July  2009, 23(3): 937-955. doi: 10.3934/dcds.2009.23.937

Entropy and variational principles for holonomic probabilities of IFS


Instituto de Matemática, UFRGS, 91509-900, Porto Alegre, Brazil, Brazil

Received  September 2007 Revised  July 2008 Published  November 2008

An IFS ( iterated function system), $([0,1], \tau_{i})$, on the interval $[0,1]$, is a family of continuous functions $\tau_{0},\tau_{1}, ..., \tau_{d-1} : [0,1] \to [0,1]$. Associated to a IFS one can consider a continuous map $\hat{\sigma} : [0,1]\times \Sigma \to [0,1]\times \Sigma$, defined by $\hat{\sigma}(x,w)=(\tau_{X_{1}(w)}(x), \sigma(w))$ where $\Sigma=\{0,1, ..., d-1\}^{\mathbb{N}}$, $\sigma: \Sigma \to \Sigma$ 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 a 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 $\mathcal H$.
For a holonomic probability $\hat{\nu}$ on $[0,1]\times \Sigma$ we define the entropy $h(\hat{\nu})$=inf$_f \in \mathbb{B}^{+} \int \ln(\frac{P_{\psi}f}{\psi f}) d\hat{\nu}\geq 0$, where, $\psi \in \mathbb{B}^{+}$ is a fixed (any one) positive potential.
Finally, we analyze the problem: given $\phi \in \mathbb{B}^{+}$, find solutions of the maximization problem $p(\phi)$= sup$_\hat{\nu} \in \mathcal{H} \{ \,h(\hat{\nu}) + \int \ln(\phi) d\hat{\nu} \,\}.$ We show an example where a holonomic not-$\hat{\sigma}$-invariant probability attains the supremum value. In the last section we consider maximizing probabilities, sub-actions and duality for potentials $A(x,w)$, $(x,w)\in [0,1]\times \Sigma$, for IFS.
Citation: Artur O. Lopes, Elismar R. Oliveira. Entropy and variational principles for holonomic probabilities of IFS. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 937-955. doi: 10.3934/dcds.2009.23.937

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