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February & March  2004, 11(2&3): 615-619. doi: 10.3934/dcds.2004.11.615

A note on periodic orbits for singular-hyperbolic flows

Carlos Arnoldo Morales 1,

1. 

Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil

Received  May 2003 Revised  February 2004 Published  June 2004

A singular-hyperbolic set for flows is a partially hyperbolic set with singularities (hyperbolic ones) and volume expanding central direction [7]. Several properties of hyperbolic systems have been conjectured for the singular-hyperbolic sets [8, p. 335]. Related to these conjectures we shall prove the existence of transitive, isolated, singular-hyperbolic set without periodic orbits on any $3$-manifold. In particular, the periodic orbits are not necessarily dense in the limit set of a isolated singular-hyperbolic set.
Keywords: Flow, partially hyperbolic set., periodic orbit.
Mathematics Subject Classification: Primary: 37D30; Secondary: 37C2.
Citation: Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
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