On infinite composition of affine mappings

• Autorzy: László Máté
• 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

EN

affine mapping; subshift; infinite tree; joint contraction

Kategorie tematyczne

• 47A35: Ergodic theory
• 26A18: Iteration
• 28A80: Fractals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1999
• Tom: 159
• Numer: 1
• Strony: 85-90

Daty

wydano

1999

otrzymano

1997-09-22

poprawiono

1998-07-03

poprawiono

1998-10-05

Twórcy

autor

László Máté

• 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.