Dynamically improper hypersurfaces for endomorphisms of projective space

TL;DR

This paper introduces a new concept related to preperiodic hypersurfaces in projective space and shows that endomorphisms with periodic critical points are densely distributed, answering a previously open question.

Contribution

It generalizes the notion of preperiodic hypersurfaces and proves the density of endomorphisms with periodic critical points in projective space.

Findings

Periodic critical points are Zariski dense among nonlinear endomorphisms.

Introduces a new class of hypersurfaces called dynamically improper hypersurfaces.

Answers a question posed by Ingram regarding the distribution of such endomorphisms.

Abstract

We introduce a new generalization of the notion of preperiodic hypersurface and explore some of its basic ramifications. We also prove that among nonlinear endomorphisms of projective space, those with a periodic critical point are Zariski dense. This answers a question of Ingram.