HAMILTONIAN SYSTEMS WITH POSITIVE TOPOLOGICAL ENTROPY AND CONJUGATE POINTS

Fei Liu; Zhiyu Wang; Fang Wang

doi:10.11948/2015038

2015 Volume 5 Issue 3

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Fei Liu, Zhiyu Wang, Fang Wang. HAMILTONIAN SYSTEMS WITH POSITIVE TOPOLOGICAL ENTROPY AND CONJUGATE POINTS[J]. Journal of Applied Analysis & Computation, 2015, 5(3): 527-533. doi: 10.11948/2015038

Citation:

Fei Liu, Zhiyu Wang, Fang Wang. HAMILTONIAN SYSTEMS WITH POSITIVE TOPOLOGICAL ENTROPY AND CONJUGATE POINTS[J]. Journal of Applied Analysis & Computation, 2015, 5(3): 527-533. doi: 10.11948/2015038

HAMILTONIAN SYSTEMS WITH POSITIVE TOPOLOGICAL ENTROPY AND CONJUGATE POINTS

1 College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, China;
2 School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China;and Beijing Center for Mathematics and Information Interdisciplinary Sciences(BCMⅡS), Beijing 100048, China

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Abstract

Abstract

In this article, we consider the geodesic flows induced by the natural Hamiltonian systems H(x, p)=1/2g^ij(x)p_ip_j + V (x) defined on a smooth Riemannian manifold (M=S¹×N, g), where S¹ is the one dimensional torus, N is a compact manifold, g is the Riemannian metric on M and V is a potential function satisfying V ≤ 0. We prove that under suitable conditions, if the fundamental group π₁(N) has sub-exponential growth rate, then the Riemannian manifold M with the Jacobi metric (h -V)g, i.e., (M, (h -V)g), is a manifold with conjugate points for all h with 0 < h < δ, where δ is a small number.
- Hamiltonian systems /
- topological entropy /
- fundamental group /
- conjugate points
MSC: 37C10;37B40

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References

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