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Mat. Zametki, 1997, Volume 61, Issue 3, Pages 332–338 (Mi mz1507)  

Compositions of linear-fractional transformations

V. I. Buslaeva, S. F. Buslaevab

a 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 $(\mathbf S_n\circ…\circ\mathbf S_1)(z)$ and $(\mathbf S_1\circ…\circ\mathbf S_n)(z)$ of linear-fractional transformations $\mathbf S_n(z)$ ($n=1,2,…$) whose fixed points have limits. In particular, if $\mathbf S_n(z)=\alpha_n(\beta_n+z)^{-1}$, then the sequence of compositions $(\mathbf S_1\circ…\circ\mathbf S_n)(z)$ 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.

DOI: https://doi.org/10.4213/mzm1507

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

Bibliographic databases:

UDC: 517.5
Received: 10.11.1996

Citation: V. I. Buslaev, S. F. Buslaeva, “Compositions of linear-fractional transformations”, Mat. Zametki, 61:3 (1997), 332–338; Math. Notes, 61:3 (1997), 272–277

Citation in format AMSBIB
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\paper Compositions of linear-fractional transformations
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\jour Math. Notes
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\pages 272--277
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