Corrigendum: The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

TL;DR

This corrigendum corrects previous assumptions about the smoothness of the functional in the Conley conjecture for Hamiltonian systems, providing a new splitting lemma applicable to a broader class of Lagrangians.

Contribution

It introduces a new splitting lemma that extends the validity of the Conley conjecture to more general Lagrangian systems beyond quadratic cases.

Findings

Corrects the previous smoothness assumptions for the functional

Provides a new splitting lemma applicable to general Lagrangians

Extends the Conley conjecture results to broader systems

Abstract

In lines 8-11 of \cite[pp. 2977]{Lu} we wrote: "For integer $m\ge 3$, if $M$ is $C^m$-smooth and $C^{m-1}$-smooth $L:\R\times TM\to\R$ satisfies the assumptions (L1)-(L3), then the functional ${\cal L}_\tau$ is $C^2$-smooth, bounded below, satisfies the Palais-Smale condition, and all critical points of it have finite Morse indexes and nullities (see \cite[Prop.4.1, 4.2]{AbF} and \cite{Be})." However, as proved in \cite{AbSc1} the claim that ${\cal L}_\tau$ is $C^2$-smooth is true if and only if for every $(t,q)$ the function $v\mapsto L(t,q,v)$ is a polynomial of degree at most 2. So the arguments in \cite{Lu} is only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in \cite{Lu} with a new splitting lemma obtained in \cite{Lu2}.