LocalCrStability for Iterative Roots of Orientation-Preserving Self-Mappings on the Interval

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 C0 stable and locally C1 stable but globally C1 unstable. Although the global C1 instability implies the general global Cr (r≥2) instability, the local C1 stability does not guarantee the local Cr (r≥2) stability. In this paper we generally prove the local Cr (r≥2) stability for iterative roots. For this purpose we need a uniform estimate for the approximation to the conjugation in Cr linearization, which is given by improving the method used for the C1 case