CFM (GE + Safran) own 39%, PW has 26%, RR has 18%, GE on it's own has 16%. So together they own 99% of the market.
Honeywell and Williams don't do anything over 10k-lb thrust (that I see). UEC is russian state owned, so I don't see anyone working with them anytime soon and AVIC is chinese state owned so kinda the same deal.
If you are talking small turbofans (smaller than 10k-lb thrust) there are a variety of companies out there but as soon as you start looking for bigger stuff to power your 737 Max or 320NEO class planes, you are talking about a very exclusive club.
As an aside, I would love to see more competition, I just think it's a really tough space to enter.
Honeywell Aerospace, Williams International, United Engine Corporation, and AVIC are some others.
There are more than you might think.
https://en.wikipedia.org/wiki/Category:Gas_turbine_manufactu...
Not all of these make aircraft engines, of course, but a fair number of them do.