# On infinite composition of affine mappings

Fundamenta Mathematicae (1999)

- Volume: 159, Issue: 1, page 85-90
- ISSN: 0016-2736

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topMáté, 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

top- [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] D. Lind and J. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, 1995. Zbl1106.37301

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