Definition of Post-critically finite map 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
 A: 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
A: 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. 
