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## Fundamenta Mathematicae

1999 | 159 | 1 | 85-90
Tytuł artykułu

### On infinite composition of affine mappings

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EN
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EN

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
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
85-90
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-09-22
poprawiono
1998-07-03
poprawiono
1998-10-05
Twórcy
autor
• Technical University of Budapest, Sztoczek u. 2 H 226 (Mathematics), H-1111 Budapest, Hungary
Bibliografia
• [1] I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl. 161 (1992), 227-263.
• [2] D. Lind and J. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, 1995.
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Bibliografia
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