Dynamical systems of the $p$-adic $(2,2)$-rational functions with two fixed points

Abstract

We consider a family of $(2,2)$-rational functions given on the set of complex $p$-adic field $\mathcal{C}_p$. Each such function $f$ has the two distinct fixed points $x_1=x_1(f)$, $x_2=x_2(f)$. We study $p$-adic dynamical systems generated by the $(2,2)$-rational functions. We prove that $x_1$ is always indifferent fixed point for $f$, i.e., $x_1$ is a center of some Siegel disk $SI(x_1)$. Depending on the parameters of the function $f$, the type of the fixed point $x_2$ may be any possibility: indifferent, attractor, repeller. We find Siegel disk or basin of attraction of the fixed point $x_2$, when $x_2$ is indifferent or attractor, respectively. When $x_2$ is repeller we find an open ball any point of which repelled from $x_2$. Moreover, we study relations between the sets $SI(x_1)$ and $SI(x_2)$ when $x_2$ is indifferent. For each $(2,2)$-rational function on $\mathcal{C}_p$ there are two points $\hat x_1=\hat x_1(f)$, $\hat x_2=\hat x_2(f)\in \mathcal{C}_p$ which are zeros of its denominator. We give explicit formulas of radiuses of spheres (with the center at the fixed point $x_1$) containing some points such that the trajectories (under actions of $f$) of the points after a finite step come to $\hat x_1$ or $\hat x_2$. We study periodic orbits of the dynamical system and find an invariant set, which contains all periodic orbits. Moreover, we study ergodicity properties of the dynamical system on each invariant sphere. Under some conditions we show that the system is ergodic iff $p=2$.