No, luckily Santa Cruz does not have rent control or anything similar to it.
The only nuisance is the “Rental Ordinance” in the CITY of Santa Cruz. I believe it applies for owners of more than 3 SFRs, or any single property with 5+ units. The rental ordinance allows the city to do inspections, and they can require the owner to make repairs. The inspection is annually, but the owner can self-inspect and the city inspects only every 5 years.
The city of SC also wants such landlords to have a business license … I think it’s ~$100 a year.
The “Rental Ordinance” can be avoided by buying in the County or in other cities.
Some areas have limitations on vacation rentals.