E-2/C-2 left thumb control?
Most likely the pilot is trimming the airplane to alleviate control pressures after he adjusts his airspeed in the pattern.  Ideally you want to trim out any kind of control pressure that you have to place on the yoke, or rudder pedals, first adjusting your speed as desired using the power levers, then trimming using elevator-rudder-aileron trimming procedure. The “coolie hat” switch on the left grip on the yoke provides both elevator and aileron trim.