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I have just started reading a little about (arithmetic) dynamics and it seems like critical points are very important - for instance, rational maps so that critical points have finite forward orbit (PCF maps?) seems to be an important object of study. For instance, the algebraic numbers $c$ so that $0$ has a finite forward orbit under $z \to z^2 + c$ seem to be quite important.

It's a little hard for me to understand why critical points should be important. I have heard for instance that PCF maps are assosciated to (infinite) Galois groups with finitely many ramified places and this is very interesting but I would be very interested in a precise reference. I have also heard that they are in some sense vaguely analogous to special CM points on modular varieties and any more elaboration along these lines would also be very welcome.

Given the interest in PCF maps, I am sure there must be any other reasons and I would be very happy to hear about any of them.

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  • $\begingroup$ "Being interesting" is a subjective matter, so it is not clear to me what a good answer to this question is. Furthermore, most branches of mathematics are in general concerned with "singularities", i.e. the critical points of some systems (in the most generic sense). $\endgroup$
    – Alex M.
    Commented Mar 28, 2021 at 16:20

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