The rate of convergence of new Lax-Oleinik type operators for time-periodic positive definite Lagrangian systems (Q2903938)

Let \(M\) be a closed, connected manifold, and let \(L : TM \times \;T \longrightarrow \mathbb{R}\) be a time-periodic Tonelli Lagrangian on \(M\). The Lax-Oleinik semi-group \((T_t)_{t \geq 0}\) is defined by NEWLINE\[NEWLINE \forall u \in C(M,\mathbb{R}),\, \forall t \geq 0,\, \forall x \in M,\, T_t u (x)= \inf_{\gamma} \int_0 ^t L( \gamma(s), \dot\gamma(s), s) ds NEWLINE\]NEWLINE where the infimum is taken over all piecewise \(C^1\) paths \(\gamma : \left[ 0,t\right] \longrightarrow M\) such that \(\gamma(t)=x\). Fathi (C.R.A.S. 1998) proved that the Lax-Oleinik semi-group converges when the Lagrangian is time-independent. On the other hand, \textit{A. Fathi} and \textit{J. N. Mather} [Bull. Soc. Math. Fr. 128, No. 3, 473--483 (2000; Zbl 0989.37035)] proved that it may not converge when the Lagrangian is time-periodic. In a previous paper [Commun. Math. Phys. 309, No. 3, 663--691 (2012; Zbl 1267.37065)] the authors introduced a new semi-group \(\tilde{T}_t\) and proved that it always converges: NEWLINE\[NEWLINE \forall u \in C(M,\mathbb{R}), \forall n \in \mathbb{N}, \forall \tau \in \left[0,1\right], \forall x \in M, \;T_n ^{\tau} u (x)= \inf_{n \leq k \leq 2n} \inf_{\gamma} \int_0 ^{t+k} L( \gamma(s), \dot\gamma(s), s) ds,NEWLINE\]NEWLINE where \(k\) is an integer and the infimum is taken over all piecewise \(C^1\) paths \(\gamma : \left[ 0,\tau +k \right] \longrightarrow M\) such that \(\gamma(\tau + k )=x\).NEWLINENEWLINEIn this paper, the authors uncover another interesting property of their new semi-group: under the hypothesis that the Aubry set consists of exactly one hyperbolic periodic orbit, the new semi-group converges exponentially fast. In 2009, \textit{R. Iturriaga} and \textit{H. Sánchez-Morgado} [J. Differ. Equations 246, No. 5, 1744--1753 (2009; Zbl 1163.37021)] had proved that under the same hypothesis, the usual semi-group converges exponentially fast. However, in the paper under review, the authors give a conter-example to the result of Iturriaga and Sanchez-Morgado, thus making their result worth being prove.