Geodesic
Persistence Theorems and Simultaneous Uniformization
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear analytic differential equations, Tome 254 (2006), pp. 196-214.

Voir la notice de l'article provenant de la source Math-Net.Ru

Among 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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Yu. S. Ilyashenko. Persistence Theorems and Simultaneous Uniformization. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear analytic differential equations, Tome 254 (2006), pp. 196-214. http://geodesic.mathdoc.fr/item/TM_2006_254_a8/

[1] Bibikov Yu.N., Local theory of nonlinear analytic ordinary differential equations, Lect. Notes Math., 702, Springer, Berlin, 1979 | MR | Zbl

[2] Buzzard G.T., Hruska S.L., Ilyashenko Yu., “Kupka–Smale theorem for polynomial automorphisms of $\mathbb C^2$ and persistence of heteroclinic intersections”, Invent. math., 161 (2005), 45–89 | DOI | MR | Zbl

[3] Collingwood E.F., Lohwater A.J., The theory of cluster sets, Cambridge Tracts Math. and Math. Phys., 56, Cambridge Univ. Press, London, 1966 | MR | Zbl

[4] Glutsyuk A., “Nonuniformizable skew cylinders. A counterexample to the simultaneous uniformization problem”, C. r. Acad. sci. Paris Sér. 1: Math. yr 2001, 332:3, 209–214 | MR | Zbl

[5] Glutsyuk A., “On simultaneous uniformization and local nonuniformizability”, C. r. Acad. sci. Paris Sér. 1: Math., 334:6 (2002), 489–494 | MR | Zbl

[6] Ilyashenko Yu.S., “Sloeniya na analiticheskie krivye”, Mat. sb., 88:4 (1972), 558–577 | Zbl

[7] Ilyashenko Yu.S., “Topologiya fazovykh portretov analiticheskikh differentsialnykh uravnenii na kompleksnoi proektivnoi ploskosti”, Tr. sem. im. I.G. Petrovskogo, 4 (1978), 83–136 | MR | Zbl

[8] Ilyashenko Yu., “Centennial history of Hilbert's 16th problem”, Bull. Amer. Math. Soc., 39:3 (2002), 301–354 | DOI | MR | Zbl

[9] Kurant R., Geometricheskaya teoriya funktsii kompleksnoi peremennoi, Gostekhteorizdat, M.; L., 1934

[10] Petrovskii I.G., Landis E.M., “O chisle predelnykh tsiklov uravneniya $dy/dx=P(x,y)/Q(x,y)$, gde $P$ i $Q$ — mnogochleny 2-i stepeni”, Mat. sb., 37:2 (1955), 209–250 | MR | Zbl