The inverse scattering problem for chaotic Hamiltonian systems

We propose an analysis of the inverse scattering problem for chaotic Hamilt onian systems. Our main goal will be the reconstruction of the structure of the chaotic saddle from asymptotic data. We will also address the question how to obtain thermodynamic measures and a partition from these data. An e ssential step in achieving this is the reconstruction of the hierarchical o rder of the fractal structure of singularities in scattering functions sole ly from knowledge of asymptotic data. This provides a branching tree which coincides with the branching tree derived from the hyperbolic component of the horseshoe in the Poincare map taken in the interaction region. We achie ve our goal explicitly for two types of systems governed by an external or an internal clock, respectively. Once we have achieved this goal, a discret e arbitrariness remains for the reconstruction of the horseshoe. Here symme try considerations can help. We discuss the implications for the inverse sc attering problem of the effects of finite resolution and the possible use o f nonhyperbolic effects. The connection between the formal development para meter of the horseshoe and the topological entropy proves helpful in the sy stems discussed. (C) 1999 Academic Press.