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Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions
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 |
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 [
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. |
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