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Flows on surfaces and exchange transformations

Abstract

This research proposal embraces open problems within the following themes: (i) Transitive interval exchange transformations with flips; (ii) Structural stability of smooth vector fields on surfaces; (iii) Asymptotic stability of continuous vector fields. In theme (i), we propose to study the geometry of the connected components of the Rauzy graph for transitive interval exchange maps with flips defined on 4 subintervals; we hope to be able to generalizing this result for the case of n subintervals; Another open problem is to consider other kind of induction called bilateral induction. In theme (ii), we propose to build examples of quasiminimal flows persistent under Cr twist perturbations of the original vector field. In theme (iii), we are going to approach the existence of robust trapping regions for asymptotically stable continuous vector fields.This research proposal embraces open problems within the following themes: (i) Transitive interval exchange transformations with flips; (ii) Structural stability of smooth vector fields on surfaces; (iii) Asymptotic stability of continuous vector fields. In theme (i), we propose to study the geometry of the connected components of the Rauzy graph for transitive interval exchange maps with flips defined on 4 subintervals; we hope to be able to generalizing this result for the case of n subintervals; Another open problem is to consider other kind of induction called bilateral induction. In theme (ii), we propose to build examples of quasiminimal flows persistent under Cr twist perturbations of the original vector field. In theme (iii), we are going to approach the existence of robust trapping regions for asymptotically stable continuous vector fields. (AU)

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Scientific publications (4)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
NOGUEIRA, ARNALDO; PIRES, BENITO. Dynamics of piecewise contractions of the interval. Ergodic Theory and Dynamical Systems, v. 35, n. 7, p. 2198-2215, OCT 2015. Web of Science Citations: 5.
PIRES, BENITO; RABANAL, ROLAND. VECTOR FIELDS WHOSE LINEARISATION IS HURWITZ ALMOST EVERYWHERE. Proceedings of the American Mathematical Society, v. 142, n. 9, p. 3117-3128, SEP 2014. Web of Science Citations: 2.
NOGUEIRA, ARNALDO; PIRES, BENITO; TROUBETZKOY, SERGE. Orbit structure of interval exchange transformations with flip. Nonlinearity, v. 26, n. 2, p. 525-537, FEB 2013. Web of Science Citations: 7.
PIRES, BENITO; TEIXEIRA, MARCO ANTONIO. On global linearization of planar involutions. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 43, n. 4, p. 637-653, DEC 2012. Web of Science Citations: 0.

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