Persistence theorems and simultaneous uniformization |
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Authors: | Yu. S. Ilyashenko |
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Affiliation: | (1) Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119992, Russia;(2) Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia;(3) Independent University of Moscow, Bol’shoi Vlas’evskii per. 11, Moscow, Russia;(4) Cornell University, Ithaca, NY, USA |
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Abstract: | One of the most intriguing problems in the theory of foliations by analytic curves is that of the persistence of complex limit cycles of a polynomial vector field, as well as related problems concerning the persistence of identity cycles and saddle connections and the global extendability of the Poincaré map. It is proved that all these persistence problems have positive solutions for any foliation admitting a simultaneous uniformization of leaves. The latter means that there exists a uniformization of leaves that analytically depends on the initial condition and satisfies certain additional assumptions, called continuity and boundedness. Thus, the results obtained are conditional, but they establish a relation between very different properties of foliations. |
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