Closed Geodesics on Positively Curved Manifolds

Closed Geodesics on Positively Curved Finsler 3-Spheres

“…If ( / )(λ/( + λ)) < K ≤ with λ < , then there exists an elliptic-parabolic closed geodesic, i.e., its linearized Poincaré map is split into two-dimensional rotations and a part whose eigenvalues are ± . Some similar results in the Riemannian case are obtained in [2,3]. Recently, Wang proved in [33] that for every Finsler n-dimensional sphere S n with reversibility λ and flag curvature K satisfying (λ/( + λ)) < K ≤ , either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp( − πμ) with an irrational μ.…”

Geodesic flows with positive topological entropy, twist maps and hyperbolicity

“…Interesting cases are obtained in W. Ballman, G. Thorgbersson and W. Ziller [4], where they give conditions under which the existence of a closed non-hyperbolic geodesic is guaranted (see specially Theorem B). Combining this result with Theorem C one obtains that any 1/4-pinched metric in S n may be approximated in the C ∞ topology by a metric with a non-trivial hyperbolic basic set.…”

Path integration over closed loops and Gutzwiller's trace formula

“…A trajectory starting at time T ′ in generic position will fulfill the condition. Non generic situations when (182) does not hold true can treated using the general formula (178) with W, W ⊥ denoting respectively the sets of Dirichlet and recurrent Jacobi fields in [T ′ , T ], [12].…”