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 $\{τ_x: Z\to Z,\, x\in x\}$ and $\{τ_y:W\to W,\, y\in Y\}$. For a fixed $α\in \mathcal{P}(X)$ with $supp(α)=X$ we define the entropy of a holonomic measure $π\in \mathcal{P}(X\times Z)$ relative to $α$, 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 $α\in \mathcal{P}(X)$ and $β\in \mathcal{P}(Y)$ with $supp(α)=X,\,supp(β)=Y$ we denote by $H_α(π), π\in Π(\cdot,\cdot,τ)$, the entropy of the $(X,Z)-$marginal of $π$ relative to $α$ and denote by $H_β(π)$, the entropy of the $(Y,W)-$marginal of $π$ relative to $β$. The marginal pressure of a continuous cost function $c \in C(X\times Y \times Z \times W)$ relative to $(α,β)$ will be defined by $P^{m}(c) = \sup_{π\inΠ(\cdot,\cdot,τ)} \int c\, dπ+ H_α(π) +H_β(π)$ 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 $-c$: \[\inf_{c -φ(x) -ψ(y)\leq 0} \int φ(x)\,dμ+\int ψ(y)\,dν= \sup_{π\inΠ(μ,ν)} \int c\, dπ.\]