Duality results for Iterated Function Systems with a general family of branches

For $X$, $Y$, $Z$ and $W$ compact metric spaces, consider two uniformly contractive IFS $\{{\tau }_x:Z\to Z,x\in x\}$ and ${\tau_y:W\to W,, y\in Y}$\mathrm{.}Forafixed$\alpha \in $\mathcal{P}$(X)$with$supp(\alpha)=X$ we define the entropy of a holonomic measure $\pi \in \mathcal{P}(X\times Z)$ relative to $\alpha$, the pressure of a continuous cost function $c(x,z)$ and show that for $c$ Lipschitz this pressure coincides with the spectral radius of the associated transfer operator. The same approach can be applied to the pair $Y,W$. For fixed probabilities $\alpha \in \mathcal{P}(X)$ and $\beta \in $\mathcal{P}$(Y)$with$supp(\alpha)=X,,supp(\beta)=Y$wedenoteby$H_\alpha(\pi), \pi \in \Pi(\cdot,\cdot,\tau)$,theentropyofthe$(X,Z)-$marginalof$\pi$relativeto$\alpha$anddenoteby$H_\beta(\pi)$, the entropy of the $(Y,W)-$marginal of $\pi$ relative to $\beta$. The marginal pressure of a continuous cost function $c\in C(X\times Y\times Z\times W)$ relative to $(\alpha ,\beta )$ will be defined by $P^{m}© = \sup_{\pi\in\Pi(\cdot,\cdot,\tau)} \int c, d\pi + H_{\alpha}(\pi) +H_{\beta}(\pi)$ and we will show the following duality result: $$\underset{P^m(c-\phi (x)-\psi (y))=0}{\inf }\int \phi (x)d\mu +\int \psi (y)d\nu =\underset{\pi \in \mathrm{\Pi }(\mu ,\nu ,\tau )}{\sup }\int cd\pi +H_{\alpha }(\pi )+H_{\beta }(\pi )\mathrm{.}$$ When $Z$ and $W$ have only one point and the entropy is unconsidered this equality can be rewritten as the Kantorovich Duality for compact spaces $X,Y$ and continuous cost $-c$: $$\underset{c-\phi (x)-\psi (y)\le 0}{\inf }\int \phi (x)d\mu +\int \psi (y)d\nu =\underset{\pi \in \mathrm{\Pi }(\mu ,\nu )}{\sup }\int cd\pi \mathrm{.}$$