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

Previous Article
Global propagation of singularities for discounted Hamilton-Jacobi equations
DCDS Home
This Issue
Next Article
Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions

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

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 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.

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 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.

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. 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.

[1]	Dario Corona. A multiplicity result for orthogonal geodesic chords in Finsler disks. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5329-5357. doi: 10.3934/dcds.2021079
[2]	Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006
[3]	Jan Prüss, Gieri Simonett. On the manifold of closed hypersurfaces in $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5407-5428. doi: 10.3934/dcds.2013.33.5407
[4]	Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[5]	Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure and Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507
[6]	Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081
[7]	Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062
[8]	Fabio Giannoni, Paolo Piccione, Daniel V. Tausk. Morse theory for the travel time brachistochrones in stationary spacetimes. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 697-724. doi: 10.3934/dcds.2002.8.697
[9]	Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148
[10]	Mark Pollicott. Closed geodesic distribution for manifolds of non-positive curvature. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 153-161. doi: 10.3934/dcds.1996.2.153
[11]	Hui Liu, Yiming Long, Yuming Xiao. The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3803-3829. doi: 10.3934/dcds.2018165
[12]	Todd Young. A result in global bifurcation theory using the Conley index. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387
[13]	Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107
[14]	Jijiang Sun, Shiwang Ma. Nontrivial solutions for Kirchhoff type equations via Morse theory. Communications on Pure and Applied Analysis, 2014, 13 (2) : 483-494. doi: 10.3934/cpaa.2014.13.483
[15]	Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112
[16]	Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629
[17]	Kung-Ching Chang, Zhi-Qiang Wang, Tan Zhang. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 809-826. doi: 10.3934/dcds.2010.28.809
[18]	Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056
[19]	Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
[20]	Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Muhammad Hamid Wei Wang
on Google Scholar
Muhammad Hamid Wei Wang

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Muhammad Hamid Wei Wang
on Google Scholar
Muhammad Hamid Wei Wang

[Back to Top]