Tero Harju
email: harju@utu.fi

Jetro Vesti

Luca Q Zamboni

On a question of Hof, Knill and Simon on palindromic substitutive systems

Keywords: Discrete one-dimensional Schrödinger operators, class P conjecture, primitive morphic words, rich words

In a 1995 paper, Hof, Knill and Simon obtain a sufficient combinatorial criterion on the subshift Ω of the potential of a discrete Schrödinger operator which guarantees purely singular continuous spectrum on a generic subset of Ω. In part, this condition requires that the subshift Ω be palindromic, i.e., contains an infinite number of distinct palindromic factors. In the same paper, they introduce the class P of morphisms f : A * → B * of the form a → pq a with p and q a palindromes, and ask whether every palindromic subshift generated by a primitive substitution arises from morphisms of class P or by morphisms of the form a → q a p. In this paper we give a partial affirmative answer to the question of Hof, Knill and Simon: we show that every rich primitive substitutive subshift is generated by at most two morphisms each of which is conjugate to a morphism of class P. More precisely, we show that every rich (or almost rich in the sense of finite defect) primitive morphic word y ∈ B ω is of the form y = f (x) where f : A * → B * is conjugate to a morphism of class P, and where x is a rich word fixed by a primitive substitution g : A * → A * conjugate to one in class P.

Introduction

In [START_REF] Hof | Singular continuous spectrum for palindromic Schrödinger operators[END_REF], Hof, Knill and Simon studied the spectral properties of discrete one-dimensional Schrödinger operators (Hφ)(n) = φ(n + 1) + φ(n -1) + V (n)φ(n), in the Hilbert space H = 2 (Z) of all square summable sequences with potential V : Z → R belonging to a subshift generated by a primitive substitution.

In physical terms, the spectral properties of H determine the "conductivity properties" of the given structure. Roughly, if the spectrum is purely absolutely continuous, then the structure behaves like a conductor. This is for instance the case when the potential V is periodic. In contrast, in the case of pure point spectrum, the structure behaves like an insulator. An intermediate spectral type, known as singular continuous spectrum, is expected to give rise to intermediate transport properties. This is for instance the case when the potential is Sturmian [START_REF] Damanik | Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity[END_REF] or generated by a substitution [START_REF] Bovier | Spectral properties of one-dimensional Schr dinger operators with potentials generated by substitutions[END_REF][START_REF] Damanik | Singular continuous spectrum for a class of substitution Hamiltonians[END_REF][START_REF] Hof | Singular continuous spectrum for palindromic Schrödinger operators[END_REF].

In [START_REF] Hof | Singular continuous spectrum for palindromic Schrödinger operators[END_REF], Hof, Knill and Simon give a sufficient combinatorial criterion for purely singular continuous spectrum in terms of a strong palindromicity property of the underlying word. More precisely, a word x ∈ A Z is said to be strongly palindromic if there exist B > 0 and a sequence (u i ) i≥1 of palindromic factors of x centered at m i -→ +∞ such that e Bm i /|u i | -→ 0. They then show that if x ∈ A Z is aperiodic and palindromic, meaning that x contains infinitely many distinct palindromes, then its subshift contains uncountably many strongly palindromic words (see Proposition 2.1 in [START_REF] Hof | Singular continuous spectrum for palindromic Schrödinger operators[END_REF]). It follows from a result of Jitomirskaya and Simon in [START_REF] Jitomirskaya | Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators[END_REF] that if x is strongly palindromic, then the point spectrum of H x is empty. They then deduce that if Ω is uniquely ergodic and generated by an aperiodic palindromic word w, then the operator H x has purely singular continuous spectrum for uncountably many x ∈ Ω. In the same paper they introduce the set P of (non-erasing) morphisms f : A * → B * of the form a → pq a where p, q a are each palindromes. Morphisms in P are said to be of class P. Actually in [START_REF] Hof | Singular continuous spectrum for palindromic Schrödinger operators[END_REF] they consider only primitive substitutions in P. As was observed in [START_REF] Hof | Singular continuous spectrum for palindromic Schrödinger operators[END_REF], substitutions in P generate palindromic subshifts. They also point out that substitutions of the form a → q a p with p and q a palindromes also generate palindromic subshifts. Thus they extend P to include also substitutions of the form f (a) = q a p and remark: Remark 1 (Hof, Knill, Simon, Remark 3 in [START_REF] Hof | Singular continuous spectrum for palindromic Schrödinger operators[END_REF]). We do not know whether all palindromic subshifts generated by primitive substitutions arise from substitutions in this extended class.

Over the years this remark has evolved into what is now called the class P conjecture. The first step in the evolution process, which is perhaps non consequential, was to convert this remark into a question. The second step, which in our minds represents a significant alteration, was to replace the entire subshift by a single element within the subshift which is fixed by a primitive substitution. The third was to give a precise interpretation to "arise from" as meaning "fixed by". The fourth and final step was to call it a conjecture.

The combinatorial criterion given by Hof, Knill and Simon in [START_REF] Hof | Singular continuous spectrum for palindromic Schrödinger operators[END_REF] appears to have very limited applications to the spectral theory of discrete one-dimensional Schrödinger operators. In fact, Damanik and Zare [START_REF] Damanik | Palindrome complexity bounds for primitive substitution sequences[END_REF] prove that the strongly palindromic sequences in a primitive substitution dynamical system form a set of measure zero. Moreover, the spectral theory of substitution defined potentials in itself is rather limited in that it would only establish behavior, expected on a set of full measure, for a set of measure zero potentials. Nevertheless the question concerning the structure of substitutions which generate palindromic subshifts is of independent interest in substitution dynamics. We regard the class P conjecture as an attempt to explain how a fixed point of a primitive substitution can contain infinitely many palindromes. Of course, a typical substitution does not preserve palindromes, hence one would expect that a palindrome generating substitution would have some particular inherent structure.

Although the original question posed by Hof, Knill and Simon concerned the entire subshift generated by a fixed point of a substitution, little is lost by restricting to the orbit of the fixed point, since while they need not all be fixed points themselves of primitive substitutions, they are always morphic images of such. In this respect, it is reasonable to widen the class of possible morphisms. In fact, if x ∈ A ω is generated by a primitive substitution f : A → A + and each of the images f (a) for a ∈ A begins or ends in a common letter, then one may conjugate each of the images by this common letter to obtain a new primitive substitution which generates the same subshift as f. However in general, if f is a class P morphism, then this new substitution need no longer be a class P morphism although it, or some power of it, will have a palindromic fixed point. For instance, Blondin Massé proved that the fixed point x of the primitive substitution a → abbab, b → abb is palindromic, but that x itself is not fixed by a primitive substitution in P (see Proposition 3.5 in [START_REF] Labbé | Propriétés combinatoires des f -palindromes[END_REF]). However this morphism is conjugate to the class P morphism a → bbaba, b → bba. Thus it is reasonable to consider the set P of all morphisms f which are conjugate to some morphism in P. Let FP denote the set of all infinite words which are fixed by some primitive substitution in P . Then the original remark of Hof, Knill and Simon was reformulated in terms of the following conjecture, called the class P conjecture: Conjecture 1 (Blondin Massé, Labbé in [START_REF] Labbé | Propriétés combinatoires des f -palindromes[END_REF]). If x is a palindromic word fixed by a primitive substitution, then x ∈ FP .

Partial results in support of the conjecture were obtained by Allouche et al. in case x is periodic (see [START_REF] Allouche | Palindrome complexity[END_REF]) and by Tan in case x is a binary word (see [START_REF] Tan | Mirror substitutions and palindromic sequences[END_REF]). Tan proves that if x is a palindromic binary word fixed by a primitive substitution f, then f 2 ∈ P .

Recently Labbé [START_REF] Labbé | A counterexample to a question of Hof, Knill and Simon[END_REF] produced a counter-example to the class P conjecture on a ternary alphabet. The counter-example is given by the fixed point

x = acabacacabacabacabacacabac • • • of the primitive substitution: f : a → ac, b → acab, c → ab.
He proves that x is palindromic but not in FP (see [START_REF] Labbé | A counterexample to a question of Hof, Knill and Simon[END_REF]). But let us remark that Labbé's counter-example to the class P conjecture does not constitute a negative answer to the original question (or remark) of Hof, Knill and Simon. In fact, it is readily verified that the second shift

T 2 (x) = abacacabacabacabacacabacabacacab • • •
is the fixed point of the class P morphism:

g : a → ab, b → acac, c → ac.
So the subshift generated by x is in fact generated by a morphism in class P. What is surprising is that Labbé's counter-example to the class P conjecture is not only palindromic, but is as rich as possible in palindromes. More precisely, Droubay, Justin and Pirillo observed that any finite word u has at most |u| + 1 distinct palindromic factors (including the empty word). Accordingly, an infinite word x is called rich if each factor u of x has |u| + 1 many distinct palindromic factors. It turns out that Labbé's counter-example to the class P conjecture is a rich word. To see this, we observe that x is obtained from the fixed point y of the morphism τ : b → ccb, c → cb by inserting the symbol a before every occurrence of each of the symbols b and c. It is readily

verified that τ = τ 3 τ 2 τ 1 where τ 1 : b → c, c → b, τ 2 : b → b, c → cb and τ 3 : b → cb, c → c,
and hence by Corollary 6.3 and Proposition 6.6 in [START_REF] Glen | Palindromic richness[END_REF] it follows that y is rich. Given that y is rich, it now follows from Corollary 6.3 in [START_REF] Glen | Palindromic richness[END_REF] that x is rich. The defect of a finite word u, defined by D(u) = |u| + 1 -|Pal(u)|, is a measure of the extent to which u fails to be rich. The defect of a infinite word x is defined by D(x) = sup{D(u)|u is a prefix of x}. This quantity can be finite or infinite, and an infinite word is rich if and only if its defect is equal to 0. Any infinite word of finite defect is necessarily palindromic, but not conversely as is evidenced, for example, by the Thue-Morse word.

In this paper we show that if y has finite defect and is generated by a primitive substitution, then there exists a morphism f ∈ P and a rich word x ∈ FP such that y = f (x). So while y itself need not be in FP (as is for instance the case of Labbé's counter-example to Conjecture 1), we show that y is the morphic image of some word in FP and furthermore the morphism is in P . Actually, our result not only applies to all fixed points of primitive substitutions having finite defect, but more generally of all words y which have finite defect and are primitive morphic, i.e., morphic images of fixed points of primitive substitutions. More precisely: Theorem 2. Let y be a primitive morphic word with finite defect. Then there exists a morphism f ∈ P and a rich word x ∈ FP such that y = f (x).

In this respect, every primitive morphic word y with finite defect is generated by not one, but two morphisms in P . The first which generates the fixed point x in Theorem 2 and the second which maps x to y. As is well known, primitive morphic words are closed under the shift map (see for instance [START_REF] Carton | The Monadic Theory of Morphic Infinite Words and Generalizations[END_REF]), something which is not true of pure primitive morphic words. A key ingredient in our proof is Durand's characterization of primitive morphic words in terms of the finiteness of the set of derived words of first returns to prefixes (see [START_REF] Durand | A characterization of substitutive sequences using return words[END_REF]). A second involves a result of Balková et al. in [START_REF] Balková | Infinite words with finite defect[END_REF] which asserts that every uniformly recurrent word of finite defect is the morphic image of a rich word. While our proof of Theorem 2 requires the added hypothesis of finite defect, we do not know of an example of a palindromic word y generated by a primitive substitution which does not verify the conclusion of Theorem 2.

Preliminaries

Given a finite non-empty set A, we denote by A * the set of all finite words u = u 1 u 2 • • • u n with u i ∈ A. The quantity n is called the length of u and is denoted |u|. The empty word, denoted ε, is the unique element in A * with |ε| = 0. We set A + = A * -{ε}. For each word v ∈ A + , let |u| v denote the number of occurrences of v in u. We denote by A ω the set of all one-sided infinite words

x = x 0 x 1 x 2 • • • with x i ∈ A. Given x ∈ A ω , let Fact + (x) = {x i x i+1 • • • x i+j | i, j ≥ 0}
denote the set of all (non-empty) factors of x. Recall that x is called recurrent if each factor u of x occurs an infinite number of times in x, and uniformly recurrent if for each factor u of x there exists a positive integer n such that u occurs at least once in every factor v of x with |v| ≥ n. An infinite word x is called periodic if

x = u ω = uuuu • • • for some u ∈ A + , and is called ultimately periodic if x = vu ω = vuuu • • • for some v ∈ A * , and u ∈ A + . The word x is called aperiodic if x is not ultimately periodic. Let x ∈ A ω and u ∈ Fact + (x). A factor v of x is called a first return to u in x if vu ∈ Fact + (x),
vu begins and ends in u and |vu| u = 2. If v is a first return to u in x, then vu is called a complete first return to u in x. We note that the two occurrences of u in vu may overlap. We denote by R u (x) the set of all first returns to u in x.

A function τ : A → A + is called a substitution. A substitution τ extends by concatenation to a morphism from A * to A * and to a mapping from A ω to A ω , i.e., τ (a

1 a 2 • • • ) = τ (a 1 )τ (a 2 ) • • • .
By abuse of notation we denote each of these extensions also by τ. A substitution τ : A → A + is primitive if there exists a positive integer N such that |τ N (a)| b > 0 for all a, b ∈ A. A word x ∈ A ω is a called a fixed point of a substitution τ if τ (x) = x. We say x ∈ A ω is pure primitive morphic if x is a fixed point of some primitive substitution τ : A → A + . A word y ∈ B ω (where B is a finite non-empty set) is called primitive morphic if there exists a morphism f : A * → B * and a pure primitive morphic word x ∈ A ω with y = f (x). It is readily verified that every primitive morphic word is uniformly recurrent.

In [START_REF] Durand | A characterization of substitutive sequences using return words[END_REF], Durand obtains a nice characterization of primitive morphic words in terms of so-called derived words. Let x ∈ A ω be uniformly recurrent. Then #R u (x) < +∞ for each u ∈ Fact + (x). Let u ∈ Pref(x) be a non-empty prefix of x. Then x induces a linear order on R u (x) as follows: given distinct v, v ∈ R u (x) we declare v < v if the first occurrence of v in x occurs prior to that of v . Let A u (x) = {0, 1, . . . , #R u (x) -1}, and let f u : A u (x) → R u (x) denote the unique order preserving bijection. We can write x uniquely as a concatenation of first returns to u, i.e., x = u 1 u 2 u 3 • • • with u i ∈ R u (x). Following [START_REF] Durand | A characterization of substitutive sequences using return words[END_REF] we define the derived word of x at u, denoted D u (x), as the infinite word with values in A u (x) given by In this example, the derived word D 0 (x) is called the ternary Thue-Morse word, or the Hall word [START_REF] Hall | Generators and relations in groups-The Burnside problem[END_REF], and is the fixed point of the substitution 0 → 012, 1 → 02, 2 → 1.

D u (x) = f -1 u (u 1 )f -1 u (u 2 )f -1 u (u 3 ) • • • .
The following result of Durand gives a characterization of primitive morphic words:

Theorem 2.1 (F. Durand, Theorem 2.5 in [START_REF] Durand | A characterization of substitutive sequences using return words[END_REF]). A word x ∈ A ω is primitive morphic if and only if the set {D u (x) | u ∈ Pref(x)} is finite.

Given a finite or infinite word u ∈ A * we denote by Pal(u) the set of all palindromic factors of u (including the empty word). Droubay, Justin and Pirillo proved that any word u ∈ A * has at most |u| + 1 many distinct palindromic factors including the empty word (see [START_REF] Droubay | Episturmian words and some constructions of de Luca and Rauzy[END_REF]). A finite word u is called rich if #Pal(u) = |u| + 1. For instance aababbab is rich while aababbaa is not. An infinite word is called rich if all of its factors are rich. For instance, every Sturmian word is rich. On the other hand, the Thue-Morse word is not rich, since it contains 00101100 as a factor, and similarly the Hall word is not rich since it contains 0120 as a factor. Rich words were first introduced by Brlek, Hamel, Nivat and Reutenauer in [START_REF] Brlek | On the Palindromic Complexity of Infinite Words[END_REF], and have since been studied in various papers (see in particular [START_REF] Glen | Palindromic richness[END_REF] for a comprehensive study of rich words).

The defect of a finite word u is defined by D(u

) = |u| + 1 -|Pal(u)|.
The defect of a infinite word x is defined by D(x) = sup{D(u)|u is a prefix of x}. This quantity can be finite or infinite. Thus an infinite word is rich if and only if its defect is equal to 0.

We will make use of the following result from [START_REF] Glen | Palindromic richness[END_REF] characterizing rich words according to complete first returns. ). An infinite word x ∈ A ω is rich if and only if all complete first returns to any palindromic factor in x are themselves palindromes.

Two morphisms f, g : A * → B * are said to be conjugate if there exists u ∈ B * such that either f (a)u = ug(a) for all a ∈ A, or uf (a) = g(a)u for all a ∈ A. For example, the morphisms f : 0 → 001, 1 → 0010010010010 is conjugate to the morphism g : 0 → 010, 1 → 0100100010010. In fact, taking u = 0010010, it is readily verified that f (a)u = ug(a) for a ∈ {0, 1}. Conjugation of morphisms is an equivalence relation (see for instance Lemma 2.3.17 in [START_REF] Lothaire | Algebraic combinatorics on Words[END_REF]). It is also compatible with composition: Lemma 2.3 (Lemma 3.10 in [START_REF] Labbé | Propriétés combinatoires des f -palindromes[END_REF]). Let f, f : A * → B * and g, g : B * → C * be morphisms such that f is conjugate to f and g is conjugate to g . Then the compositions g • f and g • f are conjugate. Proposition 2.4 (Proposition 3.17 in [START_REF] Labbé | Propriétés combinatoires des f -palindromes[END_REF]). P is closed under composition.

In contrast, the set P of class P morphisms is not closed under composition. For instance, the morphisms f : 0 → 0, 1 → 01 and g : 0 → 01, 1 → 011 are both in class P while the composition g • f : 0 → 01, 1 → 01011 is not.

In [START_REF] Balková | Infinite words with finite defect[END_REF], Balková et al. introduce a related class of morphisms, denoted P ret , defined as follows: A morphism f : A * → B * is in P ret if there exists a palindrome p such that for each a ∈ A we have that f (a)p is a palindrome, f (a)p begins and ends in p, |f (a)p| p = 2, and f (a) = f (b) whenever a, b ∈ A with a = b. We call the palindrome p the marker. For instance, the morphism f : 0 → 0, 1 → 01 is in P ∩ P ret . Here the marker is p = 0. In contrast, the morphism f : 0 → 00, 1 → 01 is in P but not in P ret . While the morphism f : 0 → 001, 1 → 0010 is in P ret (with marker p = 00100) but not in P. They show that: Proposition 2.5 (Balková et al., Proposition 5.4 in [START_REF] Balková | Infinite words with finite defect[END_REF]). P ret ⊂ P .

We note that the definition of conjugacy of two morphisms given in [START_REF] Balková | Infinite words with finite defect[END_REF] is not the same as ours. According to their definition, two morphisms f, g : A * → B * are conjugate if there exists a word u ∈ B * such that either for every letter a ∈ A, the image f (a) has u as its prefix and the image g(a) is obtained from f (a) by erasing u from the beginning and adding it to the end, or for every a ∈ A, the image f (a) has u as its suffix and the image g(a) is obtained from f (a) by erasing u from the end and adding it to the beginning. So for instance, according to their definition, the morphisms f : 0 → 001, 1 → 0010010010010 and g : 0 → 010, 1 → 0100100010010 we saw earlier are not conjugate. However, in their proof of Proposition 2.5, they actually use our earlier definition and in fact, they intended for their definition to read the same as ours [3]. Results from [START_REF] Balková | Infinite words with finite defect[END_REF] are used to prove Corollary 3.5.

Primitive morphic words of finite defect

Recall that FP denotes the set of all infinite words x which are fixed by some primitive substitution f ∈ P . We recall that the class P conjecture states that if y is a palindromic pure primitive morphic word, then y ∈ FP . Labbé's counter-example in [START_REF] Labbé | A counterexample to a question of Hof, Knill and Simon[END_REF] shows that the conjecture as stated is false even if y is rich. Instead, we show: Theorem 3.1. Let y ∈ A ω be a rich primitive morphic word. Then there exists a morphism g ∈ P and a rich word x ∈ FP such that y = g(x).

Proof. We can suppose without loss of generality that A contains the symbol 0, and that y begins in 0. Let R 0 (y) denote the set of all first returns to 0 in y, A 0 (y) = {0, 1, . . . , #R 0 (y)-1}, and f 0 : A 0 (y) → R 0 (y) be the unique order preserving bijection. Let f : A 0 (y) * → A * be the morphism defined by f (a) = f 0 (a) ∈ R 0 (y) ⊂ A + for each a ∈ A 0 (y). Writing

y = u 1 u 2 u 3 • • • with each u i ∈ R 0 (y), let D 0 (y) = f -1 0 (u 1 )f -1 0 (u 2 )f -1 0 (u 3 ) • •
• denote the derived word of y at the prefix 0. Thus f (D 0 (y)) = y. The next three lemmas are stated in terms of the prefix 0 of y since they are needed only in this special case. But in fact they hold for all palindromic prefixes u of y.

Lemma 3.2. The morphism f : A 0 (y) * → A * is in P and thus in P .

Proof. Since y is rich, by Theorem 2.2 it follows that for each v ∈ R 0 (y) there is a palindrome v ∈ A * (possibly empty) such that v = 0v . Thus, for each a ∈ A, f (a) = f 0 (a) ∈ R 0 (y) and there exists a palindrome v a such that f (a) = f 0 (a) = 0v a . Hence f ∈ P ⊂ P . Lemma 3.3. D 0 (y) ∈ A 0 (y) ω is rich and begins in 0.

Proof. Since f 0 is order preserving, f -1 0 (u 1 ) = 0, whence D 0 (y) begins in 0. Let z be a complete first return in D 0 (y) to a palindrome u ∈ Fact + (D 0 (y)). By Lemma 3.2 we deduce that f (u)0 is a palindromic factor of y and f (z)0 is a complete first return in y to f (u)0. In fact, since z begins and ends in u, it follows that f (z)0 begins and ends in f (u)0. Moreover, since f (u)0 begins and ends in 0, a third occurrence of f (u)0 in f (z)0 would give rise to a third occurrence of u in z contrary to our assumption. Since y is rich it follows from Theorem 2.2 that f (z)0 is a palindrome. In item 3. in Remark 5.2 of [START_REF] Balková | Infinite words with finite defect[END_REF] it is shown that if ϕ ∈ P ret with marker p, then ϕ(z)p is a palindrome if and only if z is a palindrome. Whence we deduce that z is a palindrome, and hence D 0 (y) is rich by Theorem 2.2. Lemma 3.4. D 0 (y) is primitive morphic.

Proof. In item 5. of Proposition 2.6 of [START_REF] Durand | A characterization of substitutive sequences using return words[END_REF], it is shown that every derived word of D 0 (y) is also a derived word of y. Since y is primitive morphic, it follows from Theorem 2.1 that y has only finitely many distinct derived words, and hence D 0 (y) has finitely many distinct derived words, and hence by Theorem 2.1, D 0 (y) is primitive morphic.

Combining the three previous lemmas we deduce that if y ∈ A ω is a rich primitive morphic word beginning in 0, then D 0 (y) ∈ A 0 (y) ω is a rich primitive morphic word beginning in 0, and if

y = u 1 u 2 u 3 • • • with u i ∈ R 0 (y), then D 0 (y) = f -1 (u 1 )f -1 (u 2 )f -1 (u 3 ) • • • where f : A 0 (y) * → A *
belongs to P and hence P .

Thus, we can inductively define a sequence of infinite words (S n (y)) n≥0 with values in finite sets (A n ) n≥0 by S 0 (y) = y, and A 0 = A, and for n ≥ 0 : S n+1 (y) = D 0 (S n (y)) and A n+1 = A 0 (S n (y)). Moreover, for each n ≥ 1 there exists a morphism g n :

A * n → A * n-1 in P ⊂ P such that writing S n-1 (y) = u 1 u 2 u 3 • • • with u i ∈ R 0 (S n-1 (y)), we have S n (y) = g -1 n (u 1 )g -1 n (u 2 )g -1 n (u 3 ) • • • .
In other words, the sequence (S n (y)) n≥0 is just the sequence of iterated derived words of y corresponding each time to the prefix 0.

By Theorem 2.1 there exist 0 ≤ m < n such that S m (y) = S n (y). Let x = S m (y). Let h = g m+1 • g m+2 • • • • • g n . Then h(x) = x and by the proof of Proposition 3.3 in [START_REF] Durand | A characterization of substitutive sequences using return words[END_REF] we deduce that h is a primitive substitution. By Proposition 2.4 the morphism h is in P . Thus

x ∈ FP . Finally let g = g 1 • g 2 • • • • • g m . Then y = g(x)
and by Proposition 2.4 we deduce that g ∈ P as required. This completes the proof of Theorem 3.1.

Corollary 3.5. Let z ∈ A ω be a primitive morphic word having finite defect. Then there exists a morphism g ∈ P and a rich word x ∈ FP such that z = g(x).

Proof. In Theorem 5.5 in [START_REF] Balková | Infinite words with finite defect[END_REF], the authors show that if z ∈ A ω is a uniformly recurrent word of finite defect, then there exists a rich word y ∈ B ω and a morphism f : B * → A * in P ret such that z = f (y). In the proof of the theorem, it is revealed that y is actually a derived word of z. Thus, if z is primitive morphic, then by Theorem 2.1 so is y, and hence by Theorem 3.1 there exists a rich word x ∈ FP and a morphism h ∈ P such that y = h(x). Let g = f • h. Then z = g(x) and by Proposition 2.4 and Proposition 2.5 we deduce that g ∈ P .

We end with a few examples. Let us begin by considering Labbé's counter-example y = acabacacabacabacabacacabac • • • fixed by the morphism a → ac, b → acab, c → ab. Then R a (y) = {ac, ab} and the derived word D a (y) ∈ {0, 1} ω is the fixed point of the morphism 0 → 01, 1 → 001 which is clearly in P . Thus, setting x = D a (y), we have that x ∈ FP and y = g(x) where g : 0 → ac, 1 → ab is in P . In this case the morphism g actually belongs to P.

Our second example is somewhat artificial but shows that the morphism g in Theorem 3.1 and Corollary 3.5 may fail to be in P. Let x be the Fibonacci word fixed by the morphism 0 → 01, 1 → 0 and set y = h(x) where h : 0 → 01, 1 → 01011. Then it is readily checked that D 0 (y) = f (x) where f : 0 → 0, 1 → 01 and y = g(D 0 (y)) where g : 0 → 01, 1 → 011. Moreover, D 0 (D 0 (y)) = x, and D 0 (x) = x. Whence y = g • f (x) with x ∈ FP and g • f = h which belongs to P but not to P.

For

  example, let x = 01101001100101101001011001101001 • • • denote the Thue-Morse word fixed by the substitution 0 → 01, 1 → 10. It is readily verified that R 0 (x) = {011, 01, 0}. So A 0 (x) = {0, 1, 2} and f 0 : A 0 (x) → R 0 (x) is given by f 0 (0) = 011, f 0 (1) = 01, and f 0 (2) = 0. Writing x as a concatenation of first returns to 0 we find x = (011)(01)(0)(011)(0)(01)(011)(01)(0)(01)(011)(0)(011)(01)(0) • • • and hence D 0 (x) = 012021012102012 • • • .

Theorem 2 . 2 (

 22 [START_REF] Glen | Palindromic richness[END_REF], Theorem 2.14 and Remark 2.15.

  Our last example illustrates what can happen when the defect is infinite. Let y = 011010011001 • • • be the Thue-Morse word fixed by the morphism 0 → 01, 1 → 10 whose square is in P. Then D 0 (y) is the ternary Thue-Morse word fixed by the morphism 0 → 012, 1 → 02, 2 → 1. Taking the next derived word we obtain D 0 (D 0 (y)) = 01230132012320130123013201301 • • • which is the fixed point of the morphism 0 → 01, 1 → 23, 2 → 013, 3 → 2. Now it is readily checked that z = D 0 (D 0 (y)) is not even palindromic, in fact the length of the longest palindromic factor of z is 3.

Acknowledgments: We are extremely grateful to both referees for their insightful remarks and suggestions which have greatly improved the exposition of the paper.

(Luca Q. Zamboni) 1 Partially supported by a FiDiPro grant (137991) from the Academy of Finland and by ANR grant SUBTILE.