What is special about the frequency $\omega_0=\pi$ that suddenly causes rate of oscillation decrease?
As a consequence of the sampling theorem, the frequency spectrum of a discrete random variable is periodic, but it turns out that signals that are inherently discrete in time and not derived from an A/D converted continuous time signal are also periodic in frequency.  As a consequence, as the frequency is increased, it goes from the positive portion of one periodic interval, to the next, i.e. it transitions to the negative image of the next interval.

In the old days, DSP was mostly a sub discipline of Electrical Engineering, and DSP was focused on doing things digitally that were previously done analog.  In order to talk about DSP, without having to reference, a sampling frequency, a convention called unit sampling was commonly used, and a period of the spectrum is from -1/2 to 1/2.

As time progressed it was noted that DSP wasn't directly tied to processing analog derived signals and unit sampling was somewhat artificial, so it was recognized that $2\pi$ was a natural periodic interval, an $2\pi$ fits between $-\pi$ and $\pi$, which is why $\omega=\pi $ is a transition point in the spectrum.