Iteration theory of L-index and multiplicity of brake orbits

Abstract

In this paper, we first establish the Bott-type iteration formulas and some abstract precise iteration formulas of the Maslov-type index theory associated with a Lagrangian subspace for symplectic paths. As an application, we prove that there exist at least n 2 + 1 geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in R 2n satisfying the reversible condition N Σ = Σ, furthermore, if all brake orbits on this hypersurface are nondegenerate, then there are at least n geometrically distinct brake orbits on it. As a consequence, we show that there exist at least n 2 + 1 geometrically distinct brake orbits in every bounded convex symmetric domain in R n , furthermore, if all brake orbits in this domain are nondegenerate, then there are at least n geometrically distinct brake orbits in it. In the symmetric case,

Seifert Conjecture in the Even Convex Case

“…Moreover it was proved that if all brake orbits on Σ are nondegenerate, then #J b (Σ) ≥ n + A(Σ), where 2A(Σ) is the number of geometrically distinct asymmetric brake orbits on Σ. Recently, in [32] the authors of this paper improved the results of [19] …”

“…The key ingredients in the proof of Theorem 1.1 are some ideas from our previous paper [19] and the following result which generalizes corresponding results of our previous papers [32,33] completely, where the iteration path γ 2 will be defined in Definition 2.5 below.…”

“…In 1989, A. Szulkin in [29] b (2n) with n ≥ 2, there holds #J b (Σ) ≥ 2. In 2009, the authors of this paper in [19] …”

“…We have Lemma 5.1. (Theorem 1.5 and of [19] and Theorem 4.3 of [25]) Let γ j ∈ P τ j (2n) for j = 1, · · · , q.…”

The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems

“…Without pinching conditions, Long et al [23] in 2006 proved that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in R n for n ≥ 2. Recently, Liu and Zhang [18] proved that there exist at least [n/2] + 1 geometrically distinct brake orbits in every bounded convex symmetric domain in R n for n ≥ 2, and there exist at least n geometrically distinct brake orbits on nondegenerate domain. For the minimal estimate of brake orbits, Liu [16] have considered the strictly convex reversible Hamiltonian systems case and proved the existence of nonconstant brake orbit with minimal period belonging to {T, T/2} for any given T > 0.…”