in this paper by “end” only the type I points.

![ []{data-label="P1brek"}](Berkline-eps-converted-to.pdf){width="6cm"}

![ []{data-label="P1brek"}](P1berk-eps-converted-to.pdf){width="6.5cm"}

Limits and Berkovich space
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Now, let us look at the branching points in $ \P^1 _{Berk}$. We saw that they occur at some points $B(a,|a-b|_\LL)$ for some $a$ and $b$ in $\LL\cup\{\infty\}$. Up to a change of coordinate in $\P^1_\LL$, we can consider that this ball is the (closed) ball $${\zeta}:=B(0,1)$$ called the Gauss point.

This point is very particular from an algebraic point of view as we will see the formalism in detail. The elements of $\zeta$ have the following form: $$c=\sum_{k\geq0}c_{k} t^{k} \text{ with all } c_k\in \C.$$ We define a map $\rho:\P^1 _{Berk}\to\P^1\C$ by $\rho(c)= c_0$ for $c\in\zeta$ and $\rho(c)=\infty$ for $c\notin \zeta$. Note that when it is well defined, applying $\rho$ corresponds to taking the limit.

Two elements of $\P^1\LL$ are in the same branch of $\zeta$ if and only if they have the same image by $\rho$. This gives a canonical identification between the branches at $\zeta$ and $\P^1\C$.

For $d\in\P^1\LL\setminus\zeta$, we have $|d-0|_\LL>1$ so $$\zeta\subset B(0,|d|_\LL)=B(0,|d|_\LL)$$ and all the elements $ B(d,0)\approx d$ with $|d|_\LL>1$ are in a same branch containing $\infty$.

For $a,b\in\zeta$, we saw before that $B(a,r)=B(b,r)\subset 