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 pℤp. 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 pℤp. 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.

中文翻译：

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关于 pℤp 上无限生成收缩映射系统的动力学

设 p ≥ 2 是素数，ℤp 是 p-adic 整数环。设 G 是由 pℤp 上的无穷多个收缩映射生成的半群。结果表明，如果 G 满足开放平铺条件，则在 G 的极限集上存在平移变换，并且平移变换相对于 pℤp 上的 Haar 测度是遍历的。作为一个应用，我们可以利用平移变换的遍历性将 p-adic Khinchin's Theorem 和 p-adic Lochs' Theorem 推广到任何无限生成的半群。