Entropy and Variational principles for holonomic probabilities of IFS

Abstract

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, $\rho \ge 0$, 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 $\nu$ on $[0,1]$ satisfying $P_u(h)=\rho h,P_u^{\ast }(\nu )=\rho \nu$. A probability $\hat{\nu }$ on $[0,1]\times \Sigma$ is called holonomic for $\hat{\sigma }$ if $\int g\circ \hat{\sigma }d\hat{\nu }=\int gd\hat{\nu },\forall g\in C([0,1])$. We denote the set of holonomic probabilities by $\mathcal{H}$. Via disintegration, holonomic probabilities $\hat{\nu }$ on $[0,1]\times \Sigma$ are naturally associated to a $\rho$-weighted system. More precisely, there exist a probability $\nu$ on $[0,1]$ and $u_i,i\in \{0,1,2,..,d-1\}$ 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 $$p(\varphi )=$$