Continuous functions between p-adic numbers [closed]

Let $p\neq q$ two prime numbers.

Are there any interesting continuous functions $\mathbb Q_p\to \mathbb Q_q$ or $\mathbb Z_p \to\mathbb Z_q$?

Or more specifically how big is the set $C( A,B)$ (continuous functions) where $A$ is $\mathbb Q_p$ or $\mathbb Z_p$ and $B$ is $\mathbb Q_q$ or $\mathbb Z_q$?

Solution 1:

• $f(\sum_{n\ge 0} c_n p^n) = \sum_{n\ge 0} c_n q^n$ is a continuous function $\Bbb{Z_p\to Z_q}$, where I mean $c_n\in 0\ldots p-1$.

• $f_\alpha(x) = \alpha f(x)$ works too for any $\alpha\in \Bbb{Z}_q$, so the cardinality of $C(\Bbb{Z}_p,\Bbb{Z}_q)$ is at least that of $\Bbb{Z}_q$.

• The cardinality of $C(\Bbb{Z}_p,\Bbb{Z}_q)$ cannot be larger: given $h\in C(\Bbb{Z}_p,\Bbb{Z}_q)$, for any $k\in \Bbb{Z}_{\ge 0}$ and $r\in \Bbb{Z}_p/p^k\Bbb{Z}_p$ take any elements $s\in \Bbb{Z}_p,H(r,k) \in \Bbb{Z}_q/q^k\Bbb{Z}_q$ such that $r\equiv s\bmod p^k, h(s)\equiv H(r,k)\bmod q^k$. Then $H$ is a countable collection of maps from finite sets to finite sets and $h(x)=\lim_{k\to \infty} H(x,k)$.