Abstract

In this paper we consider function \(f(x)={x+a\over bx+c}\), (where \(b\ne 0\), \(c\ne ab\), \(x\ne -{c\over b}\)) on three fields: the set of real, \(p\)-adic and complex numbers. We study dynamical systems generated by this function on each field separately and give some comparison remarks. For real variable case we show that the real dynamical system of the function depends on the parameters \((a,b,c)\in \mathbb R^3\). Namely, we classify the parameters to three sets and prove that: for the parameters from first class each point, for which the trajectory is well defined, is a periodic point of \(f\); for the parameters from second class any trajectory (under \(f\)) converges to one of fixed points (there may be up to two fixed points); for the parameters from third class any trajectory is dense in \(\mathbb R\). For the \(p\)-adic variable we give a review of known results about dynamical systems of function \(f\). Then using a recently developed method we give simple new proofs of these results and prove some new ones related to trajectories which do not converge. For the complex variables we give a review of known results.

中文翻译：

---

莫比乌斯变换的动力系统：实变量、$$p$$ -Adic 变量和复变量

摘要

在本文中，我们考虑函数\(f(x)={x+a\over bx+c}\)， (其中\(b\ne 0\)，\(c\ne ab\)，\(x\ ne -{c\over b}\) ）在三个字段上：实数、\(p\)进数和复数的集合。我们分别研究了该函数在每个域上生成的动力系统并给出了一些比较说明。对于实变量情况，我们表明函数的真实动力系统取决于参数\((a,b,c)\in \mathbb R^3\)。也就是说，我们将参数分为三组，并证明：对于第一类参数，轨迹明确的每个点都是\(f\)的周期点；对于第二类参数，任何轨迹（在f 下）收敛到固定点之一（最多可能有两个固定点）；对于第三类参数，任何轨迹在\(\mathbb R\)中都是密集的。对于\(p\) -adic 变量，我们回顾了有关函数\(f\)动力系统的已知结果。然后使用最近开发的方法，我们给出了这些结果的简单新证明，并证明了一些与不收敛轨迹相关的新证明。对于复杂变量，我们回顾已知结果。