How do you get out of the doomsday event and go back to the regular world?

If you are in the New Eden event for doomsday (season X) where you travel to a special state for doomsday to take plots then you can travel back and forth between the new eden state and your own state using the event menu.

Tap on the doomsday event icon on the right if you are in the new eden map or find the event in the events menu while in your city. Then select the Season tab at the top and Use the Exit button to return to your state. You repeat this process and use the Enter button to go back into new eden if you want to.