Adding the [No Encounters] ability to one of your characters equipped armor allows you to avoid random fiend encounters as you move throughout Spira. You only need to have armor of this type on one of your characters (not all three) in order to stop random encounters.
In order to add the [No Encounters] ability using the equipment customization menu, you will first need to obtain 30 Purifying Salts. The easiest way to do this is through part of the Monster Arena side quest. You will receive 99 Purifying Salts for catching 4 of every Dragon species in the game.
Make your way to the Monster Arena which is located on the far east side of the Calm Lands. Walk through the gorge and you will automatically dismount from your Chocobo. Enter the Monster Arena area and speak to the man in front of the Save Sphere. He will ask you to gather fiends from around Spira.
Buy the Training Weapons that he sells and equip them on your characters and then head to then head to the following locations. You must obtain 4 of each Dragon:
Mi’ihen Oldroad (which is after the Rin’s Travel Agency) to find Vouivre:
Mushroom Rock Road to find Lamashtu:
Thunder Plains to find Kusariqqu:
Sanubia Desert to find Mushussu:
Cavern of the Stolen Fayth or Mount Gagazet to find Nidhogg:
Use the Purifying Salt to add No Encounters to your armor.