 Return-free state in a Markov chain

A state can have no positive-probability return at any positive time, for example when its transition-matrix column is zero. Its return-time set is empty. The convention $\gcd \varnothing =0$ assigns period zero; conventions defining period only for a nonempty return set instead leave it undefined. Such a state must not be called aperiodic merely because its communicating class is a singleton.