I play Akshan to steal an opponent's gear, and my opponent plays another Akshan to steal the same gear. If my opponent's Akshan dies, who gets control of the gear?

When your opponent's Akshan dies, you regain control of the gear.

According to the ruling provided in the FAQ:

• Control-change effects are applied in timestamp order: When your opponent played their Akshan, their control-change effect depended on and overrode your earlier effect.
• Effects stop applying when their source leaves play: When your opponent's Akshan dies, their control-change effect stops applying.
• Recalculation: With your opponent's effect no longer active, the game recalculates the control of the gear. Since your original control-change effect is still active (assuming your Akshan is still on the board), you regain control of the gear.

If your Akshan were to leave the board as well, the gear would return to its original owner, as no control-change effects would remain.