Local C r stability 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 c 0 stable and locally c 1 stable but globally c 1 unstable. although the global c 1 instability implies the general global c r (r ≥ 2) instability,the local c 1 stability does not guarantee the local c r (r ≥ 2) stability. in this paper we generally prove the local c r (r ≥ 2) stability for iterative roots. for this purpose we need a uniform estimate for the approximation to the conjugation in c r linearization,which is given by improving the method used for the c 1 case. © 2014 yingying zeng.