Recurrent properties of quasi-periodic dynamical systems with multiple frequencies of p-adic Liouville numbers

Abstract

We introduce a class of p-adic Liouville numbers by using Schneider’s continued fractions with the orders, which specify the goodness of rational approximations. Considering the simultaneous rational approximations of the p-adic number and its powers, we give some inequality relations between these orders and the exponents of Diophantine approximations. As typical examples, given by using these inequality relations, we treat quasi-periodic dynamical systems, the frequencies of which are these p-adic numbers, and we investigate recurrent properties of their orbits. We obtain the positive gap values between the upper and the lower recurrent dimensions, which show some unpredictability properties of the orbits. Furthermore, we give symbolic dynamical descriptions for these properties.