Recycling of strange sets: I. Cycle expansions

The strange sets which arise in deterministic low dimensional dynamical systems are analyzed in terms of (unstable) cycles and their eigenvalues. The general formalism of cycle expansions is introduced and its convergence discussed. 1 present address: Dipartimento di Fisica dell'Universit`a and I.N.F.N., Via Celoria 16, I-20133 Milano 2 Carlsberg Fellow 1 INTRODUCTION The goal of the present series of papers is development of a perturbation theory of the low-dimensional deterministic chaos of predictive quality comparable to that of the traditional perturbation expansions for nearly integrable systems. In the traditional approach the integrable motions are used as zeroth-order approximations to physical systems, and weak non-linearities are then accounted for perturbatively. For strongly non-linear, non-integrable systems such expansions fail completely; the asymptotic time phase space exhibits amazingly rich structure which is not at all apparent in the integrable approximation...