2017/03/22 by Yasha Savelyev, Savelyev, Yasha · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DS #math.SG
paper · pdf · doi:10.48550/arxiv.1703.07801
16 pgs. Version accepted by International Journal of math. Some improvements to presentation, added references
arxiv created 2019/01/27 · arxiv updated 2019/01/29
We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields X on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at X has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on S 2k+1 . A sky catastrophe, is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller's foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.