Local Cr Stability for Iterative Roots of Orientation-Preserving Self-Mappings on the Interval

Abstract

Stability of iterative roots is important in their numerical computation. It is known that under some conditions iterative roots of orientation-preserving self-mappings are both globally $C^0$ stable and locally $C^1$ stable but globally $C^1$ unstable. Although the global $C^1$ instability implies the general global $C^{\mathrm{r}}$ ($r\ge 2$) instability, the local $C^1$ stability does not guarantee the local $C^{\mathrm{r}}$ ($r\ge 2$) stability. In this paper we generally prove the local $C^{\mathrm{r}}$ ($r\ge 2$) stability for iterative roots. For this purpose we need a uniform estimate for the approximation to the conjugation in $C^{\mathrm{r}}$ linearization, which is given by improving the method used for the $C^1$ case.