On the geometric structure of the limit set of conformal iterated function systems

Abstract: We consider infinite conformal iterated function systems on $\mathbb{R}<sup>d$. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some $l$-dimensional $C<sup>1$-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 $2$.