Random iterated function systems of one-dimensional maps on a compact interval \(X\subset \mathbb {R}\) often fail to be globally contractive in any fixed metric, even when sample Lyapunov exponents are negative. A classical workaround is to twist the geometry by a positive weight w and require a weighted-derivative condition of the form \( \sum _{k=1}^m p_k \left| f_k'(x)\,\frac{w(x)}{w(f_k(x))}\right| \le r<1, \qquad \forall x\in X, \) which ensures that the associated Markov operator is a strict contraction in a weighted Wasserstein–Kantorovich metric \(d_W^{(w)}\) . In this paper, we replace ad hoc constructions of w by a spectral theory for the derivative transfer operator \( (\mathscr {L}\varphi )(x) = \sum _{k=1}^m p_k\,|f_k'(x)|\,\varphi (f_k(x)), \qquad \varphi \in C(X). \) Our main result proves that the optimal contraction constant, minimised over all continuous weights \(w>0\) , coincides with the spectral radius \(\rho (\mathscr {L})\) . Equivalently, \(\rho (\mathscr {L})<1\) if and only if there exists a strictly positive continuous supersolution \(\phi =1/w\) of \(\mathscr {L}\phi \le r\phi \) for some \(r<1\) , and in this case the corresponding \(d_W^{(w)}\) yields exponential convergence of every trajectory to a unique invariant probability measure. Conversely, if \(\rho (\mathscr {L})\ge 1\) then no choice of w can produce global contractivity. We illustrate the framework on random logistic and Ricker families, providing parameter-dependent bounds on \(\rho (\mathscr {L})\) and showing how loss of weighted contractivity organises the transition between ergodic and null-recurrent regimes. The spectral construction also prepares the ground for controlled variants, where one seeks uniform spectral bounds for the corresponding derivative transfer operators over admissible control laws.
