一维映射迭代根的非单调性及光滑性

Non-monotonicity and smoothness of iterative roots of one-dimensional mapping

迭代根问题是动力系统嵌入流问题的弱问题,是动态插值方法的基础.然而,即使是对一维映射,迭代根的非单调性和全局光滑性都是困难的问题.本文介绍这方面的若干新结果,尤其是关于严格逐段单调连续函数的连续迭代根的存在性和构造,以及迭代根局部光滑与全局光滑的新进展.最后给出多项式迭代根这类既严格逐段单调又具光滑性的迭代根的存在条件及计算方法.

The problem of iterative roots is a weak version of embedding flows,which is a basis of dynamic interpolation.However,even for one-dimensional mappings,it is still difficult to discuss the non-monotonicity and smoothness of their iterative roots.In this paper,some new results are introduced,especially for the existence and construction of continuous iterative roots of strictly piecewise monotone and continuous mappings,and for smoothness of iterative roots about local and global cases.Finally,as a special class of strictly piecewise monotone and smooth mappings,polynomials are discussed and the conditions of existence with calculation methods of their iterative roots are given.