Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I: Basic result: Separatrices of hyperbolic periodic points (Q1817889)

It is well-known that the existence of transversally intersecting separatrices of hyperbolic periodic solutions leads, in a typical situation, to complicated and irregular dynamics. Therefore, in the case of a two-dimensional mapping or a three-dimensional flow with this transversality property, there is no non-trivial analytic or meromorphic first integral, i.e., a function constant along each trajectory of the system under consideration. The additional robust conditions are obtained and discussed that guarantee the absence of such an integral in the many-dimensional case, regardless of the finite dimension in question (the strongest analytic non-integrability). These conditions guarantee also the absence of any non-trivial analytic one-parameter symmetry group, and, more generally, analytic or meromorphic vector fields generating a local symmetry, i.e., a local phase flow commuting with the system under consideration. Furthermore, the analytic centralizer of the system is discrete in the compact-open topology. A differential-topological structure of the invariant set of ``quasi-random motions'' described by symbolic dynamics is studied for this purpose. The approach utilized is essentially geometrical. Some related topics are also discussed.