 Cycle state in a minimal deterministic finite automaton for a finite language

In a minimal deterministic finite automaton accepting a finite language, any state lying on a nonempty directed cycle is the unique rejecting sink state. It cannot reach an accept state, since traversing the cycle arbitrarily many times would produce infinitely many accepted words. Every state reachable from it also rejects every continuation, so minimality forces all such states to be the same state.