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A sectional-Anosov connecting lemma

Published online by Cambridge University Press:  21 July 2009

S. BAUTISTA  and
C. MORALES
S. BAUTISTA
Affiliation:
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogota, Colombia (email: sbautistad@unal.edu.co)
C. MORALES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970, Rio de Janeiro, Brazil (email: morales@impa.br)
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Abstract

The Anosov flows on compact manifolds M satisfy the following property: if p,q are points such that for all positive ϵ there is a trajectory from a point ϵ-close to p to a point ϵ-close to q, then there is a point whose α-limit set is that of p and whose ω-limit set is that of q. Here we give a version of this property for sectional-Anosov flows, namely, vector fields inwardly transverse to the boundary whose maximal invariant set is sectional-hyperbolic. Indeed, if in addition M is three-dimensional and p has non-singular α-limit set, then there is a point whose α-limit set is that of p and whose ω-limit set is either a singularity or that of q.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 339 - 359
Copyright
Copyright © Cambridge University Press 2009

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