Stable Properties Of Self-similar Sets And Invariant Measures

In this paper, we firstly discuss the topological properties of the shift space, which is an important tool to study fractal set, and itself is also a self-similar set. Then we prove that the fixed point of a contractive mapping is continuous with respect to a parameter. Moreover, a property of the sequence of self-similar IFS (iterated function system)isproved, that is , which is a crude proximity of the self-similar set. Inaddition , if we equip the fractal space with the Hausdorff metric, we will show that the sequence of the self-similar set is convergent in some sense with respect to the Hausdorff metric, moreover we find a method to construct a self-similar set, that isAt last, we discuss that the invariant measure, whose supportis the corresponding self-similar set, is depending continuously on the probability vectors and contractive factors.