On infinite composition of affine mappings

Máté, László

Fundamenta Mathematicae, Tome 159 (1999), p. 85-90 / Harvested from The Polish Digital Mathematics Library

Résumé

 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

Détail

Publié le : 1999-01-01

Zbl 0939.47006

EUDML-ID : urn:eudml:doc:212321

Bibliographie

[00000] [1] I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl. 161 (1992), 227-263. | Zbl 0746.15015

[00001] [2] D. Lind and J. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, 1995. | Zbl 1106.37301