A DICHOTOMY OF STRONG OPEN SET CONDITION OF SELF-CONFORMAL SETS.

Let K be a 1 + α self-conformal set. It is well known that K satisfies the strong open set condition provided it satisfies the open set condition. A point x K is called a trivial point of K if { x } is a connected component of K. Let U be a strong open set for K. In this paper, we show that whether U contains a trivial point of K determines several metrical and topological properties of K , including the maximal power law property and the dimension drop of the connected part of K. [ABSTRACT FROM AUTHOR]