A note on periodic orbits for singular-hyperbolic flows

Pages: 615 - 619, Volume 11, Issue 2/3, September/October 2004      doi:10.3934/dcds.2004.11.615

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Carlos Arnoldo Morales - Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68530, CEP 21945-970, Rio de Janeiro, Brazil (email)

Abstract: 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, periodic orbit, partially hyperbolic set.
Mathematics Subject Classification:  Primary: 37D30; Secondary: 37C27.

Received: May 2003;      Revised: February 2004;      Published: June 2004.