CRITERION FOR RAYS LANDING TOGETHER

Let f be a polynomial with degree ≥ 2 and the Julia set J f locally connected. We give a partition of complex plane C and show that, if z, z in J f have the same itinerary respect to the partition, then either z = z or both of them lie in the boundary of a Fatou component U , which is eventually iterated to a siegel disk. As an application, we prove the monotonicity of core entropy for the quadratic polynomial family {fc = z 2 + c : fc has no Siegel disks and J fc is locally connected }