P-adic dynamical systems of (2,2)-rational functions with unique fixed point

Abstract

We consider a family of (2, 2)-rational functions given on the set of complex p-adic field Cp. Each such function has a unique fixed point. We study p-adic dynamical systems generated by the (2, 2)-rational functions. We show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. Siegel disks of these dynamical systems are found. We obtain an upper bound for the set of limit points of each trajectory, i.e., we determine a sufficiently small set containing the set of limit points. For each (2, 2)-rational function on Cp there are two points x^1=x^1(f),x^2=x^2(f)∈Cp which are zeros of its denominator. We give explicit formulas of radiuses of spheres (with the center at the fixed point) containing some points such that the trajectories (under actions of f) of the points after a finite step come to x^1 or x^2. Moreover for a class of (2, 2)-rational functions we study ergodicity properties of the dynamical systems on the set of p-adic numbers Qp. For each such function we describe all possible invariant spheres. We show that if p ≥ 3 then the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. In case p=2 under some conditions we prove non ergodicity and we show that there exists a sphere on which our dynamical system is ergodic.