On the Liouville phenomenon in estimates of fractal dimensions of forced quasi-periodic oscillations

Abstract

We develop a method for studying fractal dimensions of forced almost periodic oscillations in various differential equations. The method is based on the previously introduced notion of the Diophantine dimension of an almost periodic function that is strictly connected with Diophantine approximations of its frequencies. Some estimates of the Diophantine dimension for typical quasi-periodic perturbations are obtained. For a class of control systems we state frequency-domain conditions under which the presented approach can be applied. As a result of our investigations one may observe a number-theoretic phenomenon (the Liouville phenomenon) arising within the mentioned problem. Its effect is in that we can not control fractal dimensions of quasi-periodic oscillations having well-approximable frequencies.