Let $ C $ be an algebraically closed field, complete compared to a non-Archimedean assessment (e.g. $ C = mathbb {C} _p $). The points of the adic spectrum $ operatorname {Spa} (C langle T rangle, C ^ { circ} langle T rangle) $ have been completely classified, consisting of five families of points. The following is from Scholze *Perfectoid spaces*.

- Classic points: Take $ x in C ^ { circ} $and consider the assessment $ f mapsto | f (x) | $.

Points of type 2 and 3 are the "spokes" of the tree, and they follow the following recipe. Let $ 0 leq r <1 $and correct $ x in C ^ circ $. Develop a given power series $ f $ as $ sum_n a_n (T-x) ^ n $and send it to $ sup | a_n | r ^ n $.

- Assessments as above, but with the constraint that & # 39;$ r in | C ^ { circ} | $". Branching occurs precisely at points like this. I don't know what that notation means and I would appreciate someone telling me.
- The remaining evaluations, i.e. the shelves with $ r notin | C ^ { circ} | $.
- The "dead ends" of the tree. Let $ D_1 supseteq D_2 supseteq cdots , $ be a sequence of closed discs with $ bigcap_i D_i = varnothing $ (the existence (not) of such sequences is called spherical completeness), and consider the evaluation $ f mapsto inf_i sup_ {x in D_i} | f (x) | $.
- Some rankings$ 2 $ evaluations. To fix $ x in C ^ { circ} $ and $ 0 <r <$ 1. Choose a sign $ { lessgtr} in { < , >} $. Let $ Gamma _ { lessgtr} $ to be the abelian group ordered $ mathbb {R} _ {> 0} times gamma ^ { mathbb {Z}} $ such as $ r & # 39; lessgtr gamma lessgtr r $ for everyone $ r & # 39; lessgtr r $. We are now defining our valuation. Take any power series $ f $, develop it as $ sum_n a_n (T-x) ^ n $and send it to $ sup | a_n | gamma ^ n $.

The following illustration is with the list.

I'm having a hard time with the intuition behind this photo. Here is what I can collect. The arc at the bottom represents the classic unitary disc, and one can think that the vertical direction is the & # 39;$ r $-ax & # 39;. Certainly, then, the fact that there are rays makes sense. The type 5 points close to type 2 reflect them in the closure of the neighboring type 2 point.

**Question.** An open question might be: how should I think about all this? In terms of image in particular, a more specific question is: What does branching mean? Additional questions that I would like to understand: Can I see the open disc (for example) on this image? What is the relationship between my classical intuition for diagrams? What changes if we take, say, $ C = mathbb {Q} _p $ (that is, something that is not algebraically closed)? Do the allowable openings match something in this image?

All ideas are appreciated.