On the Geometric Structure of the Limit Set of Conformal Iterated Function Systems

We consider infinite conformal function systems on Rd. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some l-dimensional C1-submanifold with positive Hausdorff t-dimensional measure, where 0 < l < d and t is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs to some l-dimensional affine subspace or geometric sphere whenever d exceeds 2 and analytic curve if d equals 2.