## Dynamical systems of correspondences on the projective line I: Moduli spaces and multiplier maps

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- by Rin Gotou
- Conform. Geom. Dyn.
**27**(2023), 294-321 - DOI: https://doi.org/10.1090/ecgd/387
- Published electronically: July 26, 2023
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## Abstract:

We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We define the moduli space $Dyn_{d,e}$ of degree $(d,e)$ correspondences. We construct a family $\rho _c : Dyn_{d,e} \dashrightarrow Dyn_{1,d+e-1}$ of rational maps representation-theoretically. Here we note that $Dyn_{1,d+e-1}$ is identical to the moduli space of the usual dynamical systems of degree $d+e-1$. We show that the moduli space $Dyn_{d,e}$ is rational by using $\rho _c$. Moreover, the multiplier maps for the fixed points factor through $\rho _c$. Furthermore, we show the Woods Hole formulae for correspondences of different degrees are also related by $\rho _c$ and obtain another representation-theoretically simplified form of the formula.## References

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## Bibliographic Information

**Rin Gotou**- Affiliation: Department of Mathematics, Graduate School of Osaka University, Toyonaka, Osaka 560-0043, Japan
- MR Author ID: 1514005
- Email: u661233h@ecs.osaka-u.ac.jp
- Received by editor(s): September 26, 2021
- Received by editor(s) in revised form: October 6, 2022, and April 13, 2023
- Published electronically: July 26, 2023
- Additional Notes: The author was supported by JSPS KAKENHI Grant-in-Aid for Research Fellow JP202122197.
- © Copyright 2023 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**27**(2023), 294-321 - MSC (2020): Primary 37F05; Secondary 12E05, 14C05, 14E08, 37F25, 37P45
- DOI: https://doi.org/10.1090/ecgd/387
- MathSciNet review: 4620883