We extend Nekrashevych's KK-duality for C∗-algebras of regular,
recurrent, contracting self-similar group actions to regular, contracting
self-similar groupoid actions on a graph, removing the recurrence condition
entirely and generalising from a finite alphabet to a finite graph.
More precisely, given a regular and contracting self-similar groupoid (G,E)
acting faithfully on a finite directed graph E, we associate two
C∗-algebras, O(G,E) and O(G,E), to it and
prove that they are strongly Morita equivalent to the stable and unstable
Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the
self-similar limit space. That these algebras are Spanier-Whitehead dual in
KK-theory follows from the general result for Ruelle algebras of irreducible
Smale spaces proved by Kaminker, Putnam, and the last author.Comment: File updated to make several corrections. Example 8.15 adde