  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 \cite{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
$O(\frac{1}{t})$.
