  In this paper, we introduce the notion of asymptotic self-similar sets on
general doubling metric spaces by extending the notion of self-similar sets,
and determine their Hausdorff dimensions, which gives an extension of Balogh
and Rohner 's result. This is carried out by introducing the notions of almost
similarity maps and asymptotic similarity systems. These notions have an
advantage of making geometric constructions possible. Actually, as an
application, we determined the Hausdorff dimension of general Sierpinski
gaskets on complete surfaces constructed by a geometric way in a natural
manner.
