Quantitative Persistence and Regularity of Invariant Graphs for Dissipative Twist Maps

In this paper we study the robustness of invariant Lagrangian graphs for dissipative twist maps on $\mathbb T^d\times\mathbb R^d$ in the $C^r$ category, for any $r\ge 1$, and show their persistence as invariant global attractors under quantitative sharp conditions involving both the size of dissipation parameter and the perturbation at stake. Note that when $d>1$ the geometric Lagrangian property of the invariant manifold may be destroyed, so that the graph may persist but fail to be Lagrangian. In general, additional dynamical requirements on the twist maps come into play if we intend to preserve also the Lagrangian property. In this line, we first study their persistence for general perturbations, then we restrict to the class of exact conformally symplectic twist maps and provide quantitative sharp conditions for the persistence of invariant exact Lagrangian graphs, without assuming any condition on the tangential dynamics of the manifold. The persistence result relies on a combination of a suitable multidimensional Herman–Mather formula proved in this setting and contraction estimates that provide sharp bounds on the $C^1$-norm of the perturbation to get a unique, globally attracting $C^r$ graph. In dimension $d=1$, by further assuming that that the circle map induced by the graph has irrational rotation number, we prove that every $C^1$ invariant graph of a $C^k$ map is $C^k$ for $k=2,3,\ldots,\infty$. Analytic maps with Lipschitz, non-$C^1$ invariant graphs at every prescribed irrational rotation number show that this initial regularity assumption is essential.