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On the common index jump theorem and further developments

Huagui Duan, Yiming Long, Wei Wang, Chaofeng Zhu

Abstract

In [LZ02] published in Annals of Mathematics, Long and Zhu established originally the common index jump theorem (CIJT) for symplectic paths in 2002, which has played an important role in later studies on periodic solution orbits for Hamiltonian systems, Reeb flows, and geodesic problems. This (CIJT) was generalized to its enhanced version (ECIJT) by Duan, Long and Wang in [DLW16] in 2016. Started from [GG20] of 2020, and finally in [CGG24] of 2024, a similar index theorem was obtained, i.e., Theorem 3.3 of [CGG24], which was given the name "index recurrence theorem" there. In this short note, we give detailed proofs to show that the major assertions, i.e., the first 4 assertions in the total of 5 assertions, in Theorem 3.3 of [CGG24] as well as all the assertions in [GG20] actually coincide completely with results in (ECIJT) of [DLW16].

On the common index jump theorem and further developments

Abstract

In [LZ02] published in Annals of Mathematics, Long and Zhu established originally the common index jump theorem (CIJT) for symplectic paths in 2002, which has played an important role in later studies on periodic solution orbits for Hamiltonian systems, Reeb flows, and geodesic problems. This (CIJT) was generalized to its enhanced version (ECIJT) by Duan, Long and Wang in [DLW16] in 2016. Started from [GG20] of 2020, and finally in [CGG24] of 2024, a similar index theorem was obtained, i.e., Theorem 3.3 of [CGG24], which was given the name "index recurrence theorem" there. In this short note, we give detailed proofs to show that the major assertions, i.e., the first 4 assertions in the total of 5 assertions, in Theorem 3.3 of [CGG24] as well as all the assertions in [GG20] actually coincide completely with results in (ECIJT) of [DLW16].

Paper Structure

This paper contains 5 sections, 54 equations.

Table of Contents

  1. Introduction
  2. Theorem 3.3 and Remark 3.6 of DLW16 as well as Theorem 3.3 of CGG24
  3. The coincidence of ${\beta}_{-}({\gamma})$ in CGG24 and $S_{{\gamma}(1)}^-(1)$ in Long's book Lo02 of 2002
  4. The (IR1) in Theorem 3.3 of CGG24 is a direct consequence of Remark 3.6 of DLW16
  5. The coincidence of (IR2) and (IR3) in Theorem 3.3 of CGG24 and (3.20)-(3.22) in Theorem 3.5 of DLW16