Summary: Let \((B, T)\) be a fibred map. A standard method for the determination of the density of an invariant measure is provided by the theory of dual maps, a generalization of backward continued fractions. A dual map \((B^{\#}, T^{\#})\) is called a natural dual if there is a differentiable map \(M\) with the property \(M \circ T=T^{\#} \circ M\). In this paper we present the surprising result of a family of fibred maps \((B, T)\) such that the set \(B^{\#}\) of every natural dual is a one-point set.