If my opponent moves Vex Apathetic into a battlefield, and I ambush with Rengar Trophy Hunter, if a tie occurs during battle, does my Rengar stay on the battlefield and the Vex gets pushed back because technically Rengar is defending?

To determine the outcome of this showdown, we must look at the specific sequence of events and the rules for combat resolution and cleanup.

Sequence of Events

1. Opponent's Move: Your opponent moves Vex, Apathetic to an empty battlefield. This initiates a non-combat showdown and makes the battlefield Contested. Your opponent is the Attacker (Rule 442.1.a.1).
2. Your Response: You play Rengar, Trophy Hunter as a [Reaction] to the move. Because there is an enemy unit (Vex) at the battlefield, you may play him there. Upon playing Rengar, you become the Defender (Rule 442.1.a.2).
3. Vex's Trigger: Vex, Apathetic's ability triggers because you played a unit while she is at a battlefield. The ability is placed on the chain, and upon resolution, Rengar, Trophy Hunter becomes Stunned and cannot move for the rest of the turn.
4. Combat Begins: The chain clears, both players pass focus, and the showdown transitions into a combat showdown.

Combat Resolution

• Combat Damage Step: Vex and Rengar exchange damage. Because Rengar is Stunned, he does not deal his Might in damage back to Vex (Rule 423.1.b).
• Combat Cleanup:
  1. Units heal.
  2. Recall: According to Rule 461.1.a.2 and RiftJudge FAQ #9995, attacking units are recalled to their controller's base if defenders are still present at the battlefield.
     • Even in the event of a "tie" (e.g., if both units survived combat), the Attacker (Vex) is recalled to your opponent's base because the Defender (Rengar) is still present at the battlefield.

Final State

• Vex, Apathetic: Recalled to your opponent's base.
• Rengar, Trophy Hunter: Remains on the battlefield.
• Conquer: Because you are the only player with a unit remaining at the battlefield during the Resolution step, you Establish Control and Conquer the battlefield (Rule 461.5).

Summary: Yes, your Rengar stays on the battlefield and the Vex is recalled to the opponent's base. This happens because Rengar is the designated Defender. Per the combat cleanup rules, the attacking unit is recalled if the defender remains at the battlefield.

This interpretation is based on Rule 442.1.a (Attacker/Defender designations), Rule 461.1.a.2 (Combat Cleanup/Recall), Rule 423.1.b (Stun interaction), and RiftJudge FAQ #9995.