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On infinite composition of affine mappings

Tom 159 / 1999

Fundamenta Mathematicae 159 (1999), 85-90 DOI: 10.4064/fm-159-1-85-90

Streszczenie

Let ${F_i = 1,...,N}$ be affine mappings of $ℝ^n$. It is well known that if there exists j ≤ 1 such that for every $σ_1,...,σ _j ∈ {1,..., N}$ the composition (1) $F_{σ1}∘...∘ F_{σ_j}$ is a contraction, then for any infinite sequence $σ_1, σ_2, ... ∈ {1,..., N}$ and any $z ∈ ℝ^n$, the sequence (2)$F_{σ1}∘...∘ F_{σ_n}(z)$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any $z ∈ ℝ^n$ and any $σ = {σ_1, σ_2,...}$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every $σ = {σ_1, σ_2,...} ∈ Σ$ the composition (1) is a contraction. This result can be considered as a generalization of the main theorem of Daubechies and Lagarias [1], p. 239. The proof involves some easy but non-trivial combinatorial considerations. The most important tool is a weighted version of the König Lemma for infinite trees in graph theory

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