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].