We show that any (1, 2)-rational function with a unique fixed point is topologically conjugate to a (2, 2)-rational function or to the function f(x) = ax/x2 + a. The case (2, 2) was studied in our previous paper, here we study the dynamical systems generated by the function f on the set of complex p-adic field ℂp. We show that the unique fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We construct the corresponding Siegel disk of these dynamical systems. We determine a sufficiently small set containing the set of limit points. It is given all possible invariant spheres.We show that the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure on the set of p-adic numbers ℚp.Moreover some periodic orbits of the system are investigated.

中文翻译：

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函数 ax/x2 + a 的 p-Adic 动力系统

我们证明任何具有唯一不动点的 (1, 2)-有理函数在拓扑上与 (2, 2)-有理函数或函数 f(x) = ax/x2 + a 共轭。情况(2, 2)在我们之前的论文中研究过，这里我们研究由函数f在复杂p-adic场ℂp集合上产生的动力系统。我们表明唯一不动点是无关紧要的，因此轨迹的收敛不是动态系统的典型情况。我们构建了这些动力系统对应的 Siegel 圆盘。我们确定一个足够小的包含极限点集的集合。给出了所有可能的不变球体。我们证明了在每个不变球体上减少的 p-adic 动力系统对于 p-adic 数集 ℚp 上的 Haar 测度不是遍历的。