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

Abstract

For $X$, $Y$, $Z$ and $W$ compact metric spaces, consider two uniformly contractive IFS $\{\tau_x: Z\to Z,\, x\in x\}$ and $\{\tau_y:W\to W,\, y\in Y\}$. For a fixed $\alpha \in \mathcal{P}(X)$ 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 $supp(\alpha)=X,\,supp(\beta)=Y$ we denote by $H_\alpha(\pi), \pi \in \Pi(\cdot,\cdot,\tau)$, the entropy of the $(X,Z)-$marginal of $\pi$ relative to $\alpha$ and denote by $H_\beta(\pi)$, 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 $P^{m}(c) = \sup_{\pi\in\Pi(\cdot,\cdot,\tau)} \int c\, d\pi + H_{\alpha}(\pi) +H_{\beta}(\pi)$ and we will show the following duality result: \[\inf_{P^{m}(c -\varphi(x) -\psi(y))=0} \int \varphi(x)\,d\mu +\int \psi(y)\,d\nu = \sup_{\pi\in\Pi(\mu,\nu,\tau)} \int c\, d\pi + H_{\alpha}(\pi) +H_{\beta}(\pi).\] 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 $-c$: \[\inf_{c -\varphi(x) -\psi(y)\leq 0} \int \varphi(x)\,d\mu +\int \psi(y)\,d\nu = \sup_{\pi\in\Pi(\mu,\nu)} \int c\, d\pi .\]