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

## TL;DR

This paper investigates the geometric structure of limit sets of infinite conformal iterated function systems in Euclidean spaces, revealing conditions under which these sets are contained within affine subspaces or spheres, depending on the dimension.

## Contribution

It characterizes the geometric structure of limit sets intersecting certain submanifolds, showing they lie within affine subspaces or spheres, extending understanding of conformal IFS in higher dimensions.

## Key findings

- Limit sets intersecting submanifolds are contained in affine subspaces or spheres for d > 2.
- In two dimensions, the closure of the limit set is contained in an analytic curve.
- The structure depends on the ambient space dimension and intersection properties.

## Abstract

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