If you run the random process many times and then look at a histogram of the vector of outputs, the distribution peaks somewhat higher than 2 [0], but the outpu

If you run the random process many times and then look at a histogram of the vector of outputs, the distribution peaks somewhat higher than 2 [0], but the output of the random process does not tend towards any particular number or sequence of numbers. Indeed, the distribution looks highly discontinuous and fractal. It's a very intriguing question. I suspect thinking in terms of rings (abstract algebra) would be helpful. [0] "somewhat higher" - I think this is because the sequence "add two then divide by two" gives you back your input when you start with 2, while "divide by two then add two" does when you start with 4; the distribution should peak between these two "stable" points. Exactly where the distribution peaks probably depends on your histogram bin width.