Exponentially Small Splitting of Separatrices Associated to 3D Whiskered Tori with Cubic Frequencies

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector omega/root epsilon, with omega=(1,Omega,(Omega) over tilde) where Omega is a cubic irrational number whose two conjugates are complex, and the components of omega generate the field Q(Omega). A paradigmatic case is the cubic golden vector, given by the (real) number Omega satisfying Omega(3)=1-Omega, and (Omega) over tilde=Omega(2). For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors < k,omega >,k is an element of Z(3). Applying the Poincare-Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on epsilon) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in epsilon, and valid for all sufficiently small values of epsilon.This estimate behaves likeexp{-h(1)(epsilon)/epsilon(1/6)}and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function h(1)(epsilon)in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector omega, and proving that it is a quasiperiodic function (and not periodic) with respect to ln epsilon. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.