Connecting orbits for families of Tonelli Hamiltonians (Q1940226)

The article under review studies the instability of Hamiltonian systems. For convex Hamiltonian systems on \(T^*\mathbb{T}\), it is well known (from Birkhoff theory for twist maps) that the unique obstruction to diffusion are the noncontractible invariant circles. The higher-dimension case of Hamiltonians on \(T^*M\), with \(M\) a compact manifold, is more complicated and is related to the phenomenon of Arnold diffusion. The weak KAM theory started by Mather and Fathi has proven to be fruitful in attacking this problem. Another possible type of generalization is to remain on \(T^*\mathbb{T}\) but study the system generated by a family of Hamiltonians. These systems are called polysystems. In this case, it is known that the unique obstruction to drift of polyorbits are the common noncontractible invariant circles. In the present paper, the author mixes the two generalizations: he studies a family of Tonelli Hamiltoninans in \(T^*M\), where \(M\) is a compact manifold of dimension \(d\). For these polysystems the existence of unstable polyorbits is investigated. The methods are in the framework of weak KAM theory and can be seen as a generalization of the work of \textit{P. Bernard} [J. Am. Math. Soc. 21, No. 3, 615--669 (2008; Zbl 1213.37089)]. The results on this paper give sufficient conditions for the existence of unstable polyorbits connecting (roughly speaking) two given cohomology classes in \(H^1(M,\mathbb{R})\). More precisely, if \(\mathcal{F}\) is a family of one-periodic Tonelli Hamiltonians, the author defines, following the ideas of Bernard [loc. cit., Zbl 1213.37089], the so-called forcing relation \(\vdash_{\mathcal{F}}\) and mutual forcing relation \(\dashv\vdash\) between cohomology classes. In Proposition 7, it is proved that the occurrence of such relations implies the existence of unstable polyorbits. The main result, Theorem 31, gives sufficient conditions for the occurrence of the relation \(\vdash_{\mathcal{F}}\). This is done using the so-called ``Mather mechanism'', after \textit{J. N. Mather} [Ann. Inst. Fourier 43, No. 5, 1349--1386 (1993; Zbl 0803.58019)].