  It has been known since Julia that polynomials commuting under composition
have the same Julia set. More recently in the works of Baker and Eremenko,
Fern\'andez, and Beardon, results were given on the converse question: When do
two polynomials have the same Julia set? We give a complete answer to this
question and show the exact relation between the two problems of polynomials
with the same Julia set and commuting pairs.
