Finite and Infinite Degree Thurston Maps with a Small Postsingular Set

Centrala begrepp

Thurston maps with at most three singular values and four postsingular values can be characterized by the existence of weakly degenerate Levy fixed curves. The Hurwitz classes of such Thurston maps exhibit interesting properties, including the existence of infinitely many realized and obstructed maps.

Sammanfattning

The paper develops the theory of Thurston maps that are defined everywhere on the topological sphere S^2 with a possible exception of a single essential singularity. The authors establish an analog of Thurston's celebrated characterization theorem for a broad class of such Thurston maps having four postsingular values.

Key highlights:

1. The authors show that for Thurston maps with at most three singular values and four postsingular values, it is sufficient to consider Levy fixed curves to determine whether the map is realized or obstructed. Specifically, the map is realized if and only if it has no weakly degenerate Levy fixed curve.
2. The authors analyze the dynamics of the corresponding pullback maps defined on the Teichmüller space, which allows them to derive various properties of Hurwitz classes of the Thurston maps in this family.
3. The authors prove that if a Thurston map in this family is not totally unobstructed, then its Hurwitz class contains infinitely many pairwise combinatorially inequivalent obstructed Thurston maps. Moreover, if the map has infinite degree, then its Hurwitz class contains infinitely many pairwise combinatorially inequivalent realized Thurston maps.
4. As an application, the authors show that the parameter space of a transcendental meromorphic map with at most three singular values contains infinitely many pairwise (topologically or conformally) non-conjugate postsingularly finite maps with four postsingular values.

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Viktiga insikter från

Finite and infinite degree Thurston maps with a small postsingular set

by Nikolai Proc... på arxiv.org 10-03-2024

https://arxiv.org/pdf/2410.01146.pdf

Djupare frågor

What are some potential applications of the results in this paper beyond the study of Thurston maps?

The results presented in this paper have significant implications beyond the immediate study of Thurston maps. One potential application lies in the classification of dynamical systems, particularly in understanding the behavior of meromorphic maps in complex dynamics. The characterization of postsingularly finite (psf) maps can lead to insights into the structure of parameter spaces for meromorphic functions, which is crucial for understanding the dynamics of entire functions and their bifurcations. Additionally, the techniques developed for analyzing pullback maps and their dynamics can be applied to other areas of complex analysis, such as the study of holomorphic self-maps of Riemann surfaces and the dynamics of rational maps. The results may also contribute to the broader field of algebraic geometry, particularly in the context of moduli spaces, where understanding the equivalence classes of maps can inform the classification of algebraic varieties.

How can the techniques developed in this paper be extended to Thurston maps with a larger number of postsingular values?

The techniques developed in this paper, particularly those involving the analysis of pullback maps and the dynamics on Teichmüller and moduli spaces, can be extended to Thurston maps with a larger number of postsingular values by adapting the framework of the analysis to accommodate additional complexity. For instance, one could investigate the dynamics of pullback maps in the context of higher-dimensional Teichmüller spaces or moduli spaces that correspond to maps with more than four postsingular values. This would involve developing new tools to handle the increased combinatorial complexity and potential obstructions that arise from additional postsingular values. Furthermore, the study of Levy cycles and their generalizations could provide a pathway to understanding the realizability of these more complex Thurston maps. By systematically exploring the relationships between the dynamics of these maps and their combinatorial properties, researchers could potentially uncover new results that extend the characterization theorems established for maps with fewer postsingular values.

Are there other families of Thurston maps that exhibit similar properties to the ones studied in this paper?

Yes, there are other families of Thurston maps that exhibit similar properties to those studied in this paper. For instance, families of Thurston maps with specific configurations of singular values and postsingular sets can be analyzed using analogous techniques. Maps that are postsingularly finite with three or more singular values, or those that exhibit certain symmetry properties, may also yield interesting results when studied through the lens of the methods developed in this paper. Additionally, the concept of Hurwitz equivalence can be applied to various families of Thurston maps, allowing for the exploration of their combinatorial structures and dynamics. The results regarding the existence of Levy fixed curves and the characterization of realizability can be extended to these families, potentially leading to new insights into their dynamics and classification. Overall, the framework established in this paper provides a robust foundation for exploring a wider array of Thurston maps and their properties.