Evaluation of the limits of maximal means

It is proved that the limit
$$ \lim_{\Delta\to\infty}\sup_\gamma\frac 1\Delta \int_0^\Delta f\bigl(\gamma(t)\bigr)\,dt, $$
where $f\colon\mathbb R\to\mathbb R$ is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation $\dot\gamma\in[\omega_1,\omega_2]$, coincides with the limit
$$ \lim_{T\to\infty}\sup_{c\ge0}\varphi_f(k,T,c), $$
where
$$ \varphi_f=\frac{(k-1)\overline I_f(T,c)} {1+(k-1)\overline\lambda_f(T,c)},\qquad k=\frac{\omega_2}{\omega_1}. $$
Here $\overline\lambda_f=\lambda_f/T$, $\overline I_f=I_f/T$, and $\lambda_f$ is the Lebesgue measure of the set
$$ \bigl\{\gamma\in[\gamma_0,\gamma_0+T]: f(\gamma)\ge c\bigr\}=A_f,\qquad I_f=\int_{A_f}f(\gamma)\,d\gamma. $$
It is established that this limit always exists for almost-periodic functions $f$.