Renormalization Group Analysis of Hamiltonian Flows

This paper is a summary of some recent work by the authors on the renormalization of Hamiltonian systems. Applications include the construction of invariant tori and related sequences of closed periodic orbits. We also discuss problems related to the breakup of invariant tori, and some numerical results. The renormalization group transformation. In this paper we describe some recent results [9–12] on the renormalization of Hamiltonian systems with two or more periodic degrees of freedom q = (q1, . . . , qd). One of the goals is to study invariant tori with certain arithmetically interesting rotation vectors, for the flow (q̇, ṗ) = (∇2H,−∇1H) associated with a Hamiltonian H = H(q, p). Here, ∇1H and ∇2H denote the partial gradients of H with respect to the first (angle) variable q and the second (action) variable p = (p1, . . . , pd), respectively. Unless specified otherwise, a rotation vector ω ∈ R is assumed to be nonzero, and parallel rotation vectors are identified. An invariant torus will be called an ω–torus, if the motion on it is conjugate to the linear flow qj(t) = qj(0) + tcωj (mod 2π), for some nonzero constant c. We are particularly interested in rotation vectors ω = (1, ω2, . . . , ωd) whose components span an algebraic number field of degree d. They can also be characterized by a “self–similarity” property [11]: There exists an integer d × d matrix T , with determinant ±1 and d − 1 simple eigenvalues of modulus less than 1, for which ω is an eigenvector with a real eigenvalue θ1 > 1. The best known example is ω = (1, θ1) with θ1 the golden mean, and T = ( 0 1 1 1 ) . We note that the matrix T can also be used to construct sequences of rational approximants to ω, by defining wn = cnTw for some nonzero w ∈ Q, where cn is a suitable normalization. In order to study ω–tori and wn–orbits, we would like to define a renormalization group (RG) transformation R, acting on a space of Hamiltonians, such that (1) R(H) = H ◦ T1 (mod G) , 1991 Mathematics Subject Classification. Primary 58F27; Secondary 58F22, 70K50.. HK was supported in part by the National Science Foundation under grants no DMS-9705095 and DMS-0088935. PW was supported in part by the Swiss National Science Foundation. c ©1997 American Mathematical Society