Proof of convergence in the problem of rectification

The behavior of the vertices $A_1(t),\dots,A_n(t)$ of a polygonal line $\mathbf A(t)$ situated in $k$-dimensional Euclidean space is considered as $t\to\infty$ (each point $A_i(t\pm1)$, $1<i<n$, is a linear combination of the points $A_{i-1}(t)$, $A_i(t)$ and $A_{i+1}(t)$; the points $A_1(t+1)$ and $A_n(t+1)$ are linear combinations of $A_1(t)$ and $A_2(t)$, and $A_{n-1}(t)$ and $A_n(t)$, respectively). It is proved that for any initial position $\mathbf A(0)$ the polygonal lines $\mathbf A(t)$ converge to one of two possible limits, namely a stationary or quasistationary polygonal line.
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