Skip to Main Content

Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Fixed points of Koch maps
HTML articles powered by AMS MathViewer

by Van Tu Le PDF
Conform. Geom. Dyn. 26 (2022), 10-33 Request permission


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.
  • 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
Similar Articles
  • Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2020): 32H50, 37F99
  • Retrieve articles in all journals with MSC (2020): 32H50, 37F99
Additional Information
  • Van Tu Le
  • Affiliation: Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1 - 00133 Roma, Italy
  • ORCID: 0000-0002-4534-6254
  • Email:
  • Received by editor(s): July 15, 2021
  • Received by editor(s) in revised form: January 12, 2022
  • Published electronically: April 25, 2022
  • Additional Notes: The author was supported by the PhD fellowship of Centre International de Mathématiques et d’Informatique de Toulouse (CIMI) and MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
  • © Copyright 2022 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 26 (2022), 10-33
  • MSC (2020): Primary 32H50, 37F99
  • DOI:
  • MathSciNet review: 4412274