March  2020, 28(1): 127-148. doi: 10.3934/era.2020008

Decomposition of spectral flow and Bott-type iteration formula

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

 

Received  September 2019 Revised  January 2020 Published  March 2020

Fund Project: Both the authors are partially supported by NSFC(No. 11790271, 11425105)

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.

Citation: Xijun Hu, Li Wu. Decomposition of spectral flow and Bott-type iteration formula. Electronic Research Archive, 2020, 28 (1) : 127-148. doi: 10.3934/era.2020008
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.  Google Scholar

[2]

M. F. AtiyahV. 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.  Google Scholar

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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.  Google Scholar

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B. Booss-BavnbekM. Lesch and J. Phillips, Unbounded Fredholm operators and spectral flow, Canad. J. Math., 57 (2005), 225-250.  doi: 10.4153/CJM-2005-010-1.  Google Scholar

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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.  Google Scholar

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S. E. CappellR. Lee and E. Y. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186.  doi: 10.1002/cpa.3160470202.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

X. HuA. Portaluri and R. Yang, Instability of semi-Riemannian closed geodesics, Nonlinearity, 32 (2019), 4281-4316.  doi: 10.1088/1361-6544/ab1c87.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Y. LongD. 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.  Google Scholar

[25]

J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. F. AtiyahV. 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.  Google Scholar

[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.  Google Scholar

[4]

B. Booss-BavnbekM. Lesch and J. Phillips, Unbounded Fredholm operators and spectral flow, Canad. J. Math., 57 (2005), 225-250.  doi: 10.4153/CJM-2005-010-1.  Google Scholar

[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.  Google Scholar

[6]

S. E. CappellR. Lee and E. Y. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186.  doi: 10.1002/cpa.3160470202.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

X. HuA. Portaluri and R. Yang, Instability of semi-Riemannian closed geodesics, Nonlinearity, 32 (2019), 4281-4316.  doi: 10.1088/1361-6544/ab1c87.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Y. LongD. 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.  Google Scholar

[25]

J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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