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 } . Also Dejean constructed an infinite (cid:16) 74+ (cid:17) -free ternary word. A word is d -directed if it does not contain both a factor of length d and its mirror image. We show that there exist exponentially many (cid:16) 74+ (cid:17) -free 180 -directed ternary words avoiding 010 . Moreover, there does not exist an infinite (cid:16) 74+ (cid:17) -free 179 -directed ternary word avoiding 010 .


Introduction
This note is about words avoiding repetitions, a well-studied area in combinatorics on words [5,6].A repetition is a factor of the form r = u n v where u is non-empty and v is a prefix of u.Then |u| is the period of the repetition r and its exponent is |r|/|u|.A word is α + -free (resp.α-free) if it contains no repetition with exponent β such that β > α (resp.β ⩾ α).
We consider ternary square-free words (i.e., 2-free words) with additional avoidance constraints.Thue [7] has shown that there exist infinite square-free words avoiding the factors {010, 212}, famous ternary word named b 3 in [2].Then b 3 avoids squares and {010, 212}, but for every finite factor f of b 3 , there exist only finitely many ternary words avoiding squares and {010, 212, f }.See also [2] for the construction of the other two morphic words.Thus the language L of (finite or infinite) ternary square-free words avoiding 010 is worth considering since it contains these three interesting words.
To refine square-freness, Dejean [3] introduced the notion of repetition threshold RT (n), which is the least real α such that there exists an infinite word over the n-letter alphabet that is α + -free.She proved that RT (3) = 7  4 by exhibiting a 19-uniform morphism whose fixed point is + -free.Our main result shows that both constraints, avoiding 010 and being 7 4 + -free, can be satisfied simultaneously.Moreover, the language L ′ of ternary 7 4 + -free avoiding 010 still contains exponentially many words.Notice that L ′ is also closed under reversal.However, there exist infinite words w ∈ L ′ such that the factor set of w is not closed under reversal, as opposed to the three cases studied by Thue.A quantitative notion of "factor set not closed under reversal" is defined in [1] as follows.A word w is d-directed if for every factor f of w of length d, the reversed word f R is not a factor of w.

Proof
A morphism is q-uniform if the image of every letter has length q.Also, a q-uniform morphism h : Σ * s → Σ * e is synchronizing if for any a, b, c ∈ Σ s and v, w ∈ Σ * e , if h(ab) = vh(c)w, then either v = ε and a = c or w = ε and b = c.We will need the following result, which corresponds to the case n = 1 of the main lemma in [4].
, then h(t) is β + -free for every (finite or infinite) α + -free word t.
Let us prove the first part of Theorem 1.1.We use Lemma 2.1 to show that the image of every  The C code to find and check this morphism is available at https://www.lirmm.fr/˜ochem/morphisms/dejean010.htm.
Now we prove the second part of Theorem 1.1.Let L 179 be the language of 7 4 + -free 179-directed ternary words avoiding 010.The standard backtracking algorithm is rather slow to show that L 179 is finite.So we use the following simple optimisation.We choose an arbitrary small factor in L 179 , namely f = 1012102120121020120212012101202.Then we use backtracking to show that: (1) The language of words in L 179 avoiding f is finite.
(2) The language of words in L 179 with prefix f is finite.
This rules out that L 179 contains an infinite word w: by item (1), w must contain f , but then by item (2), w cannot be extended indefinitely to the right of this factor f .

7 5 + 4 + 5 and β = 7 4 , 4 +
-free word over Σ 4 by the following 1557-uniform morphism is 7 -free.With α = 7 we get 2β β−α = 10, so we need to check the 7 -freeness of the image of every It is easy to check that such words avoid 010 and are 180-directed.