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

In this paper, the Conley conjecture, which were recently proved by Franks and Handel \cite{FrHa} (for surfaces of positive genus), Hingston \cite{Hi} (for tori) and Ginzburg \cite{Gi} (for closed symplectically aspherical manifolds), is proved for $C^1$-Hamiltonian systems on the cotangent bundle of a $C^3$-smooth compact manifold $M$ without boundary, of a time 1-periodic $C^2$-smooth Hamiltonian $H:\R\times T^\ast M\to\R$ which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If $H$ also satisfies $H(-t,q, -p)=H(t,q, p)$ for any $(t,q, p)\in\R\times T^\ast M$, it is shown that the time-one map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of $T^\ast M$. If $M$ is $C^5$-smooth and $\dim M>1$, $H$ is of $C^4$ class and independent of time $t$, then for any $\tau>0$ the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of $\tau$ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of $H$, $L:\R\times TM\to\R$, which is proved to be strongly convex and to have quadratic growth in the velocities yet.