The p -adic completion ℚ p of the rational numbers induces a different absolute value |⋅| p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ + → ℝ + , for which the composition with the p-adic metric d p generated by |⋅| p is still a metric on ℚ p , with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ d p is still an ultrametric on ℚ p if and only if there is a function g such that f ∘ d p = g ∘ d p and g ∘ d is still an ultrametric for every ultrametric d . Some general variants of the last statement are also proved.

中文翻译：

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P-Adic度量保留功能及其类似物

有理数的p -adic完成ℚp得出不同的绝对值|⋅|。p比典型值| ⋅| 我们有实数。在本文中，我们比较和对比函数f：ℝ+→ℝ+，对于这些函数，由|⋅|生成的具有p-adic度量dp的成分 p仍然是ℚp的度量，具有常用的度量保留功能和在on上保留欧几里得度量的函数。特别地，表明，当且仅当存在函数g使得f dp = g d d并且g d仍然是每个超度d的超度时，f dp仍然是on p的超度。最后陈述的一些一般变体也得到了证明。