The Lagrangian Conley conjecture (Q620544)

The main result of this paper is contained in Theorem 1.1 and it consists in the following Lagrangian analogue of the Conley conjecture: Let \(M\) be a smooth closed manifold, \(\mathcal{L}:{\mathbb{R}}/{\mathbb{Z}}\times TM \rightarrow \mathbb{R}\) a smooth \(1\)-periodic Tonelli Lagrangian with global flow and \(a\in \mathbb{R}\) a constant greater than \[ \max_{q\in M}\{\int_{0}^{1}\mathcal{L}(t,q,0)dt\}. \] Assume that only finitely many contractible \(1\)-periodic solutions of the Euler-Lagrange system of \(\mathcal{L}\) have action less than \(a\). Then, for each prime number \(p\), the Euler-Lagrange system of \(\mathcal{L}\) admits infinitely many contractible periodic solutions with period that is a power of \(p\) and mean action less than \(a\). The author proves this result by using a Morse theoretic argument inspired by the paper of \textit{Y. Long} [Math. Z. 233, No. 3, 443--470 (2000; Zbl 0984.37074)]. An interesting corollary is the following: If \(M\) is a smooth closed manifold and \(\mathcal{H}:{\mathbb{R}}/{\mathbb{Z}}\times T^{*}M \rightarrow \mathbb{R}\) a \(1\)-periodic Tonelli Hamiltonian with global flow \(\Phi _{\mathcal{H}}^t\), then the Hamiltonian diffeomorphism \(\Phi _{\mathcal{H}}^1\) has infinitely periodic orbits.