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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Yu. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations (Springer, Berlin, 1979), Lect. Notes Math. 702.
G. T. Buzzard, S. L. Hruska, and Yu. Ilyashenko, “Kupka-Smale Theorem for Polynomial Automorphisms of ℂ2 and Persistence of Heteroclinic Intersections,” Invent. Math. 161, 45–89 (2005).
E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets (Cambridge Univ. Press, London, 1966), Cambridge Tracts Math. Math. Phys. 56.
A. Glutsyuk, “Nonuniformizable Skew Cylinders: A Counterexample to the Simultaneous Uniformization Problem,” C. R. Acad. Sci. Paris, Sér. 1: Math. 332(3), 209–214 (2001).
A. Glutsyuk, “On Simultaneous Uniformization and Local Nonuniformizability,” C. R. Acad. Sci. Paris, Sér. 1: Math. 334(6), 489–494 (2002).
Yu. S. Il’yashenko, “Fiberings into Analytic Curves,” Mat. Sb. 88(4), 558–577 (1972) [Math. USSR, Sb. 17, 551–569 (1973)].
Yu. S. Il’yashenko, “The Topology of Phase Portraits of Analytic Differential Equations in the Complex Projective Plane,” Tr. Semin. im. I.G. Petrovskogo 4, 83–136 (1978) [Sel. Math. Sov. 5, 141–199 (1986)].
Yu. Ilyashenko, “Centennial History of Hilbert’s 16th Problem,” Bull. Am. Math. Soc. 39(3), 301–354 (2002).
R. Courant and A. Hurwitz, Vorlesungen über allgemeine Funktiontheorie und elliptische Funktionen von Adolf Hurwitz, herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant (Springer, Berlin, 1929; ONTI, Leningrad, 1934).
I. G. Petrovskii and E. M. Landis, “On the Number of Limit Cycles of the Equation dy/dx = P(x,y)/Q(x,y), where P and Q are Polynomials of the Second Degree,” Mat. Sb. 37(2), 209–250 (1955) [AMS Transl., Ser. 2, 10, 177–221 (1958)].
Author information
Authors and Affiliations
Additional information
Original Russian Text © Yu.S. Ilyashenko, 2006, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 254, pp. 196–214.
Rights and permissions
About this article
Cite this article
Ilyashenko, Y.S. Persistence theorems and simultaneous uniformization. Proc. Steklov Inst. Math. 254, 184–200 (2006). https://doi.org/10.1134/S0081543806030096
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0081543806030096