Entropy and Variational principles for holonomic probabilities of IFS

Associated to a IFS one can consider a continuous map $hat{sigma} : [0,1]times Sigma to [0,1]times Sigma$, defined by $hat{sigma}(x,w)=(tau_{X_{1}(w)}(x), sigma(w))$ were $Sigma={0,1, ..., d-1}^{mathbb{N}}$, $sigma: Sigma to Sigma$ is given by$sigma(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : Sigma to {0,1, ..., n-1}$ is the projection on the coordinate $k$. A $rho$-weighted system, $rho geq 0$, is a weighted system $([0,1], tau_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] to mathbb{R}$ and probability $ u $ on $[0,1]$ satisfying $ P_{u}(h)=rho h, quad P_{u}^{*}( u)=rho u$. A probability $hat{ u}$ on $[0,1]times Sigma$ is called holonomic for $hat{sigma}$ if $ int g circ hat{sigma} dhat{ u}= int g dhat{ u}, forall g in C([0,1])$. We denote the set of holonomic probabilities by ${cal H}$. Via disintegration, holonomic probabilities $hat{ u}$ on $[0,1]times Sigma$ are naturally associated to a $rho$-weighted system. More precisely, there exist a probability $ u$ on $[0,1]$ and $u_i, iin{0, 1,2,..,d-1}$ on $[0,1]$, such that is $P_{u}^*( u)= u$. We consider holonomic ergodic probabilities. For a holonomic probability we define entropy. Finally, we analyze the problem: given $phi in mathbb{B}^{+}$, find the solution of the maximization pressure problem $$p(phi)=$$