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1999 | 159 | 1 | 85-90

Tytuł artykułu

On infinite composition of affine mappings

Autorzy

Treść / Zawartość

Języki publikacji

EN

Abstrakty

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

Rocznik

Tom

159

Numer

1

Strony

85-90

Daty

wydano
1999
otrzymano
1997-09-22
poprawiono
1998-07-03
poprawiono
1998-10-05

Twórcy

  • 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.

Identyfikator YADDA

bwmeta1.element.bwnjournal-article-fmv159i1p85bwm