On ternary Dejean words avoiding 010

Thue has shown the existence of three types of infinite square-free words over {0, 1, 2} avoiding the factor 010. They respectively avoid {010, 212}, {010, 101}, and {010, 020}. (7+) Also Dejean constructed an infinite-free ternary word. A word is d-directed if it does not 4 contain both a factor of length d and its mirror image. We show that there exist exponentially (7+) many-free 180-directed ternary words avoiding 010. Moreover, there does not exist an 4 infinite-free 179-directed ternary word avoiding 010.