Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach
| Autor: | Gauthier, Thomas, Taflin, Johan, Vigny, Gabriel |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\mathrm{Crit}(f)$ is preperiodic, i.e. if 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 $\mathrm{End}_d^k$ of all endomorphisms of degree $d$ of $\mathbb{P}^k$, such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space $\mathrm{Poly}_d^2$ of all regular polynomial endomorphisms of degree $d\geq2$ of the affine plane $\mathbb{A}^2$, we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set. The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation. Comment: 76 pages, 2 figures. Comments are welcome! |
| Databáze: | arXiv |
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