce the notion of orbit equivalence of directed graphs, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs…
Functorial properties of Putnam's homology theory for Smale spaces
Robin J. Deeley, D. Brady Killough, Michael F. Whittaker
We investigate functorial properties of Putnam's homology theory for Smale spaces. Our analysis shows that the addition of a conjugacy condition is necessary to ensure functorialit…
Twisted C*-algebras associated to finitely aligned higher-rank graphs
Aidan Sims, Benjamin Whitehead, Michael F. Whittaker
We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties…
Topological spaces associated to higher-rank graphs
Alex Kumjian, David Pask, Aidan Sims +1
We investigate which topological spaces can be constructed as topological realisations of higher-rank graphs. We describe equivalence relations on higher-rank graphs for which the…
Zappa-Szép products of semigroups and their C*-algebras
Nathan Brownlowe, Jacqui Ramagge, David Robertson +1
Zappa-Szép products of semigroups encompass both the self-similar group actions of Nekrashevych and the quasi-lattice-ordered groups of Nica. We use Li's construction of semigroup…
Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions
Marcelo Laca, Iain Raeburn, Jacqui Ramagge +1
We consider a family of Cuntz-Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorph…
Cohomology of the Pinwheel Tiling
Dirk Frettlöh, Benjamin Whitehead, Michael F. Whittaker
We provide a computation of the cohomology of the Pinwheel tiling using the Anderson-Putnam complex. A border forcing version of the Pinwheel tiling is constructed that allows an e…
Spectral triples for hyperbolic dynamical systems
Michael F. Whittaker
Spectral triples are defined for C*-algebras associated with hyperbolic dynamical systems known as Smale spaces. The spectral dimension of one of these spectral triples is shown to…