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 $P^m(c)={\sup }_{\pi \in \Pi (\cdot ,\cdot ,\tau )}\int cd\pi +H_{\alpha }(\pi )+H_{\beta }(\pi )$ and we will show the following duality result: \[\inf_{P^{m}(c -φ(x) -ψ(y))=0} \int φ(x)\,dμ+\int ψ(y)\,dν= \sup_{π\inΠ(μ,ν,τ)} \int c\, dπ+ H_α(π) +H_β(π).\] 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 $-c$: \[\inf_{c -φ(x) -ψ(y)\leq 0} \int φ(x)\,dμ+\int ψ(y)\,dν= \sup_{π\inΠ(μ,ν)} \int c\, dπ.\]

20 pages