I am looking for examples of Markov chains for each of the following conditions:
The set of points visited by the chain issued from a point x is not a.s. constant.
The set of points visited by the chain issued from a point x is a.s. constant, the order of the first three points visited is not deterministic, and x is not recurrent.
There are probably easy examples, but I just can't think of any, so I'd be grateful for some help.