一类多项式型迭代函数方程在共振点附近的解析解

Analytic Solutions of a Polynomial-Like Iterative Functional Equation near Resonance

中文关键词:  迭代函数方程  解析解  Diophantine条件  Brjuno条件  共振点.

英文关键词:iterative functional equation  analytic solutions  diophantine condition  Brjuno condition  resonance.

中文摘要:

      本文讨论一类多项式型迭代函数方程局部解析解的存在性.通过Schr\"{o}der变换把方程转化为不含未知函数迭代的函数方程,进而求出方程的解析解.由于技术的原因,以前的工作要求Schr\"{o}der变换中的常数$\alpha$,即未知函数$f$在其不动点0处的线性化特征值$\alpha$不在单位圆周上或在单位圆周上但满足Diophantine条件.本文在$\alpha$是共振点,即$\alpha$是单位根的情形以及近共振情形且满足Brjuno条件,给出了解析解的结果.

英文摘要:

      In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied. As well as in previous work, we reduce this problem with the Schr\"oder transformation to finding analytic solutions of a functional equation without iteration of the unknown function $f$. For technical reasons, in previous work the constant $\alpha$ given in the Schr\"oder transformation, i.e., the eigenvalue of the linearized $f$ at its fixed point $O,$ is required to fulfill that $\alpha$ is off the unit circle $S^1$ or lies on the circle with the Diophantine condition. In this paper, we obtain results of analytic solutions in the case of $\alpha$ at resonance, i.e., at a root of the unity and the case of $\alpha$ near resonance under the Brjuno condition.