5 - Maps on the circle

Summary

The motion along thin tori surrounding stable periodic orbits in the Birkhoff approximation was studied in section 1.3. The Poincaré maps on such tori are translations of the circle (rotations). According to the analysis in section 4.4, the periodic Birkhoff tori and their neighbours are broken up by the nearly resonant terms of the Hamiltonian, which generate small denominators in the normal-form transformation. Even so, the survival of nonperiodic tori is not excluded, as will be confirmed by the theorem of Kolmogorov, Arnold and Moser in chapter 6. Understanding the motion near stable periodic orbits thus requires a preliminary study of general motion on tori, or their Poincaré sections – maps on the circle.

There is a didactic incentive that perhaps even outweighs the abovementioned physical motivation for the study of circle maps. In the context of these maps, many of the mathematical difficulties that beset the analysis of stable motion manifest themselves in their clearest, simplest form. We will analyse the effect of the near commensurability of frequencies, that is, rational rotation numbers. The attempt to reduce general maps to rotations leads once again to small denominators. Convergence in their presence is possible, but this result is so surprising that considerable effort will be spent in sketching proof.