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

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.