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Slices of parameter space for meromorphic maps with two asymptotic values

Part of: Holomorphic functions of several complex variables Entire and meromorphic functions, and related topics Riemann surfaces Complex dynamical systems

Published online by Cambridge University Press:  18 October 2021

TAO CHEN Open the ORCID record for TAO CHEN [Opens in a new window] ,
YUNPING JIANG Open the ORCID record for YUNPING JIANG [Opens in a new window]  and
LINDA KEEN Open the ORCID record for LINDA KEEN [Opens in a new window]
TAO CHEN
Affiliation:
Department of Mathematics, Engineering and Computer Science, Laguardia Community College, CUNY, 31-10 Thomson Ave. Long Island City, NY 11101, USA (e-mail: tchen@lagcc.cuny.edu)
YUNPING JIANG*
Affiliation:
Department of Mathematics, Queens College of CUNY, Flushing, NY 11367, USA
LINDA KEEN
Affiliation:
Department of Mathematics, CUNY Graduate School, New York, NY 10016, USA (e-mail: LKeen@gc.cuny.edu, linda.keenbrezin@gmail.com)
*
yunping.jiang@qc.cuny.edu
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Abstract

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work by Goldberg and Keen [The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math. 101(2) (1990), 335–372] on degree two rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.

Keywords

slices of parameter spacemeromorphic functionsshift locusvirtual center parametertransversality

MSC classification

Primary: 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems 37F20: Combinatorics and topology 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30F30: Differentials on Riemann surfaces 30D30: Meromorphic functions, general theory 32A20: Meromorphic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 1 , January 2023 , pp. 99 - 139
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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