Hausdorff dimension of the set of extreme points of a self-similar set

Abstract

If the system S of contracting similitudes on $ R^2$ satisfies open convex set condition, then the set F of extreme points of the convex hull $\tilde{K}$ of it's invariant self-similar set K has Hausdorff dimension 0 . If, additionally, all the rotation angles of the similitudes are commensurable with $\pi$, then the set F is finite and the convex hull $\tilde{K}$ is a convex polygon.