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Dynamical Systems of Correspondences on the Projective Line II: Degrees of Multiplier Maps
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abstract
This paper is a sequel of arXiv:2109.06394. In this paper, we consider a kind of inverse problem of multipliers. The problem is to count number of isospectral correspondences, correspondences which has the same combination of multipliers. We give a primitive explicit upper bound. In particular, for a generic rational map of degree $d$, there are at most $O(d^{10d})$ rational maps with the same combination of multipliers for the fixed points and the 3-periodic points. This paper also includes two proofs of a correction in the errata of a Hutz-Tepper's result, which states that the multipliers of the fixed and 2-periodic points determines generic cubic morphism uniquely. One is done by proceeding the computation in Hutz-Tepper's proof. The other is done by more explicit computation with the help of invariant theory.
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Cited by 1 Pith paper
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Moduli spaces of polynomial maps and multipliers at small cycles
For every degree d at least 2, the period-1 and period-2 multipliers induce a finite birational morphism from the moduli space of degree-d polynomials onto its image.
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