Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908720458366976 |
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| author | Gauthier, Thomas Taflin, Johan Vigny, Gabriel |
| author_facet | Gauthier, Thomas Taflin, Johan Vigny, Gabriel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02246 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach Gauthier, Thomas Taflin, Johan Vigny, Gabriel Dynamical Systems Complex Variables Number Theory 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. |
| title | Sparsity of postcritically finite maps of $\mathbb{P}^k$ and beyond: A complex analytic approach |
| topic | Dynamical Systems Complex Variables Number Theory |
| url | https://arxiv.org/abs/2305.02246 |