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Dynamics of hyperbolic correspondences

Part of: Complex dynamical systems

Published online by Cambridge University Press:  04 May 2021

CARLOS SIQUEIRA Open the ORCID record for CARLOS SIQUEIRA [Opens in a new window]
CARLOS SIQUEIRA*
Affiliation:
Department of Mathematics, Federal University of Bahia, SalvadorCEP 40170115, Brazil
*
e-mail: [email protected]
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Abstract

This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

Keywords

complex dynamicsholomorphic correspondencesrigidity

MSC classification

Primary: 37F15: Expanding maps; hyperbolicity; structural stability 37F05: Relations and correspondences
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2661 - 2692
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Bousch, T.. Sur quelques problèmes de dynamique holomorphe. PhD Thesis, Université d’Orsay, 1992.Google Scholar
Bullett, S.. Critically finite correspondences and subgroups of the modular group. Proc. Lond. Math. Soc. (3) 65(2) (1992), 423–448.10.1112/plms/s3-65.2.423CrossRefGoogle Scholar
Bullett, S. and Freiberger, M.. Holomorphic correspondences mating Chebyshev-like maps with Hecke groups. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1057–1090.10.1017/S0143385704001026CrossRefGoogle Scholar
Bullett, S., Lomonaco, L. and Siqueira, C.. Correspondences in complex dynamics. New Trends in One-Dimensional Dynamics. Ed. Pacifico, M. J. and Guarino, P.. Springer, Cham, 2019, pp. 51–75.10.1007/978-3-030-16833-9_5CrossRefGoogle Scholar
Bullett, S. and Penrose, C.. Mating quadratic maps with the modular group. Invent. Math. 115 (1994), 483–511.10.1007/BF01231770CrossRefGoogle Scholar
Bullett, S. and Penrose, C.. Regular and limit sets for holomorphic correspondences. Fund. Math. 167 (2001), 111–171.10.4064/fm167-2-2CrossRefGoogle Scholar
Douady, A. and Hubbard, J. H.. Étude dynamique des polynômes complexes. Partie I. Publ. Math. d’Orsay 84(2) (1984), 75.Google Scholar
Douady, A. and Hubbard, J. H.. Étude dynamique des polynômes complexes. Partie II (with the collaboration of P. Lavaurs, Tan Lei and P. Sentenac). Publ. Math. d’Orsay 85(4) (1985).Google Scholar
Dujardin, R. and Lyubich, M.. Stability and bifurcations for dissipative polynomial automorphisms of ${\mathbb{C}}^2$ . Invent. Math. 200(2) (2015), 439–511.10.1007/s00222-014-0535-yCrossRefGoogle Scholar
Mané, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. 16(2) (1983), 193–217.CrossRefGoogle Scholar
Münzner, H. F. and Rasch, H.-M.. Iterated algebraic functions and functional equations. Internat. J. Bifur. Chaos 1(4) (1991), 803–822.CrossRefGoogle Scholar
Przytycki, F. and Urbanski, M.. Conformal Fractals: Ergodic Theory Methods, 1st edn. Cambridge University Press, New York, 2010.10.1017/CBO9781139193184CrossRefGoogle Scholar
Siqueira, C.. Dynamics of holomorphic correspondences. PhD Thesis, University of São Paulo, ICMC, Digital Library USP, 2015.Google Scholar
Siqueira, C. and Smania, D.. Holomorphic motions for unicritical correspondences. Nonlinearity 30(8) (2017), 3104.10.1088/1361-6544/aa7736CrossRefGoogle Scholar
Sumi, H.. On dynamics of hyperbolic rational semigroups. J. Math. Kyoto Univ. 37(4) (1998), 717–733.Google Scholar
Urbański, M.. Geometric rigidity for class $\boldsymbol{\mathcal{S}}$ of transcendental meromorphic functions whose Julia sets are Jordan curves. Proc. Am. Math. Soc. 137(11) (2009), 3733–3739.CrossRefGoogle Scholar