On minimal decomposition of p-adic homographic dynamical systems

A homographic map in the field of p  -adic numbers QpQp is studied as a dynamical system on P1(Qp)P1(Qp), the projective line over QpQp. If such a system admits one or two fixed points in QpQp, then it is conjugate to an affine dynamics whose dynamical structure has been investigated by Fan and Fares. In this paper, we shall mainly solve the remaining case that the system admits no fixed point. We shall prove that this system can be decomposed into a finite number of minimal subsystems which are topologically conjugate to each other. All the minimal subsystems are exhibited and the unique invariant measure for each minimal subsystem is determined.