# American Institute of Mathematical Sciences

April  2022, 42(4): 1933-1948. doi: 10.3934/dcds.2021178

## A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem

 1 School of Mathematical Sciences, Peking University, Beijing 100871, China 2 School of Mathematical Sciences and LMAM, Peking University, Beijing 100871, China

* Corresponding author: Wei Wang

Received  April 2021 Revised  October 2021 Published  April 2022 Early access  November 2021

Fund Project: The second author is supported by NSFC grant 12025101, 11431001

In this paper, we prove a symmetric property for the indices for symplectic paths in the enhanced common index jump theorem (cf. Theorem 3.5 in [6]). As an application of this property, we prove that on every compact Finsler manifold $(M, \, F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\left(\frac{\lambda}{\lambda+1}\right)^2<K\le 1$, there exist two elliptic closed geodesics whose linearized Poincaré map has an eigenvalue of the form $e^{\sqrt {-1}\theta}$ with $\frac{\theta}{\pi}\notin{\bf Q}$ provided the number of closed geodesics on $M$ is finite.

Citation: Muhammad Hamid, Wei Wang. A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1933-1948. doi: 10.3934/dcds.2021178
##### References:
 [1] V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann., 346 (2010), 335-366.  doi: 10.1007/s00208-009-0401-1. [2] H. Duan, Two elliptic closed geodesics on positively curved Finsler spheres, J. Diff. Equa., 260 (2016), 8388-8402.  doi: 10.1016/j.jde.2016.02.025. [3] H. Duan and Y. Long, Multiple closed geodesics on bumpy Finsler $n$-spheres, J. Diff. Equa., 233 (2007), 221-240.  doi: 10.1016/j.jde.2006.10.002. [4] H. Duan and Y. Long, The index growth and multiplicity of closed geodesics, J. Funct. Anal., 259 (2010), 1850-1913.  doi: 10.1016/j.jfa.2010.05.003. [5] H. Duan, Y. Long and W. Wang, Two closed geodesics on compact simply-connected bumpy Finsler manifolds, J. Differential Geom., 104 (2016), 275-289.  doi: 10.4310/jdg/1476367058. [6] H. Duan, Y. Long and W. Wang, The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds, Calc. Var. Partial Differential Equations, 55 (2016), Art. 145, 28 pp. doi: 10.1007/s00526-016-1075-7. [7] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6$^{nd}$ edition, Oxford University Press, 2008. [8] A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539-576. [9] H. Liu, The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form $S^{2n+1}/\Gamma$, Calc. Var. Partial Differential Equations, 58 (2019), Art. 107, 21 pp. doi: 10.1007/s00526-019-1567-3. [10] C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China, 45 (2002), 9-28. [11] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113. [12] Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000), 76-131.  doi: 10.1006/aima.2000.1914. [13] Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math, 207, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8175-3. [14] Y. Long and H. Duan, Multiple closed geodesics on 3-spheres, Advances in Math., 221 (2009), 1757-1803.  doi: 10.1016/j.aim.2009.03.007. [15] Y. Long and W. Wang, Stability of closed geodesics on Finsler 2-spheres, J. Funct. Anal., 255 (2008), 620-641.  doi: 10.1016/j.jfa.2008.05.001. [16] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in ${\bf{R}}^2n$, Ann. of Math., 155 (2002), 317-368.  doi: 10.2307/3062120. [17] H.-B. Rademacher, A sphere theorem for non-reversible Finsler metrics, Math. Annalen, 328 (2004), 373-387.  doi: 10.1007/s00208-003-0485-y. [18] H.-B. Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems, 27 (2007), 957-969.  doi: 10.1017/S0143385706001064. [19] H.-B. Rademacher, The second closed geodesic on Finsler spheres of dimension $n>2$, Trans. Amer. Math. Soc., 362 (2010), 1413-1421.  doi: 10.1090/S0002-9947-09-04745-X. [20] Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001. doi: 10.1142/9789812811622. [21] W. Wang, Closed geodesics on positively curved Finsler spheres, Advances in Math., 218 (2008), 1566-1603.  doi: 10.1016/j.aim.2008.03.018. [22] W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617.  doi: 10.1016/j.aim.2012.04.006. [23] W. Wang, On the average indices of closed geodesics on positively curved Finsler spheres, Math. Annalen, 355 (2013), 1049-1065.  doi: 10.1007/s00208-012-0812-2. [24] W. Wang, Multiple closed geodesics on positively curved Finsler manifolds, Adv. Nonlinear Stud., 19 (2019), 495–518. doi: 10.1515/ans-2019-2043. [25] W. Ziller, Geometry of the Katok examples, Ergodic Theory Dynam. Systems, 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.

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##### References:
 [1] V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann., 346 (2010), 335-366.  doi: 10.1007/s00208-009-0401-1. [2] H. Duan, Two elliptic closed geodesics on positively curved Finsler spheres, J. Diff. Equa., 260 (2016), 8388-8402.  doi: 10.1016/j.jde.2016.02.025. [3] H. Duan and Y. Long, Multiple closed geodesics on bumpy Finsler $n$-spheres, J. Diff. Equa., 233 (2007), 221-240.  doi: 10.1016/j.jde.2006.10.002. [4] H. Duan and Y. Long, The index growth and multiplicity of closed geodesics, J. Funct. Anal., 259 (2010), 1850-1913.  doi: 10.1016/j.jfa.2010.05.003. [5] H. Duan, Y. Long and W. Wang, Two closed geodesics on compact simply-connected bumpy Finsler manifolds, J. Differential Geom., 104 (2016), 275-289.  doi: 10.4310/jdg/1476367058. [6] H. Duan, Y. Long and W. Wang, The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds, Calc. Var. Partial Differential Equations, 55 (2016), Art. 145, 28 pp. doi: 10.1007/s00526-016-1075-7. [7] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6$^{nd}$ edition, Oxford University Press, 2008. [8] A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539-576. [9] H. Liu, The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form $S^{2n+1}/\Gamma$, Calc. Var. Partial Differential Equations, 58 (2019), Art. 107, 21 pp. doi: 10.1007/s00526-019-1567-3. [10] C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China, 45 (2002), 9-28. [11] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113. [12] Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000), 76-131.  doi: 10.1006/aima.2000.1914. [13] Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math, 207, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8175-3. [14] Y. Long and H. Duan, Multiple closed geodesics on 3-spheres, Advances in Math., 221 (2009), 1757-1803.  doi: 10.1016/j.aim.2009.03.007. [15] Y. Long and W. Wang, Stability of closed geodesics on Finsler 2-spheres, J. Funct. Anal., 255 (2008), 620-641.  doi: 10.1016/j.jfa.2008.05.001. [16] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in ${\bf{R}}^2n$, Ann. of Math., 155 (2002), 317-368.  doi: 10.2307/3062120. [17] H.-B. Rademacher, A sphere theorem for non-reversible Finsler metrics, Math. Annalen, 328 (2004), 373-387.  doi: 10.1007/s00208-003-0485-y. [18] H.-B. Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems, 27 (2007), 957-969.  doi: 10.1017/S0143385706001064. [19] H.-B. Rademacher, The second closed geodesic on Finsler spheres of dimension $n>2$, Trans. Amer. Math. Soc., 362 (2010), 1413-1421.  doi: 10.1090/S0002-9947-09-04745-X. [20] Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001. doi: 10.1142/9789812811622. [21] W. Wang, Closed geodesics on positively curved Finsler spheres, Advances in Math., 218 (2008), 1566-1603.  doi: 10.1016/j.aim.2008.03.018. [22] W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617.  doi: 10.1016/j.aim.2012.04.006. [23] W. Wang, On the average indices of closed geodesics on positively curved Finsler spheres, Math. Annalen, 355 (2013), 1049-1065.  doi: 10.1007/s00208-012-0812-2. [24] W. Wang, Multiple closed geodesics on positively curved Finsler manifolds, Adv. Nonlinear Stud., 19 (2019), 495–518. doi: 10.1515/ans-2019-2043. [25] W. Ziller, Geometry of the Katok examples, Ergodic Theory Dynam. Systems, 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.
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