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We consider infinite conformal iterated function systems in the phase space R d with d > 3. Let J be the limit set of such a system. Under a mild technical assumption, which is always satisfied if the system is finite, we prove that either the Hausdorff dimension of J exceeds the topological dimension k of the closure of J or else the closure of J is a proper compact subset of either a geometric sphere or an affine subspace of dimension k. A similar dichotomy holds for conformal expanding repellers.

Finer geometric rigidity of limit sets of conformal IFS