A Renormalization Group for Hamiltonians: Numerical Results

We describe a renormalization group transformation that is related to the breakup of golden invariant tori in Hamiltonian systems with two degrees of freedom. This transformation applies to a large class of Hamiltonians, is conceptually simple, and allows for accurate numerical computations. In a numerical implementation, we find a nontrivial fixed point and determine the corresponding critical index and scaling. Our computed values for various universal constants are in good agreement with existing data for area–preserving maps. We also discuss the flow associated with the nontrivial fixed point. 1 Supported in Part by the National Science Foundation under Grant No. DMS–9705095. 2 Supported in Part by the Swiss National Science Foundation.