Nikolai G. Moshchevitin
Systems with many frequencies and
Diophantine approximations
We discuss the recurrence properties of trajectories of certain
dynamical systems with many frequencies.
Let $f : \mathbb{T}^n \to \mathbb{R}$ be a smooth function with
zero mean value. Then we prove that for any set of linearly
independent over $\mathbb{Z}$ frequencies $\omega
=(\omega_1.,,,\omega_n)$ the integral
$$
\int_0^T f(\omega t+\varphi )dt
$$
has the recurrence property and this property holds not uniformly
in initial phase $\varphi$.
By means of such integral one can
express the solutions of some degenerate Hamiltonian systems. Also
there are applications to the equations of the kind
$$
dX/dt = A(t) X,\,\,\, X \in {\rm SO}(n),\,\,\, A\subset {\rm
so}(n),
$$
where A(t) is an almost periodic or conditionally periodic
function.
The results obtained depend on Diophantine properties of
frequencies and we find out some new phenomena in general theory
of Diophantine approximations.