Riesz energy on self-similar sets

On the classification of self-similar sets

We classify binary self-similar sets, which are compact sets determined by two contractions on the plane, into four classes from the viewpoint of functional equations. In this classification, we can not only show close relationships among functions with self-similarity but also give solutions to a few open problems in other field.

On univoque points for self-similar sets

Let K ⊆ R be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence of this, we can show that the set of points with a unique coding is a graph-directed selfsimi...

Statistically Self - Similar Fractal Sets

In the present paper we define statistically self-similar sets, and, using a modification of method described in [2], [3], find a Hausdorff dimension of a statistically self-similar set.

Computability of Self-Similar Sets

We investigate computability of a self-similar set on a Euclidean space. A nonempty compact subset of a Euclidean space is called a self-similar set if it equals to the union of the images of itself by some set of contractions. The main result in this paper is that if all of the contractions are computable, then the self-similar is a recursive compact set. A further result on the case that the ...

Problems on Self-similar Sets and Self-affine Sets: an Update