Strict ergodicity of affine p-adic dynamical systems on Zp

Let p⩾2 be a prime number and let Zp be the ring of all p-adic integers. For all α,β,z∈Zp, define Tα,β(z)=αz+β. It is shown that the dynamical system (Zp,Tα,β) is minimal if and only if α∈1+prpZp and β is a unit, here rp=1 (respectively rp=2) if p⩾3 (respectively if p=2), and that when it is minimal, it is strictly ergodic and topologically conjugate to (Zp,T1,1) with an analytic and isometric conjugacy. More importantly, when the system is not minimal, we find all its strictly ergodic components. As application, monomial systems Sn,ρ(z)=ρzn on the group 1+pZp are also discussed.