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The Mahler measure of $ (x+1/x)(y+1/y)(z+1/z)+\sqrt{k} $
Decomposition of spectral flow and Bott-type iteration formula
School of Mathematics, Shandong University, Jinan, Shandong 250100, China |
Let $ A(t) $ be a continuous path of self-adjoint Fredholm operators, we derive a decomposition formula of spectral flow if the path is invariant under a matrix-like cogredient. As applications, we give the generalized Bott-type iteration formula for linear Hamiltonian systems.
References:
[1] |
V. I. Arnol'd, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen, 1 (1967), 1–14.
doi: 10.1007/BF01075861. |
[2] |
M. F. Atiyah, V. K. Patodi and I. M. Singer,
Spectral asymmetry and Riemannian geometry. Ⅲ, Math. Proc. Cambridge Philos. Soc., 79 (1976), 71-99.
doi: 10.1017/S0305004100052105. |
[3] |
W. Ballmann, G. Thorbergsson and W. Ziller, Closed geodesics on positively curved manifolds, Ann. of Math. (2), 116 (1982), 213-247.
doi: 10.2307/2007062. |
[4] |
B. Booss-Bavnbek, M. Lesch and J. Phillips,
Unbounded Fredholm operators and spectral flow, Canad. J. Math., 57 (2005), 225-250.
doi: 10.4153/CJM-2005-010-1. |
[5] |
R. Bott,
On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9 (1956), 171-206.
doi: 10.1002/cpa.3160090204. |
[6] |
S. E. Cappell, R. Lee and E. Y. Miller,
On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186.
doi: 10.1002/cpa.3160470202. |
[7] |
C.-N. Chen and X. Hu,
Maslov index for homoclinic orbits of Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 589-603.
doi: 10.1016/j.anihpc.2006.06.002. |
[8] |
A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. of Math. (2), 152 (2000), 881-901.
doi: 10.2307/2661357. |
[9] |
C. Conley and E. Zehnder,
Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37 (1984), 207-253.
doi: 10.1002/cpa.3160370204. |
[10] |
R. Cushman and J. Duistermaat,
The behavior of the index of a periodic linear Hamiltonian system under iteration, Advances in Math., 23 (1977), 1-21.
doi: 10.1016/0001-8708(77)90107-4. |
[11] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Results in Mathematics and Related Areas, 19, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3. |
[12] |
D. L. Ferrario and S. Terracini,
On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math., 155 (2004), 305-362.
doi: 10.1007/s00222-003-0322-7. |
[13] |
P. Fitzpatrick, J. Pejsachowicz and C. Stuart, Spectral flow for paths of unbounded operators and bifurcation of critical points, work in progress, (2006). Google Scholar |
[14] |
X. Hu and Y. Ou,
Collision index and stability of elliptic relative equilibria in planar $n$-body problem, Comm. Math. Phys., 348 (2016), 803-845.
doi: 10.1007/s00220-016-2695-7. |
[15] |
X. Hu and A. Portaluri, Index theory for heteroclinic orbits of Hamiltonian systems, Calc. Var. Partial Differential Equations, 56 (2017), 24pp.
doi: 10.1007/s00526-017-1259-9. |
[16] |
X. Hu, A. Portaluri and R. Yang, A dihedral Bott-type iteration formula and stability of symmetric periodic orbits, Calc. Var. Partial Differential Equations, 59 (2020).
doi: 10.1007/s00526-020-1709-7. |
[17] |
X. Hu, A. Portaluri and R. Yang,
Instability of semi-Riemannian closed geodesics, Nonlinearity, 32 (2019), 4281-4316.
doi: 10.1088/1361-6544/ab1c87. |
[18] |
X. Hu and S. Sun,
Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit, Comm. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[19] |
C. Liu and S. Tang,
Maslov (P, $\omega$)-index theory for symplectic paths, Adv. Nonlinear Stud., 15 (2015), 963-990.
doi: 10.1515/ans-2015-0412. |
[20] |
C. Liu and D. Zhang,
Iteration theory of $L$-index and multiplicity of brake orbits, J. Differential Equations, 257 (2014), 1194-1245.
doi: 10.1016/j.jde.2014.05.006. |
[21] |
C. Liu and D. Zhang,
Seifert conjecture in the even convex case, Comm. Pure Appl. Math., 67 (2014), 1563-1604.
doi: 10.1002/cpa.21525. |
[22] |
Y. Long,
Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.
doi: 10.2140/pjm.1999.187.113. |
[23] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[24] |
Y. Long, D. Zhang and C. Zhu,
Multiple brake orbits in bounded convex symmetric domains, Adv. Math., 203 (2006), 568-635.
doi: 10.1016/j.aim.2005.05.005. |
[25] |
J. Robbin and D. Salamon,
The Maslov index for paths, Topology, 32 (1993), 827-844.
doi: 10.1016/0040-9383(93)90052-W. |
[26] |
J. Robbin and D. Salamon,
The spectral flow and the Maslov index, Bull. London Math. Soc., 27 (1995), 1-33.
doi: 10.1112/blms/27.1.1. |
[27] |
C. Zhu and Y. Long,
Maslov-type index theory for symplectic paths and spectral flow. (Ⅰ), Chinese Ann. Math. Ser. B, 20 (1999), 413-424.
doi: 10.1142/S0252959999000485. |
show all references
References:
[1] |
V. I. Arnol'd, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen, 1 (1967), 1–14.
doi: 10.1007/BF01075861. |
[2] |
M. F. Atiyah, V. K. Patodi and I. M. Singer,
Spectral asymmetry and Riemannian geometry. Ⅲ, Math. Proc. Cambridge Philos. Soc., 79 (1976), 71-99.
doi: 10.1017/S0305004100052105. |
[3] |
W. Ballmann, G. Thorbergsson and W. Ziller, Closed geodesics on positively curved manifolds, Ann. of Math. (2), 116 (1982), 213-247.
doi: 10.2307/2007062. |
[4] |
B. Booss-Bavnbek, M. Lesch and J. Phillips,
Unbounded Fredholm operators and spectral flow, Canad. J. Math., 57 (2005), 225-250.
doi: 10.4153/CJM-2005-010-1. |
[5] |
R. Bott,
On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9 (1956), 171-206.
doi: 10.1002/cpa.3160090204. |
[6] |
S. E. Cappell, R. Lee and E. Y. Miller,
On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186.
doi: 10.1002/cpa.3160470202. |
[7] |
C.-N. Chen and X. Hu,
Maslov index for homoclinic orbits of Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 589-603.
doi: 10.1016/j.anihpc.2006.06.002. |
[8] |
A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. of Math. (2), 152 (2000), 881-901.
doi: 10.2307/2661357. |
[9] |
C. Conley and E. Zehnder,
Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37 (1984), 207-253.
doi: 10.1002/cpa.3160370204. |
[10] |
R. Cushman and J. Duistermaat,
The behavior of the index of a periodic linear Hamiltonian system under iteration, Advances in Math., 23 (1977), 1-21.
doi: 10.1016/0001-8708(77)90107-4. |
[11] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Results in Mathematics and Related Areas, 19, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3. |
[12] |
D. L. Ferrario and S. Terracini,
On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Invent. Math., 155 (2004), 305-362.
doi: 10.1007/s00222-003-0322-7. |
[13] |
P. Fitzpatrick, J. Pejsachowicz and C. Stuart, Spectral flow for paths of unbounded operators and bifurcation of critical points, work in progress, (2006). Google Scholar |
[14] |
X. Hu and Y. Ou,
Collision index and stability of elliptic relative equilibria in planar $n$-body problem, Comm. Math. Phys., 348 (2016), 803-845.
doi: 10.1007/s00220-016-2695-7. |
[15] |
X. Hu and A. Portaluri, Index theory for heteroclinic orbits of Hamiltonian systems, Calc. Var. Partial Differential Equations, 56 (2017), 24pp.
doi: 10.1007/s00526-017-1259-9. |
[16] |
X. Hu, A. Portaluri and R. Yang, A dihedral Bott-type iteration formula and stability of symmetric periodic orbits, Calc. Var. Partial Differential Equations, 59 (2020).
doi: 10.1007/s00526-020-1709-7. |
[17] |
X. Hu, A. Portaluri and R. Yang,
Instability of semi-Riemannian closed geodesics, Nonlinearity, 32 (2019), 4281-4316.
doi: 10.1088/1361-6544/ab1c87. |
[18] |
X. Hu and S. Sun,
Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit, Comm. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[19] |
C. Liu and S. Tang,
Maslov (P, $\omega$)-index theory for symplectic paths, Adv. Nonlinear Stud., 15 (2015), 963-990.
doi: 10.1515/ans-2015-0412. |
[20] |
C. Liu and D. Zhang,
Iteration theory of $L$-index and multiplicity of brake orbits, J. Differential Equations, 257 (2014), 1194-1245.
doi: 10.1016/j.jde.2014.05.006. |
[21] |
C. Liu and D. Zhang,
Seifert conjecture in the even convex case, Comm. Pure Appl. Math., 67 (2014), 1563-1604.
doi: 10.1002/cpa.21525. |
[22] |
Y. Long,
Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.
doi: 10.2140/pjm.1999.187.113. |
[23] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[24] |
Y. Long, D. Zhang and C. Zhu,
Multiple brake orbits in bounded convex symmetric domains, Adv. Math., 203 (2006), 568-635.
doi: 10.1016/j.aim.2005.05.005. |
[25] |
J. Robbin and D. Salamon,
The Maslov index for paths, Topology, 32 (1993), 827-844.
doi: 10.1016/0040-9383(93)90052-W. |
[26] |
J. Robbin and D. Salamon,
The spectral flow and the Maslov index, Bull. London Math. Soc., 27 (1995), 1-33.
doi: 10.1112/blms/27.1.1. |
[27] |
C. Zhu and Y. Long,
Maslov-type index theory for symplectic paths and spectral flow. (Ⅰ), Chinese Ann. Math. Ser. B, 20 (1999), 413-424.
doi: 10.1142/S0252959999000485. |
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