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March 2019, 24(3): 1143-1173. doi: 10.3934/dcdsb.2019010

## A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere

 Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain

* Corresponding author: V. Jiménez López

Received  September 2017 Revised  March 2018 Published  January 2019

Fund Project: This work has been partially supported by Ministerio de Economía y Competitividad, Spain, grant MTM2014-52920-P. The first author has been also supported by Fundación Séneca by means of the program "Contratos Predoctorales de Formación del Personal Investigador", grant 18910/FPI/13

In [15], V. Jiménez López and J. Llibre characterized, up to homeomorphism, the $\omega$-limit sets of analytic vector fields on the sphere and the projective plane. The authors also studied the same problem for open subsets of these surfaces.

Unfortunately, an essential lemma in their programme for general surfaces has a gap. Although the proof of this lemma can be amended in the case of the sphere, the plane, the projective plane and the projective plane minus one point (and therefore the characterizations for these surfaces in [15] are correct), the lemma is not generally true, see [6].

Consequently, the topological characterization for analytic vector fields on open subsets of the sphere and the projective plane is still pending. In this paper, we close this problem in the case of open subsets of the sphere.

Citation: José Ginés Espín Buendía, Víctor Jiménez Lopéz. A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1143-1173. doi: 10.3934/dcdsb.2019010
##### References:
 [1] V. W. Adkisson, Plane Peanian continua with unique maps on the sphere and in the plane, Trans. Amer. Math. Soc., 44 (1938), 58-67. doi: 10.1090/S0002-9947-1938-1501962-9. [2] V. W. Adkisson and S. Mac Lane, Extending maps of plane Peano continua, Duke Math. J., 6 (1940), 216-228. doi: 10.1215/S0012-7094-40-00616-0. [3] S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, Translations of Mathematical Monographs, 153, American Mathematical Society, Providence, 1996. [4] N. P. Bhatia and G. P. Szegő, Stability Theory of Dynamical Systems, Reprint of the 1970 original, Springer-Verlag, Berlin, 2002. [5] J. G. Espín Buendía and V. Jiménez López, Analytic plane sets are locally $2n$-stars: A dynamically based proof, Appl. Math. Inf. Sci., 9 (2015), 2355-2360. [6] J. G. Espín Buendía and V. Jiménez López, Some remarks on the $\omega$-limit sets for plane, sphere and projective plane analytic flows, Qual. Theory Dyn. Syst., 16 (2017), 293-298. doi: 10.1007/s12346-016-0192-1. [7] H. M. Gehman, On extending a continuous $(1\text{-}1)$ correspondence of two plane continuous curves to a correspondence of their planes, Trans. Amer. Math. Soc., 28 (1926), 252-265. doi: 10.2307/1989114. [8] H. M. Gehman, On extending a continuous (1-1) correspondence. Ⅱ, Trans. Amer. Math. Soc., 31 (1929), 241-252. doi: 10.2307/1989382. [9] H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. (2), 68 (1958), 460-472. doi: 10.2307/1970257. [10] M. J. Greenberg, Lectures on Algebraic Topology, W. A. Benjamin, New York-Amsterdam, 1967. [11] C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory Dynam. Systems, 6 (1986), 17-44. doi: 10.1017/S0143385700003278. [12] F. Harary, Graph Theory, Addison-Wesley, Reading-Menlo Park-London, 1969. [13] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. [14] W. Huebsch and M. Morse, Diffeomorphisms of manifolds, Rend. Circ. Mat. Palermo (2), 11 (1962), 291-318. doi: 10.1007/BF02843877. [15] V. Jiménez López and J. Llibre, A topological characterization of the $\omega$-limit sets for analytic flows on the plane, the sphere and the projective plane, Adv. Math., 216 (2007), 677-710. doi: 10.1016/j.aim.2007.06.007. [16] V. Jiménez López and G. Soler López, A characterization of $\omega$-limit sets for continuous flows on surfaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 9 (2006), 515-521. [17] U. Karimov. D. Repovš, W. Rosicki and A. Zastrow, On two-dimensional planar compacta not homotopically equivalent to any one-dimensional compactum, Topology Appl., 153 (2005), 284-293. doi: 10.1016/j.topol.2004.02.020. [18] J. R. Kline, Concerning sense on closed curves in non-metrical plane analysis situs, Ann. of Math. (2), 21 (1919), 113-119. doi: 10.2307/2007227. [19] E. A. Knobelauch, Extensions of homeomorphisms, Duke Math. J., 16 (1949), 247-259. doi: 10.1215/S0012-7094-49-01624-5. [20] K. Kuratowski, Topology. Volume Ⅱ, Academic Press, New York, 1968. [21] S. Mac Lane and V. W. Adkisson, Extensions of homeomorphisms on the sphere, in Lectures in Topology (ed. C. Chevalley), University of Michigan Press, (1941), 223–236 [22] S. Mac Lane and V. W. Adkisson, Fixed points and the extension of the homeomorphisms of a Planar Graph, Amer. J. Math., 60 (1938), 611-639. doi: 10.2307/2371602. [23] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975. [24] W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, New York, 1987. [25] R. A. Smith and E. S. Thomas, Transitive flows on two-dimensional manifolds, J. London Math. Soc. (2), 37 (1988), 569-576. doi: 10.1112/jlms/s2-37.3.569. [26] K. Spindler, Abstract Algebra with Applications. Volume Ⅱ. Rings and Fields, Marcel Dekker, New York, 1994. [27] D. Sullivan, Combinatorial invariants of analytic spaces, in Proceedings of Liverpool Singularities--Symposium, Ⅰ (1969/70) (ed. C. T. C. Wall), Springer, (1971), 165–168. [28] R. Vinograd, On the limit behavior of an unbounded integral curve (Russian), Moskov. Gos. Univ. Uč. Zap. 155, Mat., 5 (1952), 94-136. [29] J. H. C. Whitehead, Manifolds with transverse fields in euclidean space, Ann. of Math., 73 (1961), 154-212. doi: 10.2307/1970286. [30] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. doi: 10.1090/S0002-9947-1934-1501735-3. [31] H. Whitney and F. Bruhat, Quelques propriétés fondamentales des ensembles analytiques-réels, (French) [Some fundamental properties of real analytic sets], Comment. Math. Helv., 33 (1959), 132-160. doi: 10.1007/BF02565913. [32] S. Willard, General Topology, Dover Publications, Mineola, 1970.

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##### References:
 [1] V. W. Adkisson, Plane Peanian continua with unique maps on the sphere and in the plane, Trans. Amer. Math. Soc., 44 (1938), 58-67. doi: 10.1090/S0002-9947-1938-1501962-9. [2] V. W. Adkisson and S. Mac Lane, Extending maps of plane Peano continua, Duke Math. J., 6 (1940), 216-228. doi: 10.1215/S0012-7094-40-00616-0. [3] S. Kh. Aranson, G. R. Belitsky and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, Translations of Mathematical Monographs, 153, American Mathematical Society, Providence, 1996. [4] N. P. Bhatia and G. P. Szegő, Stability Theory of Dynamical Systems, Reprint of the 1970 original, Springer-Verlag, Berlin, 2002. [5] J. G. Espín Buendía and V. Jiménez López, Analytic plane sets are locally $2n$-stars: A dynamically based proof, Appl. Math. Inf. Sci., 9 (2015), 2355-2360. [6] J. G. Espín Buendía and V. Jiménez López, Some remarks on the $\omega$-limit sets for plane, sphere and projective plane analytic flows, Qual. Theory Dyn. Syst., 16 (2017), 293-298. doi: 10.1007/s12346-016-0192-1. [7] H. M. Gehman, On extending a continuous $(1\text{-}1)$ correspondence of two plane continuous curves to a correspondence of their planes, Trans. Amer. Math. Soc., 28 (1926), 252-265. doi: 10.2307/1989114. [8] H. M. Gehman, On extending a continuous (1-1) correspondence. Ⅱ, Trans. Amer. Math. Soc., 31 (1929), 241-252. doi: 10.2307/1989382. [9] H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. (2), 68 (1958), 460-472. doi: 10.2307/1970257. [10] M. J. Greenberg, Lectures on Algebraic Topology, W. A. Benjamin, New York-Amsterdam, 1967. [11] C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory Dynam. Systems, 6 (1986), 17-44. doi: 10.1017/S0143385700003278. [12] F. Harary, Graph Theory, Addison-Wesley, Reading-Menlo Park-London, 1969. [13] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. [14] W. Huebsch and M. Morse, Diffeomorphisms of manifolds, Rend. Circ. Mat. Palermo (2), 11 (1962), 291-318. doi: 10.1007/BF02843877. [15] V. Jiménez López and J. Llibre, A topological characterization of the $\omega$-limit sets for analytic flows on the plane, the sphere and the projective plane, Adv. Math., 216 (2007), 677-710. doi: 10.1016/j.aim.2007.06.007. [16] V. Jiménez López and G. Soler López, A characterization of $\omega$-limit sets for continuous flows on surfaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 9 (2006), 515-521. [17] U. Karimov. D. Repovš, W. Rosicki and A. Zastrow, On two-dimensional planar compacta not homotopically equivalent to any one-dimensional compactum, Topology Appl., 153 (2005), 284-293. doi: 10.1016/j.topol.2004.02.020. [18] J. R. Kline, Concerning sense on closed curves in non-metrical plane analysis situs, Ann. of Math. (2), 21 (1919), 113-119. doi: 10.2307/2007227. [19] E. A. Knobelauch, Extensions of homeomorphisms, Duke Math. J., 16 (1949), 247-259. doi: 10.1215/S0012-7094-49-01624-5. [20] K. Kuratowski, Topology. Volume Ⅱ, Academic Press, New York, 1968. [21] S. Mac Lane and V. W. Adkisson, Extensions of homeomorphisms on the sphere, in Lectures in Topology (ed. C. Chevalley), University of Michigan Press, (1941), 223–236 [22] S. Mac Lane and V. W. Adkisson, Fixed points and the extension of the homeomorphisms of a Planar Graph, Amer. J. Math., 60 (1938), 611-639. doi: 10.2307/2371602. [23] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975. [24] W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, New York, 1987. [25] R. A. Smith and E. S. Thomas, Transitive flows on two-dimensional manifolds, J. London Math. Soc. (2), 37 (1988), 569-576. doi: 10.1112/jlms/s2-37.3.569. [26] K. Spindler, Abstract Algebra with Applications. Volume Ⅱ. Rings and Fields, Marcel Dekker, New York, 1994. [27] D. Sullivan, Combinatorial invariants of analytic spaces, in Proceedings of Liverpool Singularities--Symposium, Ⅰ (1969/70) (ed. C. T. C. Wall), Springer, (1971), 165–168. [28] R. Vinograd, On the limit behavior of an unbounded integral curve (Russian), Moskov. Gos. Univ. Uč. Zap. 155, Mat., 5 (1952), 94-136. [29] J. H. C. Whitehead, Manifolds with transverse fields in euclidean space, Ann. of Math., 73 (1961), 154-212. doi: 10.2307/1970286. [30] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. doi: 10.1090/S0002-9947-1934-1501735-3. [31] H. Whitney and F. Bruhat, Quelques propriétés fondamentales des ensembles analytiques-réels, (French) [Some fundamental properties of real analytic sets], Comment. Math. Helv., 33 (1959), 132-160. doi: 10.1007/BF02565913. [32] S. Willard, General Topology, Dover Publications, Mineola, 1970.
The different parts of a shrub
The $5$-cusped hypocycloid (left) and the $8$-cusped hypocycloid with some arcs (right)
In the positive (counterclockwise) sense: a node with a flexible (F) leaf, a rigid (R) sprig, a bland (B) leaf and a rigid leaf (left), and a leaf with flexible (f), bland (b), rigid (r) and bland nodes (right)
From left to right, the shrubs $A$, $A'$, $A''$ and $A^*$.
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