Abstract Let $P_1,P_2,\dots , P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or Chebyshev polynomials, and $S$ the semigroup under composition generated by P_k$. We show all elements share a measure maximal entropy if only intersection principal left ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.