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Non-periodic bifurcations for surface diffeomorphisms


Authors: Vanderlei Horita, Nivaldo Muniz and Paulo Rogério Sabini
Journal: Trans. Amer. Math. Soc. 367 (2015), 8279-8300
MSC (2010): Primary 37G25, 37D25, 37G35
DOI: https://doi.org/10.1090/tran/6168
Published electronically: September 1, 2015
MathSciNet review: 3403055
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Abstract: We prove that a ``positive probability'' subset of the boundary of the set of hyperbolic (Axiom A) surface diffeomorphisms with no cycles $ \mathcal {H}$ is constituted by Kupka-Smale diffeomorphisms: all periodic points are hyperbolic and their invariant manifolds intersect transversally. Lack of hyperbolicity arises from the presence of a tangency between a stable manifold and an unstable manifold, one of which is not associated to a periodic point. All these diffeomorphisms that we construct lie on the boundary of the same connected component of $ \mathcal {H}$.


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Additional Information

Vanderlei Horita
Affiliation: Departamento de Matemática, Universidade Estadual Paulista, Rua Cristóvão Colombo 2265, 15054-000 S. J. Rio Preto, SP, Brazil
Email: vhorita@ibilce.unesp.br

Nivaldo Muniz
Affiliation: Departamento de Matemática, Universidade Federal do Maranhão, Avenida dos Portugueses, S/N, 65000-000 São Luís, MA, Brazil
Email: nivaldomuniz@gmail.com

Paulo Rogério Sabini
Affiliation: Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier, 524, 20550-900 Rio de Janeiro, RJ, Brazil

DOI: https://doi.org/10.1090/tran/6168
Received by editor(s): December 16, 2011
Published electronically: September 1, 2015
Additional Notes: This work was partially supported by CAPES, CNPq, FAPESP, INCTMat and PRONEX
Article copyright: © Copyright 2015 American Mathematical Society