Some ideas are:
- You branch off into another timeline and your actions make no difference to the previous timeline
- You’ve already taken said actions but just didn’t know about it so nothing changes
- Actions taken can have an effect (so you could suddenly erase yourself if you killed your parents)
- Only “nexus” or fixed events really matter, the timeline will sort itself out for minor changes
- something else entirely
I was just reading this article about a mathematical understanding of closed time-like curves.
In essence, the argument is that time travel to the part is possible with a degree of free will, but you would not be allowed to alter the part in such a way as to remove the motivation for traveling back in time. E.g., it would be like Futurama where Fry kills his grandfather, but he impregnates his grandmother, this allowing himself to be born. The idea is that the timeline would correct itself and ensure that your future self will always return to the past.