Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions

In our previous paper we introduced recurrent dimensions of discrete dynamical systems and we have estimated the upper and lower recurrent dimensions of discrete quasi-periodic orbits. In this paper, treating the case of 2-frequencies discrete quasi-periodic orbits, which correspond to the Poincaré sections of the 3-frequencies continuous quasi-periodic orbits, we estimate recurrent dimensions of the quasi-periodic orbits. Introducing some algebraic conditions between the two irrational frequencies, which are related to the Diophantine conditions of KAM theorem, we can estimate upper and lower recurrent dimensions of the orbits. We propose the gaps between the upper and the lower recurrent dimensions as the index parameters, which measure unpredictability levels of the orbits. Furthermore, we investigate these dimensions and their gaps for the quasi-periodic trajectories given by solutions of PDE with three periodic terms, the frequencies of which are rationally independent.