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

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

2011-09-15

arxiv.org/abs/1109.3327v1

Mathematics

Dynamical Systems

Scientific paper

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

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