$p$-adic dynamical systems of the function $\dfrac{ax}{x^2+a}$

TL;DR

This paper studies the dynamics of a specific $p$-adic rational function, revealing the structure of fixed points, invariant spheres, and the non-ergodic nature of the system on these spheres, with implications for $p$-adic dynamical theory.

Contribution

It characterizes the $p$-adic dynamical behavior of the function $f(x)=\frac{ax}{x^2+a}$, including fixed points, Siegel disks, and invariant spheres, extending previous work on related rational functions.

Findings

Unique fixed point is indifferent.

Constructed Siegel disk for the system.

System is not ergodic on invariant spheres.

Abstract

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 $f(x)={ax\over x^2+a}$. 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 $\mathbb C_p$. 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 $Q_p$. Moreover some periodic orbits of the system are investigated.