Iteration Inequalities of the Maslov-Type Index Theory with Applications

Abstract

In this paper, by using the |-index theory introduced by Y. Long in (1999, Pacific J. Math. 187, 113 149), in particular, the splitting numbers, Maslov-type mean index, and the homotopy component of symplectic matrix, we establish various inequalities of the Maslov-type index theory for iterations of symplectic paths starting from the identity. As an application, these results are used to study Rabinowitz' conjecture on the prescribed minimal period solution problem of nonlinear Hamiltonian systems. Academic Press

Precise Iteration Formulae of the Maslov-type Index Theory and Ellipticity of Closed Characteristics

“…K Remark 5.4. The necessary and sufficient condition for any equality with some m # N to hold in (5.7) and (5.9) has been proved in [LL3]. Note that the iteration equality Theorem 5.1 of [DL] for non-degenerate symplectic paths, and iteration inequality Theorem 8.3 of [DL] for degenerate symplectic paths can also be derived from our Theorem 1.3 similarly.…”

“…In [Lo10] the author extended this index theory to a new family of index functions parametrized by elements on the unit circle U in the complex plane C and established the Bott-type iteration formulae of the Maslov-type index theory in terms of these index functions. Based on these results, various sharp iteration inequalities were established by C. Liu and the author in [LL1,LL3]. These results have been applied to the study of various problems of Hamiltonian systems (cf.…”

The existence of two closed geodesics on every Finsler 2-sphere

“…To see that [17,Proposition 1], and [18], Theorem, imply Theorem 6.1, note that [31], originally proved in [27], imply that these inequalities are in fact equalities. 7 Homology of ( S 2 , 0 S 2 ) and first consequences In Ziller [43] computed the Z-homology of the free loop space of compact rank 1 symmetric spaces (with the exception of R P n ).…”

Multiple periodic points of the Poincaré map of Lagrangian systems on tori

“…(Theorem 1 of [LL1], Theorem 1.1 of [LL2]) For any path γ ∈ P τ (2n), there holds mî τ (γ) − n ≤ i mτ (γ) ≤ mî τ (γ) + n − ν mτ (γ), ∀ m ∈ N.…”

“…As a direct consequence Theorem 6.5 is proved. Based on works [DL] and [Lo5] the optimal increasing estimate Theorem 6.6 was first proved by C. Liu and the author in [LL1], and further generalized in [LL2]. where ·, · is the duality on T u M and · is the Finsler structure.…”