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Why isn't your example Markov?

It is in fact Markov; Markov just means that the probability distribution of the future depends only on the present, and so the past adds no additional information in conjunction with the present. That's certainly the case here.

This is an example of a Markov chain that is not aperiodic; what that means is that, given a starting node, at any point in time in the future, it will always be the case that it is impossible to be at a certain node. This ends up meaning that the Markov chain never ends up reaching a steady state; rather, its behavior is periodic!

fix some n in n. if my starting state is [1; 0] then the probability of the occupancy being [1; 0] after n cycles is either 1 or 0. If the starting state is [0; 1] then the probabilty of the occupancy being [1; 0] is exactly the opposite, so for a fixed point in the future, the probability is tightly past-dependent.

Yes, but I think it is still a Markov process since the information is contained in its current state.

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