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, $\rho \geq 0$, 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 $\nu $ on $[0,1]$ satisfying $ P_{u}(h)=\rho h, \quad P_{u}^{*}(\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 g d\hat{\nu}, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by ${\cal H}$. 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 $P_{u}^*(\nu)=\nu$. 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 $$p(\phi)=$$