We extend Nekrashevych’s
K
K
KK
-duality for
C
∗
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
)
(G,E)
acting faithfully on a finite directed graph
E
E
, we associate two
C
∗
C^*
-algebras,
O
(
G
,
E
)
\mathcal {O}(G,E)
and
O
^
(
G
,
E
)
\widehat {\mathcal {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
K
K
KK
-theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.