C ∗ -Algebras from Groupoids on Self-Similar Groups

Abstract

We show that the Smale spaces from self-similar groups are topologically mixing and their stable algebra and stable Ruelle algebra are strongly Morita equivalent to groupoid algebras of Anantharaman-Delaroche and Deaconu. And we show that $C^{\ast }(R_{\infty })$ associated to a postcritically finite hyperbolic rational function is an AT-algebra of real-rank zero with a unique trace state.