Groupoid Actions on Fractafolds?

Abstract. We define a bundle over a totally disconnected set such that each fiber is homeo-morphic to a fractal blowup. We prove that there is a natural action of a Renault–Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault– Deaconu groupoid itself. We also show that when the bundle is locally compact the associ-ated C∗-algebra is primitive and has a densely defined lower-semicontinuous trace. Key words: Renault–Deaconu groupoids; fractafolds; iterated function system