Discreteness of postcritically finite maps in $p$-adic moduli space
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- by Robert L. Benedetto and Su-Ion Ih;
- Trans. Amer. Math. Soc. 377 (2024), 2027-2048
- DOI: https://doi.org/10.1090/tran/9085
- Published electronically: January 3, 2024
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Abstract:
Let $p \geq 2$ be a prime number and let $\mathbb {C}_p$ be the completion of an algebraic closure of the $p$-adic rational field $\mathbb {Q}_p$. Let $f_c(z)$ be a one-parameter family of rational functions of degree $d\geq 2$, where the coefficients are meromorphic functions defined at all parameters $c$ in some open disk $D\subseteq \mathbb {C}_p$. Assuming an appropriate stability condition, we prove that the parameters $c$ for which $f_c$ is postcritically finite (PCF) are isolated from one another in the $p$-adic disk $D$ except in certain trivial cases. In particular, all PCF parameters of the family $f_c(z)=z^d+c$ are $p$-adically isolated.References
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Bibliographic Information
- Robert L. Benedetto
- Affiliation: Amherst College, Amherst, Massachusetts 01002
- MR Author ID: 647128
- ORCID: 0000-0003-0766-7927
- Email: rlbenedetto@amherst.edu
- Su-Ion Ih
- Affiliation: University of Colorado, Boulder, Colorado 80309; and Korea Institute for Advanced Study, Seoul 02455, South Korea
- MR Author ID: 703039
- ORCID: 0000-0002-3766-0972
- Email: ih@math.colorado.edu
- Received by editor(s): November 14, 2022
- Received by editor(s) in revised form: August 18, 2023
- Published electronically: January 3, 2024
- Additional Notes: The first author was supported by NSF grant DMS-150176. The second author was supported by Simons Foundation grant 622375 and the hospitality of the Korea Institute for Advanced Study during his visit.
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 2027-2048
- MSC (2020): Primary 11S82, 37P20, 37P45
- DOI: https://doi.org/10.1090/tran/9085
- MathSciNet review: 4744749