P-adic (2, 1)-rational dynamical systems

We study the dynamics of an arbitrary ( 2 , 1 ) -rational function f ( x ) = ( x 2 + a x + b ) / ( c x + d ) on the field C p of complex p -adic numbers. We show that the p -adic dynamical system generated by f has a very rich behavior. Siegel disks may either coincide or be disjoint for different fixed points of the dynamical system. Also, we find the basin of the attractor of the system. Varying the parameters, it is proven that there are periodic trajectories. For some values of the parameters there are trajectories which go arbitrary far from fixed points.