On infinite composition of affine mappings

 Let ${\displaystyle \{F_i=1,...,N\}}$ be affine mappings of ${\displaystyle {ℝ}^n}$. It is well known that if there exists j ≤ 1 such that for every ${\displaystyle {\sigma }_1,...,{\sigma }_j\in \{1,...,N\}}$ the composition (1) ${\displaystyle F_{\sigma 1}\circ ...\circ F_{{\sigma }_j}}$ is a contraction, then for any infinite sequence ${\displaystyle {\sigma }_1,{\sigma }_2,...\in \{1,...,N\}}$ and any ${\displaystyle z\in {ℝ}^n}$, the sequence (2)${\displaystyle F_{\sigma 1}\circ ...\circ F_{{\sigma }_n}\left(z\right)}$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any ${\displaystyle z\in {ℝ}^n}$ and any ${\displaystyle \sigma =\{{\sigma }_1,{\sigma }_2,...\}}$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every ${\displaystyle \sigma =\{{\sigma }_1,{\sigma }_2,...\}\in \Sigma }$ 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