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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.

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