Completely Li--Yorke chaotic homeomorphisms with positive entropy

It is an open problem whether a homeomorphism on a compact metric space satisfying that each proper pair is either positively or negatively Li–Yorke, called completely Li–Yorke chaotic, can have positive entropy. In the present paper, an affirmative answer to this question is given. In fact, for each homeomorphism $T$ with positive entropy such that each proper pair is not two-sided asymptotic, a completely Li–Yorke chaotic homeomorphism with positive entropy associated with the given homeomorphism can be constructed.