Entropy and Variational principles for holonomic probabilities of IFS

Associated to a IFS one can consider a continuous map $\hat{\sigma} : [0,1]\times \Sigma \to [0,1]\times \Sigma$,definedby$\hat{\sigma}(x,w)=(\tau_{X_{1}(w)}(x), \sigma(w))$were$\Sigma={0,1, …, d-1}^{$\mathbb{N}$}$,$\sigma: \Sigma \to \Sigma$ is given by$\sigma (w_1,w_2,w_3,\mathrm{.}\mathrm{.}\mathrm{.})=(w_2,w_3,w_4\mathrm{.}\mathrm{.}\mathrm{.})$ and $X_{k} : \Sigma \to {0,1, …, n-1}$istheprojectiononthecoordinate$k$\mathrm{.}A$\rho$-weightedsystem,$\rho \geq 0$,isaweightedsystem$([0,1], \tau_{i}, u_{i})$suchthatthereexistsapositiveboundedfunction$h : [0,1] \to $\mathbb{R}$$andprobability$\nu $on$[0,1]$satisfying$ P_{u}(h)=\rho h, \quad P_{u}^{*}(\nu)=\rho\nu$\mathrm{.}Aprobability$\hat{\nu}$on$[0,1]\times \Sigma$iscalledholonomicfor$\hat{\sigma}$if$ \int g \circ \hat{\sigma} d\hat{\nu}= \int g d\hat{\nu}, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by $\mathcal{H}$. Via disintegration, holonomic probabilities $\widehat{\nu }$ on $[0,1]\times \mathrm{\Sigma }$ are naturally associated to a $\rho$-weighted system. More precisely, there exist a probability $\nu$ on $[0,1]$ and $u_i,i\in \{0,1,2,\mathrm{.}\mathrm{.},d-1\}$ on $[0,1]$, such that is $P_u^{\ast }(\nu )=\nu$. We consider holonomic ergodic probabilities. For a holonomic probability we define entropy. Finally, we analyze the problem: given $\varphi \in {\mathbb{B}}^{+}$, find the solution of the maximization pressure problem $$p(\varphi )=$$