Multiple brake orbits in bounded convex symmetric domains

Abstract

In this paper, we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in R n for n 2.

On the normal exponential map in singular conformal metrics

“…There exists a huge amount of literature concerning the study of brake orbitssee e.g., [12,15,21,22] -and more generally on the study of periodic solutions of autonomous Hamiltonian systems with prescribed energy [13,14,16,17,18]. We also observe here that manifolds with singular boundary of the type investigated in the present paper arise naturally in the study of certain compactifications of incomplete Riemannian manifolds.…”

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

“…In [23], Long et al established two indices μ 1 (γ) and μ 2 (γ) for the fundamental solution γ of a linear Hamiltonian system by the methods of functional analysis which are special cases of the L-Maslov type index i L (γ) for Lagrangian subspaces L 0 = {0} ⊕ R n and L 1 = R n ⊕ {0} up to a constant n. This paper is divided into 3 sections. In Section 2, we give introduction to the Maslov type index theory for symplectic paths with Lagrangian boundary conditions and an iteration theory for the L 0 -Maslov type index theory.…”

“…van Groesen [11] in 1988 and Ambrosetti et al [1] in 1993 also proved the multiplicity result about brake orbits for the second order Hamiltonian systems under different pinching conditions. 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.…”

Seifert Conjecture in the Even Convex Case

“…By Proposition 6.1 of [25] and Lemma 2.8 and Definition 2.5 of [23], we give the following Definition 2.3. For any continuous path γ ∈ P τ (2n), we define the following Maslov-type indices:…”

“…The index i L (γ) for any Lagrangian subspace L ⊂ R 2n and symplectic path γ ∈ P τ (2n) was defined by the first author of this paper in [16] in a different way(see also [17] and [23]). …”