Random Pisot substitutions and their Rauzy fractals
Abstract: The Pisot conjecture is a long-standing open question concerning the dynamical spectrum of subshifts associated with Pisot substitutions. One formulation of the conjecture can be stated in terms of the so-called Rauzy fractal for the substitution, which is the attractor of a graph-directed iterated function system that can be written purely in terms of data coming from the substitution. Random substitutions generalise substitutions and can be used to interpolate between them, allowing properties of one substitution to inform properties of another. In this talk, we will introduce Rauzy fractals for Pisot random substitutions and results concerning natural measures that can be defined on the attractor. By studying these measures, we are able to prove an analogue of a multi-tiling result for Rauzy fractals but in the random setting, hinting at potential new tools for tackling the Pisot conjecture using probabilistic methods. This is joint work with Philipp Gohlke, Andrew Mitchell and Tony Samuel.