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doi: 10.3934/dcds.2021178
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A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem

Muhammad Hamid 1, and  Wei Wang 2,,

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 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.

Keywords: Finsler manifold, closed geodesic, index iteration, Morse theory.
Mathematics Subject Classification: Primary: 53C22, 53C60; Secondary: 58E10.
Citation: Muhammad Hamid, Wei Wang. A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem. Discrete & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6$^{nd}$ edition, Oxford University Press, 2008.  Google Scholar

[8]

A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539-576.   Google Scholar

[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.  Google Scholar

[10]

C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China, 45 (2002), 9-28.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001. doi: 10.1142/9789812811622.  Google Scholar

[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.  Google Scholar

[22]

W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617.  doi: 10.1016/j.aim.2012.04.006.  Google Scholar

[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.  Google Scholar

[24]

W. Wang, Multiple closed geodesics on positively curved Finsler manifolds, Adv. Nonlinear Stud., 19 (2019), 495–518. doi: 10.1515/ans-2019-2043.  Google Scholar

[25]

W. Ziller, Geometry of the Katok examples, Ergodic Theory Dynam. Systems, 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6$^{nd}$ edition, Oxford University Press, 2008.  Google Scholar

[8]

A. B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 539-576.   Google Scholar

[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.  Google Scholar

[10]

C. Liu and Y. Long, Iterated index formulae for closed geodesics with applications, Science in China, 45 (2002), 9-28.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Z. Shen, Lectures on Finsler Geometry, World Scientific, Singapore, 2001. doi: 10.1142/9789812811622.  Google Scholar

[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.  Google Scholar

[22]

W. Wang, On a conjecture of Anosov, Advances in Math., 230 (2012), 1597-1617.  doi: 10.1016/j.aim.2012.04.006.  Google Scholar

[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.  Google Scholar

[24]

W. Wang, Multiple closed geodesics on positively curved Finsler manifolds, Adv. Nonlinear Stud., 19 (2019), 495–518. doi: 10.1515/ans-2019-2043.  Google Scholar

[25]

W. Ziller, Geometry of the Katok examples, Ergodic Theory Dynam. Systems, 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.  Google Scholar

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