On Dynamics of Infinitely Generated Contractive Mapping Systems on pZ_p

Let p≥2 be a prime number and Zp be the ring of p-adic intergers.Let G be a semigroup generated by infinitely many contractive maps on p Zp.It is shown that if G satisfies the open tiling conditions,then there exists a shift transformation on the limit set of G and the shift transformation is ergodic with respect to the Haar measure on p Zp.As an application,we can generalize p-adic Khinchin’s Theorem and p-adic Lochs’Theorem to any infinitely generated semigroup by use of the ergodicity of the shift transformation.