Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 38 No. 1 (2020): Revista Integración, temas de matemáticas
Research and Innovation Articles

Examples of codification of the dynamics of a rational function into a topological tree

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  • Laura Cano,
  • Patricia Domínguez,
  • Josué Vázquez
Laura Cano
Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias Físico-Matemáticas, Puebla, México.
Patricia Domínguez
Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias Físico-Matemáticas, Puebla, México.
Josué Vázquez
Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias Físico-Matemáticas, Puebla, México.
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DOI: https://doi.org/10.18273/revint.v38n1-2020001

Published 2020-01-15

Keywords

  • Graph,
  • Holomorphic Dynamics,
  • Herman rings,
  • Quasi-conformal surgery

How to Cite

Cano, L., Domínguez, P., & Vázquez, J. (2020). Examples of codification of the dynamics of a rational function into a topological tree. Revista Integración, Temas De matemáticas, 38(1), 1–14. https://doi.org/10.18273/revint.v38n1-2020001

Copyright (c) 2020 Revista Integración, temas de matemáticas

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

In 1736 L. Euler gave solution to the famous Seven Bridges of Königsberg problem, considerin a graph consisting of nodes representing the landmasses and arcs representing the bridges. This problem is a referent of how to codify the information given of a problem into a simpler and richer structure. In the case of the Dynamics of rational functions, Shishikura in [5] explores this idea in the context of rational functions, and he stated a connection between a certain kind of topological tree with a p-cycle of Herman rings associated to a rational function. In this work we develop some examples of realizable configurations for rational functions, two of them sketched in [5], and an example of a non realizable configuration which we modify in order to be realizable.

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References

[1] Beardon A.F., Iteration of Rational Functions: Complex Analytic Dynamical Systems, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.

[2] Fatou P., “Sur les équations fonctionnelles”, Bull. Sci. Math. France 47 (1919), 161–271.

[3] Herman M.R., “Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann”, Bull. Soc. Math. France 112 (1984), No. 1, 93–142.

[4] Julia G., “Mémoire sur l’itération des fonctions rationnelles”, J. Math. Pures Appl. 1 (1918), 47–246.

[5] Shishikura M., “On the quasiconformal surgery of rational functions”, Ann. Sci. École Norm. Sup. (4) 20 (1987), No. 1, 1–29.

[6] Shishikura M., “Trees associated with the configuration of Herman rings”, Ergodic Theory Dynam. Systems 9 (1989), No. 3, 543–560.

[7] Shishikura M., “A new tree associated with Herman rings”, Complex dynamics and related fields 1269 (2002), 74–92.

[8] Sullivan D., “Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia Problem on wandering domains”, Ann. of Math. (2), 122 (1985), No. 3, 401–418.