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

Elismar R. Oliveira

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

Elismar R. Oliveira

Dynamical Systems · Submitted 18 Jul 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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What the abstract itself states, as probabilities: claims first 0.16 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.02 · states quantitative result 0.19 · compares with prior results 0.82 · mentions code or data availability 0.02.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In 1987, J. H.	background 1.00	0.03	0.01	unclear 0.95
s1	Elton, has proved the first fundamental result in convergence of IFS, the Elton's Ergodic Theorem.	background 1.00	0.02	0.03	asserted 1.00
s2	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 $(X,d)$.	contribution 0.99	0.92	0.61	asserted 1.00
s3	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.	result 0.99	0.92	0.90	asserted 1.00
s4	Additionally, we give some examples and applications to Chaos Games and Nonautonomous Dynamical Systems defined by finite difference equations.	contribution 1.00	0.90	0.04	asserted 0.99

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Original 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.

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