# On infinite composition of affine mappings

Fundamenta Mathematicae (1999)

• Volume: 159, Issue: 1, page 85-90
• ISSN: 0016-2736

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## Abstract

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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 ${\sigma }_{1},...,{\sigma }_{j}\in 1,...,N$ the composition (1) ${F}_{\sigma 1}\circ ...\circ {F}_{{\sigma }_{j}}$ is a contraction, then for any infinite sequence ${\sigma }_{1},{\sigma }_{2},...\in 1,...,N$ and any $z\in {ℝ}^{n}$, the sequence (2)${F}_{\sigma 1}\circ ...\circ {F}_{{\sigma }_{n}}\left(z\right)$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any $z\in {ℝ}^{n}$ and any $\sigma ={\sigma }_{1},{\sigma }_{2},...$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every $\sigma ={\sigma }_{1},{\sigma }_{2},...\in \Sigma$ the composition (1) is a contraction. This result can be considered as a generalization of the main theorem of Daubechies and Lagarias , 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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