Global Dynamics

Abstract

In the preceding chapters, we have achieved a detailed understanding, based on the organizing principles of renormalization and symbolic coding, of the local dynamics associated with renormalizable kicked-oscillator maps. We now wish to show how the whole recursive return map formalism can be extended (“lifted”) to the aperiodic orbits on the infinite plane of the global map W. The local scaling, characterized by the self-similar contraction of domains in the fundamental cell Ω, will be seen to have, if certain conditions are satisfied, a global counterpart in the form of asymptotic self-similar expansion. Just as the local contraction (within a single ergodic component) is associated with a scale factor ω_K < 1, the global expansivity will have its own scale factor ω_W > 1. Given that the temporal behavior scales with a factor ω_T, it is not surprising that orbits will be found to tend to infinity for asymptotically long times with a power-law exponent log ω_W/logω_T.