The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds

Abstract

In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold $(M,F)$ with a bumpy, irreversible Finsler metric $F$ and $H^*(M;{\bf Q})\cong T_{d,n+1}(x)$ for some even integer $d\ge 2$ and integer $n\ge 1$, there exist at least $\frac{dn(n+1)}{2}$ distinct non-hyperbolic closed geodesics with odd Morse indices, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic satisfies $i(c)>0$. Note that the last non-zero index condition is satisfied if the flag curvature $K$ satisfies $K\ge 0$. For an odd-dimensional bumpy Finsler sphere $(S^d,F)$, there exist at least $(d+1)$ distinct prime closed geodesics with even Morse indices, and at least $(d-1)$ of which are non-hyperbolic, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic $c$ satisfies $i(c)\ge 2$. Note that the last index condition $i(c)\ge 2$ is satisfied if the reversibility $\lambda$ and the flag curvature $K$ of $(M,F)$ satisfy $\frac{\lambda^2}{(1+\lambda)^2}<K\le 1$. Note that the first two in the above three lower bound estimates are sharp due to Katok's examples. In addition, we also prove that either there exists at least one non-hyperbolic closed geodesic, or there exist infinitely many distinct closed geodesics on a compact simply connected bumpy Finsler $(M,F)$ satisfying the above cohomological condition with some even integer $d\ge 2$ and integer $n\ge 1$.