Can you get demoted from legendary arenas in Clash Royale (not counting the season reset)

Great question. According to this video around the 1:55 mark once you enter a new arena or leagues (4000+ trophies) you can not go back.

Here is the official website mentioning it as well:

Introducing... Trophy Gates! Trophy Gates allow you to save your Trophy progress, each time you reach a new Arena! When you pass through a Trophy Gate, you will NOT drop below that threshold. For example, when you hit 3600 Trophies and enter Spooky Town, you will not drop below 3600 Trophies again!

Once you hit 4000+ Trophies, Trophy Gates do not exist. You can drop Leagues but you will never drop below 4000 Trophies (Legendary Arena).

You didn't ask why, but in case anyone was wondering, it sounds like this was probably done to prevent people from intentionally dropping trophies and bullying lower arena players.