The locus of post-critically finite maps in the moduli space of self-maps of $\mathbb{P}^n$

Abstract: A degree $d>1$ self-map $f$ of $\mathbb{P}^n$ is called post critically finite (PCF) if its critical hypersurface $C_f$ is pre-periodic for $f$, that is, if there exist integers $r \geq 0$ and $k>0$ such that $f^{r+k}(C_f)$ is contained in $f^{r}(C_f)$.

I will discuss the question: what does the locus of PCF maps look like as a subset of the moduli space of degree $d$ maps on $\mathbb{P}^n$? I’ll give a survey of many known results and some conjectures in dimension $1$. I’ll then present a result, joint with Patrick Ingram and Joseph Silverman, that suggests that in dimensions two or greater, PCF maps are comparatively scarce in the moduli space of all self-maps.

algebraic geometrydynamical systems

Audience: researchers in the topic

( paper )

Comments: The discussion for Rohini Ramadas’s talk is taking place not in zoom-chat, but at tinyurl.com/2020-05-15-rr (and will be deleted after 3-7 days).

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Stanford algebraic geometry seminar

Series comments: The seminar was online for a significant period of time, but for now is solely in person. More seminar information (including slides and videos, when available): agstanford.com

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