On Lattès Maps
نویسندگان
چکیده
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
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