On the Dynamics of Invariant Graphs for Dissipative Twist Maps
We study the persistence and internal dynamics of invariant graphs for dissipative twist maps. Explicit $C^1$ persistence and parameter estimates allow every real rotation number to be realized by varying either translation parameter, while arbitrarily small zero-mean analytic perturbations at fixed irrational horizontal parameters can yield analytic graphs with dynamics conjugate to rational rotations. With a fixed Brjuno horizontal parameter and fixed-strip analyticity, normally attracting $C^r$ graphs persist under perturbations of size $(1-λ)^γ$, for any $γ>0$ and prescribed finite $r$, as $λ\to1^-$. At the critical Lipschitz bound, the invariant graphs can carry Denjoy dynamics and have countably many nondifferentiable points, with perturbations smooth outside a null Cantor set and tending to zero in every Hölder norm of order less than one.