Turing Machine state going to itself
Yes, a state can go to itself when seeing any symbol. (As Steven's answer says, the analogous question for head movement - whether the head of the Turing machine has to move at each stage, or whether it can stay put at a given moment - varies from definition to definition, but I've never seen a definition which prohibits a state from looping to itself.)


Note that this is a particularly natural thing to allow: it's basically a "while" command.


However, as with many aspects of Turing machines, tweaking the definition to rule it out isn't significant: Turing machines which prohibit states from looping to themselves are no weaker than Turing machines which allow states to loop to themselves. This is a good exercise.