Noncommutative Geometry and Topology in Prague

Self-similarity of substitution tiling semigroups

Substitution tilings arise from graph iterated function systems. Adding a contraction constant, the attractor recovers the prototiles. On the other hand, without the contraction one obtains an infinite tiling. In this talk I'll introduce substitution tilings and an associated semigroup defined by Kellendonk. I'll show that this semigroup defines a self-similar action on a topological Markov shift that's conjugate to the punctured tiling space. The limit space of the self-similar action turns out to be the Anderson-Putnam complex of the substitution tiling and the inverse limit recovers the translational hull. This was joint work with Jamie Walton.