What question did this study set out to answer?

The aim is to show that the moduli space of rational maps of degree at least 2 is Carathéodory hyperbolic when the map is not flexible Lattès.

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February 22, 2026

The moduli space of a rational map is Carathéodory hyperbolic

Key Points

The aim is to show that the moduli space of rational maps of degree at least 2 is Carathéodory hyperbolic when the map is not flexible Lattès.
Introduced moduli space ${M}_f$ of rational maps of degree $d \geq 2$
Constructed normal affine variety $X_f$ of dimension $2d - 2$
Established a holomorphic injection from ${M}_f$ to $X_f$
Proved that ${M}_f$ is Carathéodory hyperbolic for non-flexible Lattès maps
Showed that $i({M}_f)$ is precompact in $X_f$ for any rational map in this category
Provided a construction for $X_f$ when $d \geq 4$ as the normalization of the image of the reciprocal multiplier spectrum morphism

Abstract

Abstract Let f be a rational map of degree d 2. The moduli space Mf, introduced by McMullen and Sullivan, is a complex analytic space consisting of all quasiconformal conjugacy classes of f. For f that is not flexible Lattès, we show that there is a normal affine variety Xf of dimension 2d-2 and a holomorphic injection i: Mf Xf such that i (Mf) is precompact in Xf. In particular, Mf is Carathéodory hyperbolic (that is, bounded holomorphic functions separate points in Mf), provided that f is not flexible Lattès. This solves a conjecture of McMullen. When d 4, we give a concrete construction of Xf as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism.

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The moduli space of a rational map is Carathéodory hyperbolic

Key Points

Abstract

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