Summary: Given \(X, Y, Z\) and \(W\), compact metric spaces, we consider two iterated function systems \(\{\tau_x:Z \to Z,x\in X\}\) and \(\{\tau_y:W \to W,y \in Y\}\), where \(\tau_x\) and \(\tau_y\) are contractions. Let \(\Pi (\cdot,\cdot,\tau)\) be the set of probabilities \(\pi \in \mathcal P (X \times Y \times Z \times W)\) with \((X,Z)\)-marginal being holonomic with respect to \(\tau_x\) and \((Y,W)\)-marginal being holonomic with respect to \(\tau_y\). Given \(\mu \in \mathcal P(X)\) and \(\nu \in \mathcal P(Y)\), let \(\Pi(\mu,\nu,\tau)\) be the set of probabilities in \(\Pi (\cdot,\cdot,\tau)\) having \(X\)-marginal \(\mu\) and \(Y\)-marginal \(\nu\). Let \(H_\alpha(\pi)\) be the relative entropy of \(\pi\) with respect to \(\alpha\) and \(H_\beta(\pi)\) be the relative entropy of \(\pi\) with respect to \(\beta\). Given a cost function \(c\in C(X\times Y \times Z \times W)\), let \(P_{\alpha,\beta} (c)=\sup_{\pi\in \Pi(\cdot,\cdot,\tau)} \int cd \pi + H_\alpha(\pi)+H_\beta(\pi).\) We will prove the duality equation: \[ \underset { P_{\alpha,\beta} (c-\varphi(x)-\psi(y))=0} {\inf} \int \varphi(x)d \mu + \int \psi(y)d \nu = \underset{\pi \in \Pi(\mu,\nu \tau)} {\sup} \int cd \pi+H_\alpha(\pi)+H_\beta(\pi). \] In particular, if \(Z\) and \(W\) are single points and we drop the entropy, the equation above can be rewritten as the Kantorovich duality for the compact spaces \(X,Y\) and a continuous cost function \(-c\).