How does an oscilloscope synchronize its screen refresh to the waveform?
For a digital scope, the capture and the display is mostly decoupled.
For your example of the whole capture period being 1us long, the scope will trigger, capture the whole waveform into the buffer, and then display the buffer to the screen. Then it will retrigger sometime later, capture that, and display that. The refresh rate of the screen isn't really relevant, except in relation to how often the scope can show you a new waveform. It's up to the trigger circuit (and the user adjusting the trigger settings) to make sure that the trigger occurs at the same point in the waveform each time, so that successive triggers show a stable picture.
If you use a slower timebase (say, 100ms/div), then the scope can start pulling samples out of the buffer and displaying them on screen before the whole capture is done, so that you can see the waveform &quot;as it happens&quot;. But the samples are still flowing into the buffer at a nice constant rate, and again, the refresh rate of the screen doesn't really matter.
For an analog scope, things are pretty different. There is no refresh rate, the timebase generator controls the horizontal deflection, the incoming waveform controls the vertical deflection, the trigger gets the party started, and phosphor persistence keeps things on screen for long enough for you to see a whole waveform at once. Each time the trigger fires, the beam starts moving from the left of the screen to the right at a controlled rate, wiggling up and down as it goes... and of course the trigger circuit won't let it start over at the beginning before it gets to the end.