A dynamical systems approach to Birkhoff's theorem
Consider a monotone twist mapping of the cylinder. This means a diffeomorphism \(\phi\) of the cylinder which is isotopic to the identity and area-preserving and satisfies \[ \biggl|{\partial \phi_1 \over \partial y}\biggr|\geq \delta >0. \] Here \(\phi =(\phi_1 ,\phi_2)\) depends on the angular variable \(x\) and the vertical variable \(y\). An invariant curve \(\Gamma\) is a Jordan curve in the cylinder which is homotopically nontrivial and satisfies \(\phi (\Gamma)=\Gamma\). A classical result by Birkhoff says that every invariant curve must be a graph of the type \(y=\varphi (x)\) with \(\varphi\) continuous (later one proves that \(\varphi\) is even Lipschitz-continuous). Birkhoff's proof is very intuitive but difficult to formalize. Several authors have given complete proofs using Birkhoff's ideas. In this paper the author presents a different proof. He proceeds by contradiction and assumes that an invariant curve is not a graph. Then he proves that there is a Jordan domain \(\Omega_0\) whose boundary is composed by an arc of \(\Gamma\) and a vertical segment. The procedure can be iterated to produce a sequence of domains \(\Omega_n\) of this type and such that the measure \(|\Omega_n |\) tends to infinity. This is the searched contradiction.
scientific article; zbMATH DE number 1558726