The convergence of Lax-Oleinik semigroup for time-periodic Lagrangian

In this article, the convergence of so-called Lax-Oleinik semigroup is studied for time-periodic Lagrangian systems when the degree of freedom is greater than 2. Under certain conditions, we show that the Lax-Oleinik semigroup converges if the rotation vector is completely irrational. Removing such conditions, we will give another kind of convergence of the sequence F _c ((x, s), (x′, s′ +T _n )), the convergence of which is closely related to the Lax-Oleinik semigroup.