Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach

• DOI: 10.48550/arxiv.2305.02246
• Publication Date: 2023-01-01
• Authors: Thomas Gauthier; Johan Taflin; Gabriel Vigny
• Keywords: Degree (music)

An endomorphism $f:\mathbb{P}^k\to\mathbb{P}^k$ of degree $d\geq2$ is said to be postcritically finite (or PCF) if its critical set $\mathrm{Crit}(f)$ preperiodic, i.e. there are integers $m>n\geq0$ such that $f^m(\mathrm{Crit}(f))\subseteq f^n(\mathrm{Crit}(f))$. When $k\geq2$, it was conjectured by Ingram, Ramadas and Silverman that, in the space $\mathrm{End}_d^k$ all endomorphisms $d$ $\mathbb{P}^k$, not Zariski dense. We prove this conjecture. Further, $\mathrm{Poly}_d^2$ regular polynomial affine plane $\mathbb{A}^2$, we construct a dense open subset where have uniform bound on number preperiodic points lying set. The proofs combination theory heights arithmetic dynamics methods from real produce subsets with maximal bifurcation.