I want to make my first self bow and I'm looking for woods that don't require chasing a ring. Can someone give me a list of woods I could use.

Hickory

pretty much any white wood doesn't require chasing a ring.

What Sam said, white sap wood just under the bark is back if your bow. I like pig nut hickory best.

I say Hickory too.

Considering Hesperia is dry .... I don't wanna be impolite and call it a desert .... Lol, hickory would probably be my go to.

It really performs well in a dry climate. Plenty of tension strength, no ring chasing, and when kept at relatively low moisture content will be a snappy shooter.

I would also suggest hickory. Hickory, unlike other woods, loves dry climates. Hickory can take 6% M/C where most other woods need 9% to 11% or they would explode at that low M/C.
Hickory should be cut during the growing season so the bark will peel off. If cut during the dormant season the bark will be hard to remove cleanly.

No one says Vine maple?

I've never used vine maple. I have a few staves of VM but haven't gotten to use them yet.

the vine maple bows i have seen, have been phenomenal shooters.
also just peel bark and build!