The Ergodic Theorem for a new kind of attractor of a GIFS

TL;DR

This paper extends Elton's Ergodic Theorem to a new class of attractors associated with GIFS, demonstrating convergence of averages along trajectories to a projected Hutchinson measure in compact metric spaces.

Contribution

It generalizes Elton's Ergodic Theorem to GIFS with projected Hutchinson measures, providing new convergence results and applications to chaos games and nonautonomous systems.

Findings

Convergence of time averages to the projected Hutchinson measure.

Extension of Elton's Ergodic Theorem to GIFS.

Applications to chaos games and difference equations.

Abstract

In 1987, J. H. Elton, has proved the first fundamental result in convergence of IFS, the Elton's Ergodic Theorem. In this work we prove the natural extension of this theorem to the projected Hutchinson measure $\mu_{\alpha}$ associated to a GIFSpdp $\mathcal{S}=\left(X, (\phi_j:X^{m} \to X)_{j=0,1, ..., n-1}, (p_j)_{j=0,1, ..., n-1}\right),$ in a compact metric space $(X,d)$. More precisely, the average along of the trajectories $x_{n}(a)$ of the GIFS, starting in any initial points $x_0, ..., x_{m-1} \in X$ satisfies, for any $f \in C(X , \mathbb{R})$, $$\lim_{N\to +\infty} \frac{1}{N}\sum_{n=0 }^{N-1} f(x_{n}(a)) = \int_{X} f(t) d\mu_{\alpha}(t),$$ for almost all $a \in \Omega=\{0,1, ..., n-1\}^{\mathbb{N}}$, the symbolic space. Additionally, we give some examples and applications to Chaos Games and Nonautonomous Dynamical Systems defined by finite difference equations.