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. DuanY. 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.

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.

[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. DuanY. 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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