How is directional control during takeoff and landing maintained in tailwheel airplanes equipped with skis?
I have a fair amount of ski time on Cessna 180s and Citabrias. It's a bit like flying floats. The main thing is, you have no brakes.
For lateral control, the skis have varying amounts of lateral compliance depending on the snow compaction, so you slip and slide more or less for each situation, but generally, the laggy reactions makes it easier to control direction on takeoff and landing compared to wheels.  The compliance is like being on wheels on wet grass; instead of darting to the side when the plane swings, it just kind of sashays laterally giving you more time to make corrections.
You may have a ski on the tail, which provides a limited amount of steering control to the extent that the ski design is able to bite sideways (a little skeg on the bottom really helps).  A decent tail ski on lightly compacted snow provides pretty good steering control without having to use large blasts of power to blow the rudder.  Otherwise, you are dependent on large blasts of power to blow the rudder to turn.
On ice or very deep powdery snow, you will be depending more on power to turn, and this is quite a challenge because the shot of power to get the tail to swing to turn also accelerates you forward, and remember, you have no brakes.
Bottom line is you can't really maneuver in tight spaces when on skis without using huge bursts of power, and annoying the local owner whose airplane may be in your prop wash, so it requires a lot more preplanning and thinking things through.
Some airplanes do without a tail ski, keeping the normal tailwheel, and this makes it easier to maneuver on compacted snow where the tire has some bite.  On deep loose snow, the tailwheel becomes an anchor, and you have to lift it out of the snow with shots of power and elevator, on top of lots of rudder to try to maneuver.  That can get exciting.  A tailwheel-only ski plane in deep snow is a pain.  For aircraft with wheel-skis, you normally see a tail ski with a penetrating wheel to do double duty.
So, you have no brakes. So you have to do run-ups on the move, either in a big circle if you have the space, or on the takeoff &quot;roll&quot; if you don't.
Then there's landing at the end of a sunny warm day on a compacted runway and the temperature drops below freezing in the late afternoon as the sun sets, and the runway has a bit of a downhill slope to it, and it's essentially compacted snow with an ice glaze.  Did I mention that you have no brakes?  Been there, done that.