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

2015/07/18 by Elismar R. Oliveira, Oliveira, Elismar R. · 2 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1507.05140

9 figures, 17 pages

arxiv created 2015/10/11 · arxiv updated 2015/10/13

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 μ_α associated to a GIFSpdp S=(X, (ϕ_j:X^m → X)_{j=0,1, ..., n-1}, (p_j)_{j=0,1, ..., n-1}), 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 x₀, ..., x_m-1 ∈ X satisfies, for any f ∈ C(X , ℝ), lim_{N→ +∞} (1)/(N)∑_n=0 ^N-1 f(x_n(a)) = ∫_X f(t) dμ_α(t), for almost all a ∈ Ω=\0,1, ..., n-1\^ℕ, the symbolic space. Additionally, we give some examples and applications to Chaos Games and Nonautonomous Dynamical Systems defined by finite difference equations.

Cited by

• Blending attractors of Iterated Function Systems
• The Hutchinson-Barnsley theory for iterated function systems with general measures

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