On Lattès Maps

By John W. Milnor. 2004

154 citations recorded by the source corpus.

An exposition of the 1918 paper of Lattès, together with its historical antecedents, and its modern formulations and applications. 1. The Lattès paper. 2. Finite Quotients of Affine Maps 3. A Cyclic Group Action on C/Λ . 4. Flat Orbifold Metrics 5. Classification 6. Lattès Maps before Lattès 7. More Recent Developments 8. Examples References §1. The Lattès paper. In 1918, some months before his death of typhoid fever, Samuel Lattès published a brief paper describing an extremely interesting class of rational maps. Similar examples had been described by Schröder almost fifty years earlier (see §6), but Lattès’ name has become firmly attached to these maps, which play a basic role as exceptional examples in the holomorphic dynamics literature. His starting point was the “Poincaré function” θ : C → Ĉ associated with a repelling fixed point z0 = f(z0) of a rational function f : Ĉ → Ĉ . This can be described as the inverse of the Kœnigs linearization around z0 , extended to a globally defined meromorphic function.1 Assuming for convenience that z0 6=∞ , it is characterized by the identity

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

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Citations in the corpus

• Une caractérisation des endomorphismes de Lattès par leur mesure de Green (2005, unknown): “Ils sont traditionnellement appelés «exemples de Lattès» et font l’objet d’une étude détaillée dans l’article de revue de Milnor [22].”
• Bifurcation currents in holomorphic dynamics on (2007, background): “Of course the behaviour of ddgFλ + ω is very far from the desidered regularity, but there is at least one case in which holomorphic dynamics may give interesting examples: let M = P1 and {fλ}λ∈X be a family of flexible Lattès maps (see [21], Ch.”
• Prym Varieties and Teichmüller Curves (2006, unknown): “4], [Mil].”
• Bifurcation currents in holomorphic dynamics on ${\bf P}^k$ (2005, background): “f0 is the map associated to an imaginary quadratic number field, see [22] Lemma 5.”