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I'm studying the dynamic of the post-critically finite for my master thesis and my professor gave me the problem concerning generalization of post-critically finite ration map. Concretely, let $f : \mathbb{P}^n \rightarrow \mathbb{P}^n$ be a holomorphic map. The critical set consist of points sucht that $$ C(f)=\{z \in \mathbb{P}^n | \, rank T_z f < n\}$$ The post-critical set of $f$ $PC(f)$ is forward image of $C(f)$ $$ PC(f)= \bigcup\limits_{i=0}^{\infty} f^i (C(f))$$ A map is called post-critically finite if $PC(f)$ is algebraic, at least my professor told me so. It's clear that if $PC(f)$ is algebraic, it is closed. My question is: Is that the converse statement is true, i.e if $f$ has $\overline{PC(f)}$ is algebraic, then $PC(f)$ is algebraic?

At first glance, it would be wrong for the statement that "$\overline{D} \subset \mathbb{P}^n$ is algebraic then $D$ is algebraic". For example, take D is a countable dense subset of $\mathbb{P}^n$. but it is not actually the counter example of my question. Thank you for any suggestion. This question is asked at math.stackexchange.com but there no one interested so I hope MO could help me out. Thanks again

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You can probably find what you want, and more, in the following article, although I'm not sure if it's available on the web:

Fornæss & N. Sibony Critically Finite Rational Maps on $\mathbb P^2$, Proceedings of the Madison Symposion honoring Walter Rudin, AMS series in Contemporary Mathematics: (1992) 245-260.

Two more recent articles that discuss post-critically finite maps on $\mathbb P^n$ and are available on the web are:

Sarah Koch, Teichmüller theory and critically finite endomorphisms Advances in Mathematics Vol. 248, 2013. http://www-personal.umich.edu/~kochsc/endo.pdf

Dynamics of post-critically finite maps in higher dimension, Matthieu Astorg, 2016. https://arxiv.org/abs/1609.02717

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  • $\begingroup$ Thank you Prof Silverman, my duty is work out the result in the paper of Matthieu Astorg. And in fact all the technical work out with the definition that my professor gave. But I still being confused about the part "taking closure". For example, at the first of Matthieu's paper, he said the post-critically finite (PCF) is the same as mine, but in the definition 3.3 he add the closure operator there. In many other context, take closure or not pops out everywhere so I got the feeling maybe they are equivalent. But I found it is not that obvious. $\endgroup$ – Curiosity Mar 25 '17 at 10:01
  • $\begingroup$ @Curiosity it is not true. even in dimension 1, it is possible for a rational map $f$ to have critical points with dense orbits in the Riemann sphere. Such maps satisfy $\overline{D}$ algebraic but are not PCF $\endgroup$ – glougloubarbaki May 9 '17 at 12:52
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Thanks to the keyword in the comment of Glougloubarbaki, I made a search and found the following paper

Mary Rees, Ergodic rational maps with dense critical point forward orbit, (1984)

So there does exist a rational map such that the $\overline{PC(f)}$ is algebraic but $PC(f)$ itself is not algebraic. So the best way to state a map $f$ is post-critically finite is that $PC(f)$ is algebraic, then so does its closure.

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