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On infinite composition of affine mappings. (English) Zbl 0939.47006
Let \(\mathcal J\) be the set of infinite sequences of the symbols \(1,\dots{},N\) i.e. \(\mathcal J=\{1,\dots{},N\}^{\infty}\) and let \(s\) be the shift operator on \(\mathcal J\) which is defined for each \(\sigma=(\sigma_1,\dots{},\sigma_n,\dots{})\in \mathcal J\) as follows \[ s(\sigma_1,\dots{},\sigma_n,\dots{})=(\sigma_2,\sigma_3,\dots{},\sigma_{n+1},\dots,); \quad \sigma_i\in \{1,\dots{},N\}, \;i\in \mathbb N. \] The dynamical system \([\mathcal J,s]\) thus obtained with the usual metric \[ d_c[\omega,\sigma]=\sum_{i=1}^{\infty} \frac{|\omega_i-\sigma_i|}{N^i};\quad \omega,\sigma\in \mathcal J, \] is called the full \(N\)-shift. A closed shift-invariant subspace \(\mathcal K\) of the full \(N\)-shift is called a subshift. Denote \(B(\mathcal K)=\{\{(\sigma_i,\dots{},\sigma_j)\}_{j\geq i}:\;\sigma=(\sigma_1,\dots{},\sigma_n,\dots{})\in \mathcal K\}.\) Let us suppose that a positive functional \(\Phi\) is defined on \(B(\mathcal K)\) such that \(\Phi\) is submultiplicative i.e. the inequalities \[ \Phi(\sigma_1,\dots{},\sigma_n)\leq\Phi(\sigma_1,\dots{}\sigma_j)\Phi(\sigma_{j+1},\dots{}, \sigma_n),\;1\leq j<n. \] hold. Define a number \(\Phi^*=\Phi^*(K,\Phi)\) as follows \[ \Phi^*=\lim_{n\rightarrow\infty} (\Phi^*_n)^{1/n} \text{ where }\Phi^*_n= \max\{\Phi(\sigma_1,\dots{},\sigma_n): \sigma=(\sigma_1,\dots{},\sigma_n,\dots{})\in \mathcal K\}. \] The main result of the paper is the following
Theorem 3. Let \(\{F_{\sigma_i}:i=1,\dots{},N\}\) be affine mappings of \(\mathbb R^n\) and \(\mathcal K\) be a subshift of the full N-shift \(\mathcal J\). Then the sequence \(\{F_{\sigma_1}\circ \dots{}\circ F_{\sigma_n}\}\) is convergent for every \(\sigma\in \mathcal K\) and \(z\in \mathbb R^n\) to a limit \(x(\sigma)\) independent of \(z\) if and only if \(\Phi^*<1\).
The proof is based on a generalization of the well-known König lemma.

47A35 Ergodic theory of linear operators
28A80 Fractals
26A18 Iteration of real functions in one variable
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