In the short story "A Rose for Emily," is Homer Barron a static or a developing character?
Homer Barron is, in literary terms, a static character. Homer doesn't change in the story. He doesn't develop or grow.
For instance, he shows interest in Emily, comes and goes as workers often do, shows up one last time, then disappears. But nothing is revealed about his thought processes or his decision making. Even the statement that Homer likes men is hearsay and doesn't amount to any type of growth on Homer's part.
The reader doesn't even know, as Emily must have presumed, that Homer decides not to marry her or remain in a relationship with her. Very little about Homer is actually revealed.
Homer is important because he represents the North and the new, because he has a relationship with Emily, and because of what Emily does to him. In fact, in terms other than literary, he does develop--into a corpse, or course. But he is not a dynamic character and does not develop, in characterization terms, in the story. He is an important character in the story, but he is a static character.