Deoxys can be captured on Birth Island, but getting there requires an event item that may be difficult to obtain without cheating. Players will need the Aurora Ticket, which was only given out at Nintendo events.
The Aurora Ticket was given away at the "Pokmon Rocks America" events held between 2003 and 2005. If it was not obtained there, it must be traded for with another player. Another possibility is to receive it as a Mystery Gift in one of the Nintendo DS games and then transfer it back to "Pokmon Emerald" by way of a DS model with a Game Boy Advance cartridge slot, but this process will require a lot of trial and error. The easiest method is to obtain one by way of a Gameshark or an Action Replay device.
Once the Aurora Ticket is obtained, the player can reach Birth Island by riding the S.S. Tidal. Birth Island is a tiny triangle-shaped island with only a single rock on it. When players approach the rock, it will move, and the player must then follow it in the minimum number of steps possible. After several rounds of this, Deoxys appears to battle the player and can be captured.