2011/11/03 by Duanzhi Zhang, Zhang, Duanzhi, Chungen Liu +2
Mathematics · #34C25 #58E05 #70H05 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DS #msc:34C25 #msc:58E05 #msc:70H05
paper · pdf · doi:10.48550/arxiv.1111.0722
35 pages
arxiv created 2011/11/03 · openalex publication_date 2011/11/03 · arxiv updated 2011/11/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper, we prove that there exist at least [(n+1)/(2)]+1 geometrically distinct brake orbits on every C2 compact convex symmetric hypersurface \Sg in \R2n for n≥ 2 satisfying the reversible condition N\Sg=\Sg with N=\diag (-In,In). As a consequence, we show that there exist at least [(n+1)/(2)]+1 geometrically distinct brake orbits in every bounded convex symmetric domain in \Rn with n≥ 2 which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for n=3. As an application, for n=4 and 5, we prove that if there are exactly n geometrically distinct closed characteristics on \Sg, then all of them are symmetric brake orbits after suitable time translation.