Santa Barbara was not exactly its first choice. Santa Barbara daisy, Erigeron karvinskianus, is not even native to California. It is actually from Central America. It just happens to do very well here, and can naturalize if conditions are right. It can be rather grungy through summer in the wild, but with a bit of watering, it can bloom nicely all year.
The thin stems can spread a few feet without getting more than a foot deep. If even shallower growth is preferred, older plants can be cut down or pulled up as they get replaced by their own offspring. The narrow leaves are quite tiny. The white or slightly pinkish flowers are not much bigger, less than half an inch wide, with prominent yellow centers.
Santa Barbara daisy is also known as Mexican fleabane, both because it is actually native to Mexico, and also because it is supposedly useful for repelling fleas. The problem with using it to repel fleas is that only its smoke is effective. There are probably other herbal alternatives that work just as well without being a fire hazard.