The rate of convergence of new Lax-Oleinik type operators for time-periodic positive definite Lagrangian systems

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Assume that the Aubry set of the time-periodic positive definite Lagrangian L consists of one hyperbolic 1-periodic orbit. We provide an upper bound estimate of the rate of convergence of the family of new Lax-Oleinik type operators associated with L introduced by the authors in W-Y. In addition, we construct an example where the Aubry set of a time-independent positive definite Lagrangian system consists of one hyperbolic periodic orbit and the rate of convergence of the Lax-Oleinik semigroup cannot be better than O(1/t).