  In the first part of the paper, we introduce notions of asymptotic continuous
orbit equivalence and asymptotic conjugacy in Smale spaces and characterize
them in terms of their asymptotic Ruelle algebras with their dual actions. In
the second part, we introduce a groupoid $C^*$-algebra which is an extended
version of the asymptotic Ruelle algebra from a Smale space and study the
extended Ruelle algebras from the view points of Cuntz--Krieger algebras. As a
result, the asymptotic Ruelle algebra is realized as a fixed point algebra of
the extended Ruelle algebra under certain circle action.
