Fixed points of Koch maps

Le, Van Tu

doi:10.1090/ecgd/370

Fixed points of Koch maps
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by Van Tu Le

Conform. Geom. Dyn. 26 (2022), 10-33

DOI: https://doi.org/10.1090/ecgd/370

Published electronically: April 25, 2022

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Abstract:

We study endomorphisms constructed by Sarah Koch in [Teichmüller theory and critically finite endomorphisms, Advances in Mathematics 248 (2013), 573–617] and we focus on the eigenvalues of the differential of such maps at its fixed points. To each post-critically finite unicritical polynomial, Koch associated a post-critically algebraic endomorphism of ${\mathbb {CP}}^k$. Koch showed that the eigenvalues of the differentials of such maps along periodic cycles outside the post-critical sets have modulus strictly greater than $1$. In this article, we show that the eigenvalues of the differentials at fixed points are either $0$ or have modulus strictly greater than $1$. This confirms a conjecture proposed by the author in his thesis. We also provide a concrete description of such values in terms of the multiplier of a unicritical polynomial.

References

Matthieu Astorg, Dynamics of post-critically finite maps in higher dimension, Ergodic Theory Dynam. Systems 40 (2020), no. 2, 289–308. MR 4048294, DOI 10.1017/etds.2018.32
Xavier Buff, Adam L. Epstein, and Sarah Koch, Eigenvalues of the Thurston operator, J. Topol. 13 (2020), no. 3, 969–1002. MR 4100123, DOI 10.1112/topo.12145
Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
William Floyd, Daniel Kim, Sarah Koch, Walter Parry, and Edgar Saenz, Realizing polynomial portraits, Preprint, arXiv:2105.10055, 2021.
Thomas Gauthier and Gabriel Vigny, The geometric dynamical northcott and bogomolov properties, Preprint, arXiv:1912.07907, 2019.
Patrick Ingram, Rohini Ramadas, and Joseph H Silverman, Post-critically finite maps on ${\mathbb {P}^n}$ for ${n \ge 2}$ are sparse, Preprint, arXiv:1910.11290, 2019.
Zhuchao Ji, Structure of julia sets for post-critically finite endomorphisms on $\mathbb {P}^2$, Mathematische Annalen (2021), 1–24.
Sarah Koch, Teichmüller theory and critically finite endomorphisms, Adv. Math. 248 (2013), 573–617. MR 3107522, DOI 10.1016/j.aim.2013.08.019
Sarah Colleen Koch, A new link between Teichmuller theory and complex dynamics, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Cornell University. MR 2712719
Van Tu Le, Dynamique des endomorphismes post-critiquement algébriques, Ph.D. Thesis, Université de Toulouse, Université Toulouse III-Paul Sabatier, 2020.
Van Tu Le, Periodic points of post-critically algebraic holomorphic endomorphisms, Ergod. Theory Dyn. Syst. (2020), 1–33.
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
John Milnor, Collected papers of John Milnor. VII, American Mathematical Society, Providence, RI, 2014. Dynamical systems (1984–2012); Edited by Araceli Bonifant. MR 3013944
Feng Rong, The Fatou set for critically finite maps, Proc. Amer. Math. Soc. 136 (2008), no. 10, 3621–3625. MR 2415046, DOI 10.1090/S0002-9939-08-09358-1