2002-2003 Convergence of Koenigs’ sequences.
D. J. Dewsnap, P. Fischer
Real Anal. Exchange 28(1): 111-120 (2002-2003).

## Abstract

Let $f$ be an interval map defined on a neighborhood of a fixed point $0$ with $$f'(0)=\lambda$$ where $$0<|\lambda|<1$$ and let $$\phi_{n}(x)=f^{n}(x)/\lambda^{n}$$. It is shown that if $f(x)=\lambda x+\mathcal{O}\!\left(\frac{|x|}{y\log{(y)}\cdots \log^{p-1}{\!(y)}(\log^{p}(y))^{1+\varepsilon}}\right)$ for some $$\varepsilon>0$$ and nonnegative integer $p$ where $$y=|\!\log{(|x|)}|$$, then the Koenigs' sequence $$\{\phi_{n}\}$$ of $f$ converges uniformly on a neighborhood of $0$ to a limit $$\phi$$ with $$\phi(0)=0$$ and $$\phi'(0)=1$$. On the other hand, if $f(0)=0$ and $f(x)=x\!\left(\!\lambda -\frac{1}{\log(x)\log(-\log(x)) \cdots\log^{p}{(-\log(x))}}\right)$ for sufficiently small $x>0$ where $0<\lambda<1$ and $p$ is a nonnegative integer, then the Koenigs' sequence of $f$ diverges on a small right-neighborhood of $0$. It is illustrated by examples that when $$\varepsilon=0$$ in the first equation for $f$ given above, the Koenigs' sequence of $f$ can also converge to zero on a neighborhood of $0$ or converge to a limit $$\phi$$ that is nondifferentiable at $0$. It is also shown that when the Koenigs' sequence of a map $f$ converges to a limit $$\phi$$ that is differentiable at $0$, then $$\phi'(0)$$ is either $0$ or $1$.

## Citation

D. J. Dewsnap. P. Fischer. "Convergence of Koenigs’ sequences.." Real Anal. Exchange 28 (1) 111 - 120, 2002-2003.

## Information

Published: 2002-2003
First available in Project Euclid: 12 June 2006

zbMATH: 1050.39026
MathSciNet: MR1973973

Subjects:
Primary: 37E05 , 39B12

Keywords: Koenigs' sequence , Schr\"{o}der equation 