On a nonlinear p-adic dynamical system

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.