Entropy and Variational principles for holonomic probabilities of IFS

TL;DR

This paper develops entropy and variational principles for holonomic probabilities associated with iterated function systems (IFS), introducing weighted systems and analyzing the maximization of pressure for these probabilistic structures.

Contribution

It introduces a framework for holonomic probabilities in IFS, defining entropy and pressure, and explores their properties and maximization in weighted systems.

Findings

Defined holonomic probabilities for IFS.

Established entropy and pressure concepts for these probabilities.

Analyzed maximization problems related to pressure in weighted systems.

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 $$p(\phi)=$$