On a nonlinear p-adic dynamical system

Abstract

We investigate the behavior of trajectories of a (3, 2)-rational p-adic dynamical system in the complex p-adic field ℂ p , when there exists a unique fixed point x 0. We study this p-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point x 0). We show that there exists a radius r depending on parameters of the rational function such that: when x 0 is an attracting point then the trajectory of an inner point from the ball U r (x 0) goes to x 0 and each sphere with a radius > r (with the center at x 0) is invariant; When x 0 is a repeller point then the trajectory of an inner point from a ball U r (x 0) goes forward to the sphere S r (x 0). Once the trajectory reaches the sphere, in the next step it either goes back to the interior of U r (x 0) or stays in S r (x 0) for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of U r(x 0) it will stay (for all the rest of time) in the sphere (outside of U r(x 0)) that it reached first.