# 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 [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

## How to cite

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Máté, László. "On infinite composition of affine mappings." Fundamenta Mathematicae 159.1 (1999): 85-90. <http://eudml.org/doc/212321>.

@article{Máté1999,
abstract = { 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},
author = {Máté, László},
journal = {Fundamenta Mathematicae},
keywords = {affine mapping; subshift; infinite tree; joint contraction; shift invariante subspaces; König lemma; submultiplicative; affine mappings},
language = {eng},
number = {1},
pages = {85-90},
title = {On infinite composition of affine mappings},
url = {http://eudml.org/doc/212321},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Máté, László
TI - On infinite composition of affine mappings
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 1
SP - 85
EP - 90
AB -  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
LA - eng
KW - affine mapping; subshift; infinite tree; joint contraction; shift invariante subspaces; König lemma; submultiplicative; affine mappings
UR - http://eudml.org/doc/212321
ER -

## References

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1. [1] I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl. 161 (1992), 227-263. Zbl0746.15015
2. [2] D. Lind and J. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, 1995. Zbl1106.37301

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