A degenerate Hartman theorem
The importance of the Hartman theorem in the study of dynamical systems and particulary for the three-body problem is well-known. The author considers the case of a real analytic diffeomorphism, \(f:\mathbb{R}^2\to \mathbb{R}^2\), with \(q\) as a non-hyperbolic fixed point and \(\Delta f(q)= \text{Id}\). The author also announces several ideas as a basis for further study. He considers a vector bundle with base space a 2-manifold with fiber dimension 2, and he constructs the vector cone by fixing an element of the base space and by choosing a particular subset within the fiber over this element. He proves that this subset is closed under addition and scalar multiplication and he studies an orbit for the element of the base space and its associated cone field. It is also shown that the tangent space of the stable manifold of the equilibrium point acts on by the derivative of the diffeomorphism, and it is observed how this derivative influences the vector cone. A coordinate system with domain the base space is obtained by applying these ideas to the inverse diffeomorphism. Briefly, imposing nontrivial sufficient conditions on a 2-dimensional diffeomorphism ensures that the diffeomorphism is topologically conjugate to a decoupled map.
scientific article; zbMATH DE number 952562