Compositions of linear-fractional transformations

V. I. Buslaev^a, S. F. Buslaeva<sup>b

a</sup> Steklov Mathematical Institute, Russian Academy of Sciences
^b Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: We study the asymptotic behavior of the compositions $(\mathbf S_n\circ\dots\circ\mathbf S_1)(z)$ and $(\mathbf S_1\circ\dots\circ\mathbf S_n)(z)$ of linear-fractional transformations $\mathbf S_n(z)$ ($n=1,2,\dots$) whose fixed points have limits. In particular, if $\mathbf S_n(z)=\alpha_n(\beta_n+z)^{-1}$, then the sequence of compositions $(\mathbf S_1\circ\dots\circ\mathbf S_n)(z)$ at the point $z=0$ coincides with the sequence of convergents of the formal continued fraction
$$ \frac{\alpha_1}{\beta_1+\dfrac{\alpha_2}{\beta_2+\dotsb}}. $$
The result obtained can be applied in the study of convergence of formal continued fractions.

UDC: 517.5

Received: 10.11.1996

DOI: 10.4213/mzm1507

 English version: Mathematical Notes, 1997, 61:3, 272–277

Bibliographic databases:

• 
• 
• 
•