Decidability on Dube’s Self-similar Fractals

Abstract

Dube proved some undecidability on self-affine fractals. In this paper, we obtain the decidability for self-similar fractal of Dube’s type. In fact, we prove that the following problems are decidable to test if the Hausdorff dimension of a given Dube’s self-similar set is equal to its similarity dimension, and to test if a given Dube’s self-similar set satisfies the strong separation condition.