Renormalization, invariant tori, and periodic orbits for Hamiltonian flows

textConsideration is given to a family of renormalization transformations developed to study the existence of invariant tori in Hamiltonian systems. These transformations are used to construct invariant tori with self–similar frequency vectors, as well as sequences of periodic orbits approximating them, for near– integrable Hamiltonians in two or more degrees of freedom. Results on the location of the periodic orbits, and accumulation rates of these orbits to the invariant tori are presented. A numerical implementation of one of these transformations is used to to search for non–trivial fixed points in a reduced space of Hamiltonians with two degrees of freedom. Evidence supporting the existence of a fixed point that seems to be related to the break–up of golden invariant tori is provided. The critical indexand scaling found for this point are in good agreement with previous numerical experiments carried out for area preserving maps. Finally, a simple space of Hamiltonians in two degrees of freedom, all having a common invariant torus, is analyzed to gain insight vii into the dynamics of the flow on the invariant torus as the conjugacy to linear motion breaks down. By studying a particular one-parameter family of Hamiltonians, it is observed how the flow on the torus changes from being dense quasi–periodic to having an invariant quasi–periodic Cantor set. This transition is triggered by the appearance of a fixed point for the flow on the torus. Accumulation ratios of periodic orbits approaching the torus with this particular flow are found numerically, and shown to be different from those associated with known renormalization fixed pointsMathematic