The Lagrangian Conley Conjecture

TL;DR

This paper proves a Lagrangian version of the Conley conjecture, demonstrating that certain dynamical systems have infinitely many periodic solutions with specific properties, expanding understanding of periodic orbits in Lagrangian systems.

Contribution

It establishes the existence of infinitely many periodic solutions for a class of Lagrangian systems, a novel extension of the Conley conjecture to the Lagrangian setting.

Findings

Existence of infinitely many contractible periodic solutions.

Solutions have bounded mean action.

Either infinitely many are 1-periodic or have unbounded period.

Abstract

We prove a Lagrangian analogue of the Conley conjecture: given a 1-periodic Tonelli Lagrangian with global flow on a closed configuration space, the associated Euler-Lagrange system has infinitely many periodic solutions. More precisely, we show that there exist infinitely many contractible integer periodic solutions with a priori bounded mean action and either infinitely many of them are 1-periodic or they have unbounded period.