As noted, $\mathbb{Q}_p$ and its finite extensions are locally compact, but the Julia set is often empty. Indeed, in this case that the Fatou set is always non-empty, another difference from the compact case. However, since $\mathbb{Q}_p$ isn't algebraically closed, one might best compare dynamics over $\mathbb{Q}_p$ as being analogous in some ways to dynamics over $\mathbb{R}$. So the analogue of $\mathbb{C}$ is $\mathbb{C}_p$, the completion of the algebraic closure of $\mathbb{Q}_p$. Unfortunately, $\mathbb{C}_p$ is not locally compact (and of course, totally disconnected), so one can't use measure-theoretic arguments. For example, it's not easy to make sense of equidistribution. The modern solution is to instead look at Berkovich space. This is a locally compact and connected space that includes $\mathbb{P}^1(\mathbb{C}_p)$ as a sort of boundary. A good introduction to Berkovich spaces is listed below. And in Berkovich space over $\mathbb{C}_p$, we're back to the situation where the Julia set is always non-empty.

I'll mention one other interesting difference between the complex and $p$-adic cases. A famous theorem of Sullivan says that a rational map has no wandering domains in $\mathbb{P}^1(\mathbb{C})$. In opposition to this, Benedetto has constructed rational maps that do have wandering domains in $\mathbb{P}^1(\mathbb{C}_p)$. However, it is not known if wandering domains can exist in $\mathbb{P}^1(\mathbb{Q}_p)$.

• Baker, Matthew; Rumely, Robert, Potential theory and dynamics on the Berkovich projective line. Mathematical Surveys and Monographs, 159. American Mathematical Society, Providence, RI, 2010

As noted, $\mathbb{Q}_p$ and its finite extensions are locally compact, but the Julia set is often empty. Indeed, in this case that the Fatou set is always non-empty, another difference from the compact case. However, since $\mathbb{Q}_p$ isn't algebraically closed, one might best compare dynamics over $\mathbb{Q}_p$ as being analogous in some ways to dynamics over $\mathbb{R}$. So the analogue of $\mathbb{C}$ is $\mathbb{C}_p$, the completion of the algebraic closure of $\mathbb{Q}_p$. Unfortunately, $\mathbb{C}_p$ is not locally compact (and of course, totally disconnected), so one can't use measure-theoretic arguments. For example, it's not easy to make sense of equidistribution. The modern solution is to instead look at Berkovich space. This is a locally compact and connected space that includes $\mathbb{P}^1(\mathbb{C}_p)$ as a sort of boundary. Two good introductions to Berkovich spaces are listed below. And in Berkovich space over $\mathbb{C}_p$, we're back to the situation where the Julia set is always non-empty.

I'll mention one other interesting difference between the complex and $p$-adic cases. A famous theorem of Sullivan says that a rational map has no wandering domains in $\mathbb{P}^1(\mathbb{C})$. In opposition to this, Benedetto has constructed rational maps that do have wandering domains in $\mathbb{P}^1(\mathbb{C}_p)$. However, it is not known if wandering domains can exist in $\mathbb{P}^1(\mathbb{Q}_p)$.

• Robert L. Benedetto, Dynamics in One Non-Archimedean Variable. Graduate Studies in Mathematics, Volume: 198. American Mathematical Society, Providence, RI, 2019;

• Matthew Baker and Robert Rumely, Potential theory and dynamics on the Berkovich projective line. Mathematical Surveys and Monographs, Volume 159. American Mathematical Society, Providence, RI, 2010