The study of dynamical systems from a probabilistic perspective is called ergodic theory. Have a look at my other comment for some specific references for more

The study of dynamical systems from a probabilistic perspective is called ergodic theory. Have a look at my other comment for some specific references for more on this. And yes it is very interesting. One can actually model iid coin flips (or dice rolls) as a type of expanding dynamical system called a full-shift.