WEI Xiao-qin.The Problem of n-th Iteration of Quadratic Fractional Functions[J].Journal of Chongqing Technology and Business University(Natural Science Edition）,2021,38(4):63-67

The Problem of n-th Iteration of Quadratic Fractional Functions

DOI：

 作者 单位 魏小琴 重庆师范大学 数学科学学院，重庆 401331

离散动力系统是对常微分方程解族进行离散化之后得到的系统，因其形式简洁并易于反映问题的本质，从20世纪60年代开始在Smale等著名数学家的倡导下蓬勃发展起来，对函数n次迭代的研究有助于了解离散动力系统轨道的长期行为；关于二次分式函数的n次迭代将在已有结果的基础上利用共轭相似法研究的前人未解决的3类特殊情形，即：b1=0且a1c1=0；b1≠0且a1c1=0；a1b1c1≠0且a1=a2+1，b2=b1+2，c1=c2+1；共轭相似法的原理是找到一个可逆桥函数将二次分式函数转化为二次函数，再根据二次函数已有的n次迭代结果解决问题；方法的关键在于寻找桥函数，但这没有一个固定的方法，针对每一类特殊情形，将寻找不同的桥函数.

The discrete dynamical system is obtained by discretizing the solution cluster of ordinary differential equations. Due to the simple form and easy way of reflecting the nature of the problem， it has flourished since the 1960s under the initiative of Smale and other famous mathematicians. The study on n-th iteration of functions can help us understand long-term behaviors of the orbits of the discrete dynamical systems. There have been plentiful results on n-th iteration of the quadratic fractional functions.In this paper， based on the previous results， three kinds of unsolved cases are investigated by using the conjugation similarity method， i.e.， (I) b1=0 and a1c1=0； (II) b1≠0 and a1c1=0； (III) a1b1c1≠0 and a1=a2+1，b2=b1+2，c1=c2+1. The principle of conjugation similarity method is to find reversible bridge function to convert the quadratic fractional functions into quadratic functions, then， according to the existing n-th iteration results on quadratic functions， we can solve the problem. The key point of this method is to find bridge functions， but there is no general method. Thus， for every case, we will find different bridge functions.