Milnor and Thurston [On iterated maps of the interval, Dynamical Systems (J. Alexander, ed.), Lecture Notes in Math. 1342, Springer, Berlin (1988), 465-563] proved that a logistic map with positive topological entropy is monotone semi-conjugate to a tent map with the same entropy. In this paper, we show that depending on whether the turning point of a tent map with positive entropy is periodic or not, the logistic maps with the same entropy are either semi-conjugate or conjugate to the tent map. Furthermore, logistic maps with different entropy are never monotone semi-conjugate to the tent map.

logistic family, tent family, conjugacy, topological entropy.