D Jamet

G Paquin

G Richomme
email: gwenael.richomme@univ-montp3.fr

L Vuillon

On the fixed points of the iterated pseudopalindromic closure operator

Keywords: Sturmian word, palindromic closure, pseudopalindrome, pseudostandard word, fixed point, involutory antimorphism

First introduced in the study of the Sturmian words by de Luca in 1997, iterated palindromic closure was generalized to pseudopalindromes by de Luca and De Luca in 2006. This operator allows one to construct words with infinitely many pseudopalindromic prefixes, called pseudostandard words. We provide here several combinatorial properties of the fixed points under iterated pseudopalindromic closure.

Introduction

The Sturmian words form a well-known class of infinite words over a 2-letter alphabet that occur in many different fields, for instance in astronomy, symbolic dynamics, number theory, discrete geometry, crystallography, and of course, in combinatorics on words (see [START_REF] Lothaire | Algebraic Combinatorics on Words, volume 90 of Encyclopedia of Mathematics and its Applications[END_REF], chap. 2). These words have many equivalent characterizations whose usefulness depends on the context. In discrete geometry, they are exactly the words that code the discrete approximations of lines with irrational slopes, using horizontal and diagonal moves. In symbolic dynamics, Sturmian words are obtained by the exchange of 2 intervals. They are also known as the balanced aperiodic infinite words over a 2-letter alphabet. A subclass of the Sturmian words is formed by the standard Sturmian ones. For each Sturmian word, there exists a standard one having the same language, i.e., the same set of factors. A standard Sturmian word is, in a sense, the representative of all Sturmian words having the same language. All the words in this subclass can be obtained by a construction called iterated palindromic closure [START_REF] De Luca | Sturmian words: structure, combinatorics, and their arithmetics[END_REF]. This operation gives a bijection between standard Sturmian words and non-eventually constant infinite words over a 2-letter alphabet.

Some other fixed points of functions are famous in combinatorics on words. As an example, the self-generating word introduced in [START_REF] Kolakoski | Self Generating Runs, Problem 5304[END_REF], called the Kolakoski word, is the fixed point under the run-length encoding function; it has raised some challenging problems. For instance, we still do not know what are its letter frequencies, if they exist. The question of recurrence of the Kolakoski word as well as the closure of its set of factors under complementation or reversal are other open problems.

In this context, it is a natural problem to try to characterize the fixed points under the iterated palindromic closure operator, and more generally, under the iterated pseudopalindromic closure operator, introduced in [START_REF] De Luca | Pseudopalindrome closure operators in free monoids[END_REF]. In this paper, we study these words and show some of their properties. It is organized as follows. We first give the definitions and notation used. Then we recall the definition of the iterated palindromic closure operator and introduce the iterated pseudopalindromic closure operator, which generalizes the first one using a generalization of palindromes. In Section 3, we prove the existence of fixed points under the iterated pseudopalindromic closure operator and we show them explicitly: there are 3 families of fixed points. In Section 4, we give some of their combinatorial properties, while in Section 5, we characterize the prefixes of these fixed points.

Since the words in the first family of fixed points are standard Sturmian, we use known results about standard Sturmian words in order to prove that they are not ultimately periodic, not fixed points under nontrivial morphisms and we also show some repetition properties like the greatest integer power avoided. Since the other two families of fixed points are not Sturmian words, proving their combinatorial properties is more difficult. We use a powerful theorem of de Luca and De Luca [START_REF] De Luca | Pseudopalindrome closure operators in free monoids[END_REF] that links a Sturmian word and an episturmian word to the fixed points of the respective second and third families, using a morphism. Thus, we first prove properties of their associated Sturmian and episturmian words and then we apply them to the fixed points.

Notice that this paper is an extended and enhanced version of a paper presented in Salerno during the 7-th International Conference on Words [START_REF] Jamet | On the fixed points of the iterated pseudopalindromic closure[END_REF].

Some iterated closures

We first recall notions about words (for more details, see for instance [START_REF] Lothaire | Algebraic Combinatorics on Words, volume 90 of Encyclopedia of Mathematics and its Applications[END_REF]).

An alphabet A is a finite set of symbols called letters. A word over A is a sequence of letters from A. The empty word ε is the empty sequence. Equipped with the concatenation operation, the set A * of finite words over A is a free monoid with neutral element ε and set of generators A, and we set A + = A * \ ε. We denote by A ω the set of (right)-infinite words over A. For the sake of clarity, we denote in bold character a letter denoting an infinite word, to distinguish from a finite word. The set A ∞ is defined as the set of finite and infinite words:

A ∞ = A * ∪ A ω .
If, for some words u, s ∈ A ∞ , v, p ∈ A * , u = pvs, then v is a factor of u, p is a prefix of u and s is a suffix of u. If v = u (resp. p = ε and s = ε, and p = ε and s = ε), v is called a proper factor (resp. proper prefix and proper suffix). The set of factors of the word u is denoted by F (u). Two words u and v are prefix comparable if u is a prefix of v or v is a prefix of u. For u = vw, with v ∈ A * and w ∈ A ∞ , v -1 u denotes the word w and uw -1 denotes the word v.

As usual, for a finite word u and a positive integer n, the n-th power of u, denoted u n , is the word ε if n = 0; otherwise u n = u n-1 u. A word v which is a power of a letter a is called a block of a's in u if u = pvs where p does not end with a and s does not begin with a. If u = ε, u ω denotes the infinite word obtained by infinitely repeating u. An infinite word u is periodic (resp. ultimately periodic) if it can be written as u = w ω (resp. u = vw ω ), with v ∈ A * and w ∈ A + . Given a finite or infinite word u, we denote by u[i] the ith letter of u and by u[i . . . j] the word u

[i]u[i + 1] • • • u[j]. Given a nonempty finite word u = u[1]u[2] • • • u[n]
, the length |u| of u is the integer n. One has |ε| = 0. The last letter of the word u is denoted by last(u). The number of occurrences of the letter a in the word u is denoted by |u| a . The frequency of the letter a in a finite word w is |w| a /|w|. For an infinite word w, the frequency of a letter a is defined as

lim n→∞ |w[1 . . . n]| a /n if it exists.
If |u| a = 0, then u is called an a-free word. If for some integer k ≥ 2 the word u ∈ A ∞ does not contain any k-th power, then u is called a k-th power-free word. The rational power of a word u is defined by u q = u ⌊q⌋ p, with q ∈ Q such that q|u| ∈ N and p is the prefix of u of length |u|(q -⌊q⌋). The critical exponent of an infinite word w, denoted by E(w), is the supremum of the rational powers of all its (finite) factors. There exist words such that the critical exponent is never reached. For instance, the Fibonacci word f has critical exponent E(f ) = 2 + φ, where φ is the golden ratio, but none of the factors of f realize E(f ) (see [START_REF] Mignosi | Repetitions in the Fibonacci infinite word[END_REF]).

The reversal of the finite word

u = u[1]u[2] • • • u[n], also called the mirror image, is R(u)= u[n]u[n -1] • • • u[1] and if u = R(u), then u is called a palindrome.
The right-palindromic closure (palindromic closure, for short) of the finite word u denoted by u (+) is defined by u (+) = u•R(p), with u = ps and s is the maximal palindromic suffix of u. In other words, u (+) is the shortest palindromic word having u as prefix.

Iterated palindromic closure

Sturmian words may be defined in many equivalent ways (see Chapter 2 in [START_REF] Lothaire | Algebraic Combinatorics on Words, volume 90 of Encyclopedia of Mathematics and its Applications[END_REF] for more details). For instance, they are the non-ultimately periodic infinite words over a 2-letter alphabet that have minimal complexity, that is the number of distinct factors of length n is (n + 1), for each positive integer n. They are also the set of non-ultimately periodic binary balanced words. Recall that a word w over A is balanced if for all factors f, f ′ having same length, and for all letters a ∈ A, one has

||f | a -|f ′ | a | ≤ 1.
The Sturmian words are also infinite words that describe discrete approximations of irrational slopes (see [START_REF] Lothaire | Algebraic Combinatorics on Words, volume 90 of Encyclopedia of Mathematics and its Applications[END_REF]). More precisely, an infinite word

s = s[0]s[1]s[2] • • • is Sturmian if
and only if there exist α, ρ ∈ R, with 0 ≤ α < 1 and α irrational, such that s is equal to one of the following two infinite words s α,ρ , s ′ α,ρ ∈ {a, b} ω , defined by:

s α,ρ [n] = a if ⌊α(n + 1) + ρ⌋ = ⌊αn + ρ⌋, b otherwise,
and

s ′ α,ρ [n] = a if ⌈α(n + 1) + ρ⌉ = ⌈αn + ρ⌉, b otherwise.
Note that ρ is the intercept and α is the slope of the line approximated by the word s.

A Sturmian word is called standard (or characteristic) if ρ = α. All Sturmian words considered in this paper belong to this particular class of Sturmian words. Let us see how the iterated palindromic closure operator is hidden in the structure of the standard Sturmian words.

Given a finite word w, let us denote by Pal(w) the word obtained by iterating the palindromic closure:

Pal(ε) = ε;
Pal(wa) = (Pal(w)a) (+) , for all words w and letters a.

Note that the Pal operator is also denoted by ψ in the works of de Luca (see for instance [START_REF] De Luca | Sturmian words: structure, combinatorics, and their arithmetics[END_REF]). By the definition of the iterated palindromic closure Pal, for any finite word w and letter a, Pal(w) is a prefix of Pal(wa). One can then extend the iterated palindromic closure to any infinite word w = (a[n]) n≥1 as follows:

Pal(w) = lim n→∞ Pal(a[1] • • • a[n]).
We then say that the word w directs the word Pal(w). From the works of [START_REF] De Luca | Sturmian words: structure, combinatorics, and their arithmetics[END_REF], we know that Pal gives a bijection between the set of infinite words over {a, b} not of the form ua ω or ub ω , for some u ∈ {a, b} * , and the set of standard Sturmian words over {a, b}. The word w is then called the directive word of the standard Sturmian word Pal(w). Note that words of the form Pal(ua ω ) are periodic (see Lemma 4.1 below recalled from [START_REF] Droubay | Episturmian words and some constructions of de Luca and Rauzy[END_REF]).

The Pal operator is also well-defined over a k-letter alphabet, with k ≥ 3. In this case, it is known [START_REF] Droubay | Episturmian words and some constructions of de Luca and Rauzy[END_REF] that Pal(A ω ) is exactly the set of standard episturmian words, a generalization over a k-letter alphabet, k ≥ 3, of the family of standard Sturmian words (for more details, see [START_REF] Glen | Episturmian words: a survey[END_REF]). When w is a word over A containing each letter infinitely often, then Pal(w) is called a strict standard episturmian word. The set of strict (standard) episturmian words corresponds to the set of (standard) Arnoux-Rauzy words [3]. As we will do in what follows, we have underlined in the previous examples the letters corresponding to the letters of the directive words, for the sake of clarity.

Iterated pseudopalindromic closure

A few years ago, de Luca and De Luca [START_REF] De Luca | Pseudopalindrome closure operators in free monoids[END_REF] extended the notion of palindrome to what they call pseudopalindrome, using involutory antimorphisms. In order to define it, let us first recall that a map ϑ : A * → A * is called an antimorphism of A * if for all u, v ∈ A * one has ϑ(uv) = ϑ(v)ϑ(u). Moreover, an antimorphism is involutory if ϑ 2 = id. A trivial involutory antimorphism is the reversal function R. Any involutory antimorphism ϑ of A * can be constructed as ϑ = τ • R = R • τ , with τ an involutory permutation of the alphabet A. From now on, in order to describe an involutory antimorphism ϑ, we will then only give the involutory permutation τ of the alphabet A. The two antimorphisms E and H defined respectively over {a, b} and {a, b, c} by

E = R • τ with τ (a) = b, τ (b) = a, H = R • τ with τ (a) = a, τ (b) = c, τ (c) = b
will play, in addition to R, an important role in our study. The antimorphism E will be called, as usual, the exchange antimorphism. We propose to name the antimorphism H the hybrid antimorphism, hence the notation, since it contains both an identity part and an exchange part.

We can now define the generalization of palindromes given in [START_REF] De Luca | Pseudopalindrome closure operators in free monoids[END_REF]: a word w ∈ A * is called a ϑ-palindrome if it is the fixed point of an involutory antimorphism ϑ of the free monoid A * : ϑ(w) = w. When the antimorphism ϑ is not mentioned, we call w a pseudopalindrome.

Remark 2.3. Notice here that, when f is an antimorphism which is not involutory, it is possible that f has a finite fixed point (for instance the word acb is a fixed point of the antimorphism f defined by f (a) = cb, f (b) = a and f (c) = ε). Nevertheless we observe that a non-erasing antimorphism f has a non-empty finite fixed point w if and only if f is involutory on the alphabet of w. Indeed if f (w) = w for a non-empty word w and a non-erasing antimorphism f , then it is directly verified that w

[i] = f (w[n -i + 1]) for all i = 1, . . . , n.
Analogous to the palindromic closure (+) , the ϑ-palindromic closure of the finite word u, also called the pseudopalindromic closure when the antimorphism is not specified, is defined by u ⊕ = sqϑ(s), where u = sq, with q the longest ϑ-palindromic suffix of u. The pseudopalindromic closure of u is the shortest pseudopalindrome having u as prefix.

∈ A ω , Pal ϑ (w) = lim n→∞ Pal ϑ (w[1] • • • w[n]
). This limit exists since by the definition of Pal ϑ , for any involutory antimorphism ϑ, w ∈ A * and a ∈ A, Pal ϑ (w) is a prefix of Pal ϑ (wa). The infinite word obtained by the Pal ϑ operator is a ϑ-standard word, also called a pseudostandard word when the antimorphism is not specified. This new class of words is a general one that includes the standard Sturmian and the standard episturmian ones and was first introduced in [START_REF] De Luca | Pseudopalindrome closure operators in free monoids[END_REF].

Example 2.6. Over A = {a, b}:

Pal E (aab) = (Pal E (aa)b) ⊕ = ((Pal E (a)a) ⊕ b) ⊕ = ((ab • a) ⊕ b) ⊕ = (abab • b) ⊕ = ababbaabab. Example 2.7. Over A = {a, b, c}, the H-standard word directed by (abc) ω is Pal H ((abc) ω ) = abcacbabcaabcacbabcabcacbabcaabcacbabca • • • .
In Section 3, we are interested in the fixed points under the Pal ϑ operator: we are looking for the words u ∈ A ω and the antimorphisms ϑ such that Pal ϑ (u) = u. Notice that the study of the fixed points under the operator Pal ϑ includes the ones under the operator Pal, since Pal = Pal R . Indeed, one easily sees that the R-palindromes are exactly the usual ones, as we saw in Example 2.5.

Existence of fixed points

In this section, we prove the existence of fixed points under the iterated pseudopalindromic closure and we show which forms they can have. We 1. Using the antimorphism R:

Pal R (abx • • • ) = abax • • • Pal 2 R (abx • • • ) = abaabax • • • Pal 3 R (abx • • • ) = abaabaababaabaabaababaabaabax • • • .
2. Using the antimorphism E:

Pal E (abx • • • ) = abbaabx • • • Pal 2 E (abx • • • ) = abbaabbaababbaabbaababbaabbaab baababbaabbaababbaabbaabx • • • .
In both examples, we see that the position of the letter x of the directive word w in Pal k R (w) and Pal k E (w) grows with the value of k. We also observe that the common prefix of Pal k ϑ (w) and Pal k+1 ϑ (w) also seems to grow with k, either for ϑ = R or for ϑ = E. Lemmas 3.2 and 3.3 show that only a short prefix of w is required to determine the word obtained by infinitely iterating the Pal ϑ operator and from these two lemmas we get Theorem 3.4 (to follow). Lemma 3.2. Let ϑ = R • τ be an involutory antimorphism and let (u k ) k≥1 be a sequence of finite words defined by

u 1 = a n b if τ (a) = a, a if τ (a) = b,
and for k ≥ 2, u k = Pal ϑ (u k-1 ), with a, b ∈ A and a = b, n ≥ 1. Then lim k→∞ u k exists. Proof. Let us first show by induction that for all k ≥ 1, u k is a proper prefix of u k+1 . If τ (a) = a, then u 1 = a n b is a proper prefix of u 2 = Pal ϑ (a n b) = (Pal ϑ (a n )b) ⊕ = (a n b) ⊕ = a n b. Otherwise, τ (a) = b and then, u 1 = a is a proper prefix of u 2 = Pal ϑ (a) = ab. Let us suppose u k-1 is a proper prefix of u k for 2 ≤ k ≤ n.
Then by induction and using the definition of u n , we get

u n+1 = Pal ϑ (u n ) = Pal ϑ (Pal ϑ (u n-1 )) = Pal ϑ (u n-1 w) = u n w ′ ,
with w, w ′ ∈ A + . Thus, u n is a proper prefix of u n+1 . Hence the sequence (u k ) k≥1 tends to a limit.

The limit of the sequence defined in Lemma 3.2 will be denoted by s ϑ,n if τ (a) = a and s ϑ otherwise. Lemma 3.3. Let (u k ) k≥1 be the same sequence as in Lemma 3.2 and let us consider an infinite word w having u 1 as a prefix. Then for all k ≥ 1, u k is a proper prefix of Pal k-1 ϑ (w).

Proof. By the hypothesis, u 1 is a proper prefix of w = Pal 0 ϑ (w). Let us suppose that for

1 ≤ k ≤ n, u k is a proper prefix of Pal k-1 ϑ (w). Let us consider Pal n ϑ (w): Pal n ϑ (w) = Pal ϑ (Pal n-1 ϑ (w)) = Pal ϑ (u n v) = Pal ϑ (u n )v ′ = u n+1 v ′ , for v, v ′ ∈ A ω . Then, u n+1 is a proper prefix of Pal n ϑ (w)
. This shows that the sequence Pal k ϑ (w) converges to s ϑ or s ϑ,n , depending on τ . Remark 3.5. In [START_REF] De Luca | Pseudopalindrome closure operators in free monoids[END_REF], it is mentioned that the number of involutory antimorphisms of a k-letter alphabet A equals the number of involutory permutations over k elements, which is known to be

s R,n,a,b = lim i→∞ Pal i (a n b) = a n ba n (aba n ) n+1 b(a n+1 b) n+1 a n a • • • .
k! ⌊k/2⌋ i=0 1 2 i (n -2i)!i! .
Even if there exist many involutory antimorphisms for arbitrary k-letter alphabets, fixed points over the Pal ϑ operators contain at most three letters. More precisely, the fixed points under a 3-letter alphabet {a, b, c} starting with a can only be obtained by the antimorphism H (we recall that H = R•τ , with τ (a) = a and τ (b) = c). Indeed, τ (a) = b yields s E,a,b and τ (a) = a and τ (b) = b yield s R,n,a,b . Moreover, for the antimorphism E, the fixed point cannot start with a 2 , since a 2 is not a prefix of Pal E (a 2 ) = abab.

Examples 3.6. For the first few values of n and for the antimorphisms R and H, we obtain the following fixed points:

s R,1,a,b = abaabaababaabaabaababaabaabaababaabaababaabaabaabab • • • s R,2,a,b = aabaaabaaabaaabaabaaabaaabaaabaaabaabaaabaaabaaaba • • • s R,3,a,b = aaabaaaabaaaabaaaabaaaabaaabaaaabaaaabaaaabaaaaba • • • s H,1,a,b,c = abcacbabcaabcacbabcacbabcaabcacbabcabcacbabcaabcacbabc • • • s H,2,a,b,c = aabcaacbaabcaaabcaacbaabcaaabcaacbaabcaacbaabcaaabca • • • .

Combinatorial properties of the fixed points

In this section, we consider successively the fixed points s R,n,a,b , s E,a,b and s H,n,a,b,c of the Pal ϑ operator and we give some of their combinatorial properties. We will see that words s R,n,a,b are Sturmian and s E,a,b is related to a Sturmian word, whereas the words s H,n,a,b,c cannot be such, since they contain the three letters a, b and c. This explains why we consider the word s E,a,b before words s H,n,a,b,c contrary to their order of introduction in Theorem 3.4.

Study of the fixed point s R,n,a,b

Here, we consider the first fixed point of the Pal ϑ operator, with ϑ = R. Thus, in what follows, instead of writing Pal R , we will write Pal, since it is equal.

Before stating our first property, we need the following lemma. Theorem 4.4. [START_REF] Crisp | Substitution invariant cutting sequences[END_REF] The standard Sturmian word of slope α is a fixed point of some nontrivial morphism if and only if α has a continued fraction expansion of one of the following kinds:

1. α = [0; 1, a 0 , a 1 , . . . , a k ], with a k ≥ a 0 , 2. α = [0; 1 + a 0 , a 1 , . . . , a k ], with a k ≥ a 0 ≥ 1. Recall that a morphism (endomorphism if A = B) f from A * to B * , A, B
alphabets, is a mapping from A * to B * such that for all words u, v over A, f (uv) = f (u)f (v). Given an endomorphism f and a letter a, if lim n→∞ f n (a) exists, then this limit is denoted f ω (a) and is a fixed point of f . (Morphisms f n are naturally defined by f 0 is the identity and

f n = f n-1 • f )
In the past, some fixed points of nontrivial morphisms, that is morphisms different from the identity, showed interesting properties such as the Fibonacci word f and the Thue-Morse one T [START_REF] Thue | Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen[END_REF] (see also for instance [2,[START_REF] Berstel | Combinatorics on Words: Christoffel Words and Repetitions in Words[END_REF][START_REF] Cassaigne | On extremal properties of the Fibonacci word[END_REF]). The first one is obtained by lim i→∞ ϕ i A wide literature is devoted to the study of these fixed points and the known results about their generating morphism are often used in order to find some of their properties. It is thus natural to wonder if the fixed points of the Pal operator are also fixed points of some nontrivial morphisms. The answer is: Proposition 4.5. For a fixed n, s R,n,a,b is not a fixed point of a nontrivial morphism.

Proof. From Theorem 4.4, a standard Sturmian word which is a fixed point of a nontrivial morphism has a slope with an ultimately periodic continued fraction expansion. If its period has even length, then it implies, using Theorem 4.3, that its directive word is also ultimately periodic and hence s R,n,a,b itself, which is impossible by Proposition 4.2. If the period of the continued fraction expansion has odd length, then the same argument holds, using twice the period. Lemma 4.6 in itself is not that interesting, but we will see that it will be useful to state repetition properties of the fixed point s R,n,a,b in Proposition 4.8. Notice that the first part of Lemma 4.7 is due to [START_REF] Mignosi | Infinite words with linear subword complexity[END_REF] and the second is due to [10]. By direct computation, we easily obtain arbitrarily large prefixes of the word s R,n,a,b for a fixed n. The continued fraction expansion of α R,n,a,b is then obtained. For the first values of n, we get:

α 1,a,b = [0; 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, . . .] = 0.366095116093540422949960571470577467087211123077286 . . . α 2,a,b = [0; 3, 1, 3, 1, 3, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 1,
P = {i ∈ N \ 0 | s R,n,a,b [i + 1] = a} and P ′ = {Pal(s R,n,a,b [1 . . . i]) | i ∈ P }.
Both sets are infinite. Moreover, by its construction, any palindrome in the set P ′ is followed by an a at its first occurrence in s R,n,a,b . By Theorem 4.3 and since s R,n,a,b equals its directive word, if . By the construction of the palindromes in P ′ , we know that they all are palindromes such that ξ = 1. That implies that for any n, the continued fraction expansion of the slope of s R,n,a,b begins by infinitely many palindromes. We conclude using Theorem 4.9: since the continued fraction expansion of the slope is not ultimately periodic, it cannot be quadratic (see [START_REF] Ya | Continued fractions[END_REF]); hence, it is transcendental.

a i 1 b i 2 • • • b i 2 a i 1 is a palindromic prefix of s R,n
Notice that the previous proof works since s R,n,a,b equals its directive word. Otherwise, the result it not necessarily true.

Notice also that the standard Sturmian words whose slopes have continued fraction expansions beginning with arbitrary long palindromes are harmonic (see [START_REF] Carpi | Harmonic and gold sturmian words[END_REF]). So the fixed points s R,n,a,b form a special class of harmonic standard Sturmian words.

Study of the fixed point s E,a,b

We have seen in the previous subsection that since the s R,n,a,b are standard Sturmian words, some properties follow directly. Here, we study the fixed point

s E,a,b = abbaabbaababbaabbaababbaabbaab • • • .
Recall that Sturmian words are known to be balanced. It is sufficient to consider the factors bb and aa to be convinced that s E,a,b is not balanced, and consequently, that it is not a Sturmian word.

We now recall a powerful result of de Luca and De Luca. For ϑ = τ • R an involutory antimorphism over an alphabet A, with τ an involutory permutation of A, µ ϑ is the morphism defined for all a in A, by µ ϑ (a) = a if a = τ (a) and by µ ϑ (a) = aτ (a) otherwise. Since we cannot use the known results about Sturmian words in order to prove combinatorial properties of the fixed point s E,a,b , the idea here is to first consider the word Pal(s E,a,b ) that will further appear to be standard Sturmian, and then to extend the properties to µ E (Pal(s E,a,b )) which is the fixed point s E,a,b , by Theorem 4. [START_REF] De Luca | Sturmian words: structure, combinatorics, and their arithmetics[END_REF].

In what follows, w E will denote Pal(s E,a,b ), that is

w E = ababaababaabababaababaabababaababaababa • • • .
Notice that here, µ E is the Thue-Morse morphism, that is µ E (a) = ab and µ E (b) = ba. Note also that s E,a,b = µ E (w E ), and so s E,a,b ∈ {ab, ba} ω . Proposition 4.12. w E is not ultimately periodic, and consequently w E is a Sturmian word.

Proof. By Lemma 4.1, w E is ultimately periodic if and only if s E,a,b = uα ω , for u ∈ A * and α ∈ A. By its construction, s E,a,b has infinitely many Epalindromes prefixes having the form Pal i E (ab) = abu i ab, with u i ∈ {a, b} * , for i arbitrarily large. Analogous to the proof of Proposition 4.2, we conclude that s E,a,b = uα ω and hence, w E is not ultimately periodic: it is a standard Sturmian word. Lemma 4.13. Let ϑ be an involutory antimorphism over an alphabet A. An infinite word w over A is ultimately periodic if and only if µ ϑ (w) is ultimately periodic.

Proof. The "only if" part is immediate. Assume µ ϑ (w) = uv ω for words u ∈ A * and v ∈ A + . When v begins with a letter a such that µ ϑ (a) = a, then a occurs in no word µ ϑ (b) with b = a, implying that u = µ ϑ (u ′ ), v = µ ϑ (v ′ ) for some words u ′ , v ′ . Then µ ϑ (w) = µ ϑ (u ′ v ′ω ). It is quite immediate that the morphism µ ϑ is injective on infinite words (and also on finite ones). Hence w = u ′ v ′ω is ultimately periodic. Assume now that v begins with a letter a such that a = τ (a). In what follows, we denote by α the complementary letter of the letter α over the 2-letter alphabet {a, b}, that is a = b and b = a. From It is known by [START_REF] Shur | The structure of the set of cube-free z-words in a two-letter alphabet[END_REF] that, for all rational q ≥ 2, a word w in a twoletter alphabet avoids repetition u q if and only if µ E (w) also avoids them. Proposition 4.19 then has the following corollary: Corollary 4.20. s E,a,b contains 4-th powers, but is 5-th power-free.

= µ ϑ (u ′ ) and v = µ ϑ (v ′ ), or, ua = µ ϑ (u ′ ) and a -1 va = µ ϑ (v ′ ). Once again µ ϑ (w) = µ ϑ (u ′ v ′ω ) and so w = u ′ v ′ω is ultimately periodic. Since s E,a,b = µ E (w E ),
µ E (w E,a,b ) = s E,a,b = µ E • η(s E,a,b ) = µ E (η • µ E (w E ))
Since s E,a,b is not a Sturmian word, it does not have a known geometrical interpretation. Thus, the notion of slope does not apply here. However, since s E,a,b ∈ {ab, ba} ω , we observe that the frequencies of the letters in s E,a,b are both 1/2.

Study of the fixed point s H,n,a,b,c

Let us now study the properties of the last kind of fixed points. Since the words s H,n,a,b,c do not have a separating letter (a letter a that precedes or follows each other letter b = a), they are not episturmian. As in the previous subsection, let w H,n denote the episturmian word associated by Theorem 4.11 to the fixed point s H,n,a,b,c , that is:

w H,n = Pal(s H,n,a,b,c ) = a n ba n ca n ba n aba n ca n ba n • • • .
As in the proofs of Propositions 4.2 and 4.12, one can see that the three letters a, b, c occur infinitely often in s H,n,a,b,c . Thus by Proposition 4.15 and by their construction, the w H,n satisfy: From Propositions 4.22 and 4.23, we get: Proposition 4.24. For a fixed n, the word w H,n is not a fixed point of a nontrivial morphism.

To go further, we need to recall basic relations between so-called epistandard morphisms and palindromic closure. For any letter a, we denote by L a the morphism defined by L a (a) = a and L a (b) = ab when b is a letter different from a. We extend this notation to arbitrary word: L ε is the identity morphism, L ua = L u • L a for any word u and letter a.

Morphisms L u are known to be the pure standard episturmian morphisms, or pure epistandard morphisms for short (see [START_REF] Justin | Episturmian words and episturmian morphisms[END_REF]).

The Pal operator is strongly related to epistandard morphisms by the following formula [START_REF] Justin | Episturmian words and episturmian morphisms[END_REF]:

Pal(uv) = L u (Pal(v))Pal(u)
, for all words u, v and letter a.

(

Also when at least two letters occur infinitely often in w,

Pal(w) = lim n→∞ L a 1 ...an (a n+1 ). (2)
Now we come to repetitions in w H,n . In [START_REF] Justin | Episturmian words and episturmian morphisms[END_REF], Justin and Pirillo provided important tools about fractional powers in episturmian words. In particular they proved the following result which is a slightly different version of the original one. The equivalence between the two formulations is discussed in the Appendix.

2).

Let A be an alphabet containing at least two different letters. Let s be a strict standard episturmian word over A directed by ∆. Assume that ∆ = v ω (in particular ∆ is periodic and s is the fixed point of a morphism) with v ∈ A + and let:

• ℓ = max{i | α i is a factor of ∆ with α ∈ A}, • L be the set of all 3-uples (x, a, y) such that xaya ℓ is a prefix of ∆, a ∈ A, |v| ≤ |xay| < |v 2 |, y = ε, a ∈ alph(y).
The critical exponent of s is

ℓ + 2 + sup (x,a,y)∈L lim i→∞ |Pal(v i xa)| |L v i xay (a)| .
With the notation of the previous result, one can observe that L v i xay (a) = L v i xay ′ (ba), where y = y ′ b begins with Pal(v i xa), b = a ∈ A. Indeed ba contains two different letters and Lemma 4.21 in [START_REF] Glen | Quasiperiodic and Lyndon episturmian words[END_REF] states that for any word u containing at least two different letters and for any other word w, there exists a word u w containing at least two different letters such that L w (u) = Pal(w)u w . Hence in the situation of the previous theorem, the critical exponent lies between ℓ + 2 and ℓ + 3. In particular s is (ℓ + 3)-rd power-free but contains an (ℓ + 2)-nd power. This can be extended to a larger class of episturmian words, as follows.

Proposition 4.26. Let s be a strict standard episturmian word directed by a word ∆ and let ℓ denote the greatest integer i such that α i is a factor of ∆ with α a letter. Assume ∆ contains at least one factor aua ℓ va with a a letter and u, v non-empty words that do not contain the letter a. Then s is (ℓ + 3)-rd power-free but contains an (ℓ + 2)-nd power. Proof. Let (v i ) i≥1 be the sequence of prefixes of s having a first letter different from the last letter (it is infinite since s is a strict standard episturmian word). For i ≥ 1, let s i denote the standard episturmian word directed by v ω i . It is straightforward that s = lim i→∞ s i (since s and s i share as prefix Pal(v i ) whose length grows with i). By choice of v i , we know that max{j | α j ∈ F (v ω i ), α ∈ A} ≤ ℓ. Hence by Theorem 4.25 each s i is (ℓ + 3)-rd power-free (see the discussion before the proposition). Consequently s is also (ℓ + 3)-rd power-free. Now by the hypotheses, ∆ = waua ℓ va∆ ′ with a ∈ A and u, v ∈ A + such that |u| a = |v| a = 0. Let s ′ be the standard episturmian word directed by va∆ ′ . The letter a occurs in s ′ and considering b the first letter of v, we see that b = a and ab is a factor of the infinite word s ′ . Since Pal(w) = lim n→∞ L a 1 ...an (a n+1 ) (by Eq. ( 2)), s contains as a factor the word L waua ℓ (ab) = L wau (a ℓ+1 b) and so s contains L wa (L u (a) ℓ+1 Pal(u)b). By [START_REF] Justin | Episturmian words and episturmian morphisms[END_REF], since a does not occur in u, L u (a) = Pal(u)a. Consequently 1. By its construction, s H,n,a,b,c is a pseudostandard word, but is not episturmian as already said at the beginning of the subsection.

2. Let φ be a morphism such s H,n,a,b,c = φ(s H,n,a,b,c ). We are going to prove that φ is the identity. Notice that since s H,n,a,b,c = µ H (w H,n ), the word s H,n,a,b,c can be uniquely factorized over {a, bc, cb}.

We now prove that words φ(a), φ(bc) and φ(cb) all belong to {a, bc, cb} * . We denote by u a , u bc , u cb the words such that φ(a) = µ H (u a ), φ(bc) = µ H (u bc ), φ(cb) = µ H (u cb ). We denote by η the morphism from {a, bc, cb} * to {a, b, c} * defined by η(a) = u a , η(bc)= u bc , η(cb) = u cb . We have

µ H (w H,n ) = s H,n,a,b,c = µ H • η(s H,n,a,b,c ) = µ H • η • µ H (w H,n ).
The injectivity of µ H over the set of infinite words implies that w H,n = η • µ H (w H,n ). By Proposition 4.24 this implies that η • µ H is the identity morphism over {a, b, c} * . Thus η(a) = a, η(bc) = b, η(cb) = c and so φ(a) = a, φ(bc) = bc, φ(cb) = cb which implies that φ is the identity morphism.

(α)| b = |µ H (α)| c .
5 About prefixes of fixed points (+) and that Pal ϑ (w ′ ) is a prefix of s, we conclude: w ′ x is a prefix of s.

We now continue exploring links between fixed points under the iterated palindromic closure Pal and their prefixes. Thus, we consider here only the first kind of fixed points, denoted s R,n,a,b . We will use pure epistandard morphisms (defined before Theorem 4.25) and their relations (1) and ( 2) with Pal.

The next proposition provides a second characterization of prefixes of fixed points of Pal using morphisms. ( 3 ⇒ 2 is immediate except if w is a power of a letter. But in this case, a being the letter such that w = a |w| and α being any other letter, w is a prefix of L w (α).

) a. If p 2 = ε. Then Pal(ap 2 ) = ax 1 ax 2 . . . ax k a, k ≥ 1, with x i ∈ A \ {a} for 1 ≤ i ≤ k, and consequently, L p (a) = L ap 1 (a)L ap 1 (x 1 ) • • • L ap 1 (x k )L ap 1 (a). Since L p (a) is a palindrome, L ap 1 (x 1 )[1] = last(L ap 1 (x k )). Moreover, L ap 1 (x 1 )[1] = a and x k = a implies last(L ap 1 (x k )) = a: contradiction. 3
2 ⇒ 1. If α occurs in w, then by Lemma 5.6, L w (α) is a prefix of Pal(w) and so by hypothesis, w is a prefix of Pal(w). When α does not occur in w, Lemma 5.6 implies L w (α) = Pal(w)α and consequently w is a prefix of Pal(w). In both cases, it follows from Proposition 5.3 that w is a prefix of a word s R,n,a,b for different letters a, b and an integer n ≥ 1.

Here is a third characterization of the prefixes of fixed points of Pal.

Conclusion

Let us summarize three problems raised by the content of this paper.

It is easy to see that any infinite word which is k-th power-free for an integer k has a critical exponent. This is the case for all words studied in this paper. An open question is to find closed formulas of the values of the critical exponent of words s R,n,a,b , s H,n,a,n and s E,a,b .

Another direction of research would be to find a geometric interpretation of palindromic closure. It may help to find further properties of the fixed points of the operation we considered here.

Finally since the study of the pseudostandard words which are fixed points of the Pal ϑ operator raises numerous intriguing questions, it might be interesting to also work with the more general families of words introduced in [START_REF] De Luca | Pseudopalindrome closure operators in free monoids[END_REF] and [START_REF] Bucci | On some problems related to palindromic closure[END_REF] (see also [START_REF] Bucci | On different generalizations of episturmian words[END_REF]). The first one is called the generalized pseudostandard words, that is the pseudostandard words directed by 2 words: the usual directive word and a word describing the antimorphism to use at each iteration. The second one is the pseudostandard words with seeds, that is the words obtained by iteration of the ⊕ ϑ operator starting with a non-empty word, called the seed.

We gratefully thank the referees for a careful reading of the paper and for their valuable suggestions.

we deduce that P (q + r + 1) = |xa| and d(r) = |y| + 1 (or equivalently |xa| = r + q + 1 -d(r)). We also have v i

Example 2 . 1 .Example 2 . 2 .

 2122 The Fibonacci infinite word f = Pal((ab) ω ) = abaababaabaababaabab • • • is a standard Sturmian word directed by the word (ab) ω . Indeed: Pal(a) = a Pal(ab) = (Pal(a)b) (+) = aba Pal(aba) = (Pal(ab)a) (+) = abaaba Pal(abab) = (Pal(aba)b) (+) = abaababaaba • • • The infinite word abcabaac • • • directs the standard episturmian word w = Pal(abcabaac • • • ) = abacabaabacababacabaabacabaa • • • .

Example 2 . 4 .Example 2 . 5 .

 2425 Over the alphabet {a, b}, since the longest E-palindromic suffix of w = aaba is ba, w ⊕ = aaba • E(aa) = aababb. Over the alphabet {a, b, c}, let us now consider ϑ such that τ (a) = a, τ (b) = b and τ (c) = c, and let w = aacbcb. Since the longest ϑ-palindromic suffix of w is bcb, w ⊕ = aacbcb • ϑ(aac) = aacbcbcaa. Notice that in this example, ϑ = R • τ = R • id = R, i.e. ϑ is ordinary palindromic closure. Extending the Pal operator to pseudopalindromes, the Pal ϑ operator is naturally defined by Pal ϑ (ε) = ε and Pal ϑ (wa) = (Pal ϑ (w)a) ⊕ , for w ∈ A * and a ∈ A. Then, for w

Examples 3 . 1 .

 31 naturally set Pal 0 ϑ (w) = w and Pal n ϑ (w) = Pal ϑ (Pal n-1 ϑ (w)), for any w ∈ A ω , involutory antimorphism ϑ and n ≥ 1. Let us see some examples of the iteration of the Pal ϑ operator over infinite words. Over a 2-letter alphabet A = {a, b}, there are only two possible involutory antimorphisms: the reversal antimorphism R and the exchange antimorphism E. Let us consider the iteration of the Pal ϑ operator over the word w = abx • • • , with x ∈ A.

Theorem 3 . 4 . 1 .

 341 Over a k-letter alphabet, with k ≥ 2, there are exactly 3 kinds of fixed points of Pal ϑ having at least 2 different letters, only depending on the first letters of the word and the involutory antimorphism ϑ = R • τ considered. When τ (a) = a and τ (b) = b, with a, b ∈ A and a = b, Pal ϑ has a unique fixed point beginning with a n b, for all n ≥ 1, denoted s R,n,a,b , which equals

2 .

 2 When τ (a) = a and τ (b) = c for pairwise different letters a, b, c ∈ A, Pal ϑ has a unique fixed point beginning with a n b, for all n ≥ 1, denoted by s H,n,a,b,c , which equals s H,n,a,b,c = lim i→∞ Pal i H (a n b) = a n bca n cba n bca n (abca n cba n bca n ) n c • • • .

3 .

 3 When τ (a) = b and τ (b) = a, with a, b ∈ A, a = b, Pal ϑ has a unique fixed point beginning with a n b only if n = 1. It is denoted by s E,a,b and equals s E,a,b = lim i→∞ Pal i E (a) = abbaabbaababbaabbaababbaabbaabb • • • . Proof. The proof is obtained by combining Lemmas 3.2 and 3.3. Theorem 3.4 characterizes all possible fixed points of Pal ϑ except the trivial fixed point of the form a ω , which is a fixed point of Pal ϑ using any antimorphism ϑ = R • τ with τ (a) = a. This trivial fixed point corresponds to the words obtained in parts 1 and 2 of Theorem 3.4 with n = ∞.

  Lemma 4.1 ([13], Theorem 3). An infinite word obtained by the Pal operator is ultimately periodic if and only if its directive word has the form ua ω , with u ∈ A * and a ∈ A.Proposition 4.2. For a fixed positive n ∈ N, s R,n,a,b is not ultimately periodic, and consequently s R,n,a,b is a standard Sturmian word. Proof. By definition of the word s R,n,a,b , (Pal i (a n b)) i≥0 forms a sequence of prefixes of s R,n,a,b . The sequence of lengths of these prefixes is strictly increasing by the definition of the Pal operator. Since ba n is a suffix of Pal i (a n b), both letters a and b occur infinitely often in s R,n,a,b . Hence s R,n,a,b is not of the form uα ω for a word u and a letter α. Since by its definition, s R,n,a,b equals its directive word, Lemma 4.1 implies that s R,n,a,b is not ultimately periodic. Proposition 4.2 is very useful, since it allows us to use properties of standard Sturmian words in order to characterize the fixed points s R,n,a,b . Let us recall some of them.

Theorem 4 . 3 (

 43 [START_REF] Fraenkel | Determination of [nθ] by its sequences of differences[END_REF], see also Section 2.2.2 in[START_REF] Lothaire | Algebraic Combinatorics on Words, volume 90 of Encyclopedia of Mathematics and its Applications[END_REF]).Let ∆(w) = a d 1 b d 2 a d 3 b d 4 • • • be the directiveword of an infinite standard Sturmian word w, with d i ≥ 1. Then the slope of w has the continued fraction expansion α = [0; 1 + d 1 , d 2 , d 3 , d 4 , . . .]. Recall that the slope of a Sturmian word s ∈ {a, b} ω is α = lim n→∞ |s[1 . . . n]| b /n and refers to the geometric interpretation of a Sturmian word.

  (a), with ϕ(a) = ab and ϕ(b) = a, while the second one is defined by lim i→∞ µ i (a), with µ(a) = ab and µ(b) = ba. The computation of the Fibonacci word using morphisms yields ϕ(a) = ab, ϕ 2 (a) = aba, ϕ 3 (a) = abaab, ϕ 4 (a) = abaababa, . . . , f = abaababaabaababaaba • • • as in Example 2.1, and the one of the Thue-Morse word yields µ(a) = ab, µ 2 (a) = abba, µ 3 (a) = abbabaab, . . . , T = abbabaabbaababbabaababbaabbabaab • • • .

  Let α n,a,b denote the slope associated to s R,n,a,b . Lemma 4.6. The continued fraction expansion of α n,a,b has bounded partial quotients. Proof. One can easily see that, for the continued fraction expansion [0; 1 + d 1 , d 2 , . . .] of α n,a,b , each d i belongs to {1, n, n + 1}. Indeed, the 0 appears only once, as the first value of the expansion. Since s R,n,a,b is standard Sturmian, there is one letter having only blocks of length 1 and the other letter has blocks of two consecutive lengths, n and n + 1, with n the length of the first block prefix of s R,n,a,b . Thus, using Theorem 4.3, we get d 1 = n, d 2k = 1 and d 2k+1 ∈ {n, n + 1} for k ≥ 1. Since n is fixed, the conclusion follows.

Lemma 4 . 7 (

 47 [START_REF] Vandeth | Sturmian words and words with a critical exponent. Theoret[END_REF],Theorem 17). Let α > 0 be an irrational number withd α = [d 0 ; d 1 , d 2 , .. .], its continued fraction expansion. Then the standard Sturmian word of slope α denoted by w α is k-th power-free for some integer k if, and only if, d α has bounded partial quotients. Moreover, if d α has bounded partial quotients, then w α is k-th power-free but not (k -1)-th power-free for k = 3 + max i≥0 d i .Combining Lemmas 4.6 and 4.7, we directly get: Proposition 4.8. s R,n,a,b is (n + 4)-th power-free, but contains (n + 3)-th powers.

Theorem 4 . 9 . [ 1 ]Proposition 4 . 10 .

 491410 3, 1, 2, . . .] = 0.263762936248362388488733270234476572992585105341587 . . . α 3,a,b = [0; 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 3, 1, 4, 1, 4, 1, 4, . . .] = 0.207106782338295017398506080110904477388983761913715 . . . α 4,a,b = [0; 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 4, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 4, 1, 5, . . .] = 0.170820393253126826628272040633095783457508409253431 . . . α 5,a,b = [0; 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 5, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, . . .] = 0.145497224367909729164797535715036805731190553212987 . . . A problem that naturally arises is whether α n,a,b is transcendental. The answer is positive as shown in Proposition 4.10 whose proof uses the following result of Adamczewski and Bugeaud [1]. Let a = (a ℓ ) ℓ≥1 be a sequence of positive integers. If the word a begins with arbitrarily long palindromes, then the real number α = [0; a 1 , a 2 , . . . , a ℓ , . . .] is either quadratic irrational or transcendental. For any n ≥ 1, α n,a,b is transcendental. Proof. By Proposition 4.2, s R,n,a,b is not ultimately periodic. Consequently, there are infinitely many occurrences of a's and b's in s R,n,a,b . Let

Theorem 4 .

 4 11 ([12], Theorem 7.1). For any w ∈ A ω and for any involutory antimorphism ϑ, one has Pal ϑ (w) = µ ϑ (Pal(w)).

  Let b = τ (a) with b = a. We have µ ϑ (a) = ab, µ ϑ (b) = ba and neither a nor b occurs in µ ϑ (c) for c ∈ A \ {a, b}. Possibly replacing v by v 2 , we can assume that |v| a + |v| b is even. Depending on the parity of |u| a +|u| b , two cases are possible: u

Proposition 4 .Case 2 ,

 42 [START_REF] Jamet | On the fixed points of the iterated pseudopalindromic closure[END_REF]. s E,a,b is not a fixed point of some nontrivial morphism. Proof. Let us suppose by contradiction that there exists a nontrivial morphism φ such that s E,a,b = φ(s E,a,b ). Four cases can hold: Case 1, |φ(a)| and |φ(b)| are odd. Since φ(a) is a prefix of s E,a,b which belongs to {ab, ba} ω , φ(a) = µ E (u)α for a word u and a letter α ∈ {a, b}. Since φ(a)φ(b) is a prefix of s E,a,b , there exists a word v such that φ(b) = αµ E (v). Since φ(abba) = µ E (uαv)αµ E (vu)α is a prefix of s E,a,b , there exists a word w such that µ E (w) = αµ E (vu)α. Necessarily w ∈ α * and there exist integers k and ℓ such that φ(a) = (αα) k α and φ(b) = α(αα) ℓ . Since abb is a prefix of s E,a,b , we have k = ℓ = 0, and thus φ(a) = a, φ(b) = b which contradicts the fact that φ is not the identity. |φ(a)| is odd and |φ(b)| is even. Since φ(ab) is a prefix of s E,a,b , we deduce that φ(a) = µ E (u)α and φ(b) = αµ E (v)β for words u, v and letters α, β. Since φ(abba) = µ E (uαv)βαµ E (v)βµ E (u)α is a prefix of s E,a,b , β = α and βµ E (u)α = αµ E (u)α ∈ {ab, ba} * which is not possible since any word in {ab, ba} * contains the same number of occurrences of a's as b's.

Case 3 ,

 3 |φ(a)| is even and |φ(b)| is odd. Acting as previously, we deduce that φ(a) = µ E (u), φ(b) = µ E (v)α for words u, v and a letter α. Moreover αµ E (v)α must belong to {ab, ba} * which once again is impossible. Case 4, |φ(a)| and |φ(b)| are even. In this case, φ = µ E • η for a morphism η that maps the letters a and b to words of even length.

Proposition 4 . 21 .

 421 The words w H,n are not ultimately periodic, and consequently they are strict standard episturmian words.Since by definition, s H,n,a,b,c = µ H (w H,n ), Lemma 4.13 implies: Proposition 4.22. The words s H,n,a,b,c are not ultimately periodic. Let us recall a useful result from Justin and Pirillo. Proposition 4.23. [19] A strict standard episturmian word is a fixed point of a nontrivial morphism if and only if its directive word is periodic.

Theorem 4 .

 4 25 ([19], Theorem 5.

L

  wa (L u (a) ℓ+1 Pal(u)b) = L wa ((Pal(u)a) ℓ+1 Pal(u)b) = L w (L a (Pal(u)a) ℓ+1 L a (Pal(u))ab) = L w (L au (a) ℓ+2 b).Hence s contains the (ℓ + 2)-nd powers (L wau (a)) ℓ+2 .The previous proposition can be viewed as a generalization of Lemma 4.7. As a direct consequence, we have: Corollary 4.27. The words w H,n are (n + 4)-th power-free but contain (n + 3)-rd powers. We now deduce from what precedes properties of the words s H,n,a,b,c . Proposition 4.28. Let s H,n,a,b,c be a fixed point of the Pal H operator, for a fixed n. Then s H,n,a,b,c satisfies the following properties:1. It is not an episturmian word, but is a pseudostandard word.2. It is not a fixed point of some nontrivial morphism.

3 .

 3 It is (n + 4)-th power-free but contains (n + 3)-rd powers.4. The frequencies of the letters b and c are equal.Proof.

  First assume that φ(a) = ε. Then by Proposition 4.22, s H,n,a,b,c is not ultimately periodic which implies φ(b) = ε and φ(c) = ε. Then φ(b) begins with a. Moreover since φ(bca n b) and φ(bca n cba n b) are factors of s H,n,a,b,c , we deduce that φ(bc) and φ(cb) both belong to {a, bc, cb} + . Assume now that φ(a) = ε. Then φ(a) begins with a. Since φ(a)φ(a), φ(a)φ(bc)φ(a) and φ(a)φ(cb)φ(a) are factors of s H,n,a,b,c , we deduce that all words φ(a), φ(bc) and φ(cb) belong to {a, bc, cb} + .

Theorem 3 .Fact 5 . 1 . 1 .Fact 5 . 2 .Proposition 5 . 3 .

 35115253 [START_REF] Berstel | Combinatorics on Words: Christoffel Words and Repetitions in Words[END_REF] shows that the fixed points s R,n,a,b and s H,n,a,b,c (resp. s E,a,b ) of the Pal ϑ operator are the limit of the sequence of finite words u 1 = a n b (resp. u 1 = a), u k = Pal ϑ (u k-1 ) for k ≥ 2. In particular, we can observe the following property:For any prefix p of a fixed point of the Pal ϑ operator, p is also a prefix of Pal ϑ (p).Before proving the converse (which leads to a characterization of prefixes of fixed points of the Pal ϑ operator), let us observe a property that follows directly from the definition of the Pal ϑ operator. For a finite word w and an involutory antimorphism ϑ = R • τ , the following assertions are equivalent: Pal ϑ (w) = w; 2. |Pal ϑ (w)| = |w|; 3. w = a |w| for a letter a such that τ (a) = a. Proof. The only difficulty concerns 2 ⇒ 3 and this can be proved by induction on |w| using the immediate property "|u| ≤ |Pal ϑ (u)| for any word u". Let w be a finite word starting with a letter a and ϑ = R • τ be an involutory antimorphism such that τ (a) = b, with b = a. Then |w| < |Pal ϑ (w)|. Proof. For w = a, Pal ϑ (w) = ab and then, |w| < |Pal ϑ (w)|. Since Pal ϑ (wα) = (Pal ϑ (w)α) ⊕ and |wα| = |Pal ϑ (wα)| if and only if |w| = |Pal ϑ (w)|, we conclude. Let A be an alphabet of cardinality at least 2, and let ϑ = R • τ be an involutory antimorphism over A. A finite word w over A is a prefix of Pal ϑ (w) if and only if w is a prefix of a fixed point of Pal ϑ not of the form a ω , with a ∈ A. Proof. As mentioned at the beginning of the section, we just have to prove the "only if" part. We act by induction on |w|. The case |w| = 0 is trivial. If w = a n for a n ≥ 1 and a letter a, since w is a prefix of Pal ϑ (w), τ (a) = a or n = 1 and τ (a) = b, for a = b. If τ (a) = a, w is a prefix of s R,n,a,b or s H,n,a,b,c for distinct letters a, b, c, otherwise w is a prefix of s E,a,b . Assume now that w = a n b with a = b and n ≥ 1. If τ (a) = a and τ (b) = b, then w is a prefix of s R,n,a,b . If τ (a) = a and τ (b) = b, then w is a prefix of s H,n,a,b,τ (b) . If τ (a) = a, then since w is a prefix of Pal ϑ (w), n = 1 and w is a prefix of s E,a,b . The remaining case is w = w ′ x, with x a letter and w ′ containing at least two different letters. Since w ′ x is a prefix of Pal ϑ (w ′ x), then w ′ is a prefix of Pal ϑ (w ′ x). By the definition of the Pal ϑ operator, Pal ϑ (w ′ ) is also a prefix of Pal ϑ (w ′ x). From |w ′ | ≤ |Pal ϑ (w ′ )|, we deduce that w ′ is a prefix of Pal ϑ (w ′ ) and so by induction w ′ is a prefix of a nontrivial fixed point s of Pal ϑ . Facts 5.1 and 5.2 imply |w ′ | < |Pal ϑ (w ′ )| and consequently, |w ′ x| ≤ |Pal ϑ (w ′ )|. Using this last inequality and the fact that w ′ x is a prefix of Pal ϑ (w ′ x) = (Pal ϑ (w ′ )x)

Proposition 5 . 4 .Proposition 5 . 7 . 2 ⇒ 1 .Proposition 5 . 8 . 1 =

 545721581 For any finite word w over an alphabet of cardinality at least two, the following assertions are equivalent: Let p ∈ A * and a ∈ A. Then the following are equivalent:1. L p (a) = Pal(p); 2. |p| a = 1 and p[1] = a. Proof. 1 ⇒ 2. i) If p[1] = a, then L p (a)[1] = p[1] = a and last(L p (a)) = a: contradiction, since L p (a) is palindromic. ii) If p[1] = aand |p| a ≥ 2, then p can be written as p = ap 1 ap 2 , with |p 2 | a = 0. By Lemma 5.6, we have: L p (a) = Pal(p)Pal(ap 1 ) -1 . Since Pal(ap 1 ) is not empty, we get a contradiction: L p (a) = Pal(p). Thus, the only possibility is p[1] = a and |p| a = 1. If |p| a = 1 and p[1] = a, then p = ap ′ , with |p ′ | a = 0. By Lemma 5.6, L p (a) = Pal(p)Pal(ε) -1 = Pal(p). Let p ∈ A * and a ∈ A. If L p (a) is palindromic, then 1. either L p (a) = Pal(p); 2. or L p (a) = a. Proof. We have seen in the proof of Proposition 5.7 (see i)), that L p (a) palindromic implies p[1] = a. If |p| a = 1 and p[1] = a, then by Proposition 5.7, L p (a) = Pal(p). Assume |p| a ≥ 2 and p[1] = a. Then there exist words p 1 and p 2 such that p = ap 1 ap 2 , |p 2 | a = 0. By Lemma 5.6 and Eq. (1) (for the second equality), we have: L p (a) = Pal(p)Pal(ap 1 ) -1 = L ap 1 (Pal(ap 2 ))Pal(ap 1 )Pal(ap 1 ) -L ap 1 (Pal(ap 2 )).

b. If p 2 =

 2 ε and p is not a power of a, then let us rewrite p = p 11 ap 12 a n , for some n > 0 and words p 11 and p 12 such that |p 12 | a = 0 and p 12 = ε. We have L p (a) = L p 11 ap 12 a n (a) = L p 11 ap 12 (a) which is not a palindrome (by case a.) c. If p 2 = ε and p = a n for some n, we easily see that L p (a) = a and Pal(p) = a n . Proof of Proposition 5.4. 1 ⇒ 3. By Proposition 5.3, if w is a prefix of a word s R,n,a,b for different letters a, b and an integer n ≥ 1, then w is a prefix of Pal(w). Assume w is not a power of a letter and let α be a letter occurring in w such that α = last(w). Then one can verify by induction on the length of w that |w| ≤ |L w (α)|. Since by Lemma 5.6, L w (α) is a prefix of Pal(w), we deduce that w is a prefix of L w (α).

Proposition 5 . 9 . 1 ⇒ 2 . 2 ⇒ 3 .

 591223 Let w be a word which is prefix comparable to a n b where a, b are two different letters and n ≥ 1 is an integer. The following assertions are equivalent: Proof of Proposition 5.9. When w is a prefix of s R,n,a,b , since s R,n,a,b is a fixed point of Pal, Pal(w) is a prefix of s R,n,a,b . Since |w| ≤ |Pal(w)|, w is a prefix of Pal(w). Moreover by Eq. (1), L w (Pal(w)) is a prefix of Pal(ww), and by definition of Pal, Pal(w) is a prefix of Pal(ww). Since |Pal(w)| < |L w (Pal(w))|, Pal(w) is a prefix of L w (Pal(w)), and so w is a prefix of L w (Pal(w)). Finally from w prefix of Pal(w), we deduce that L w (w) is a prefix of L w (Pal(w)). It is straightforward that |w| ≤ |L w (w)| so that w is a prefix of L w (w). One can easily verify that |w| < |L w (a n b)|. When a n b is a prefix of w, w and L w (a n b) are both prefixes of L w (w), and so w is a prefix of L w (a n b). Otherwise, since w and a n b are prefix comparable, w is a power of a, and L w (a n b) = wa n b. 3 ⇒ 4 is immediate with c = a, d = b and m = n. 4 ⇒ 1. By Lemma 5.10, Pal(w) is a prefix of L w (c m d). Since |w| ≤ |Pal(w)|, w is a prefix of Pal(w). Hence by Proposition 5.3, w is a prefix of a fixed point of Pal. Since w and a n b are prefix comparable, we can deduce that w is a prefix of s R,n,a,b .

  -1 xa = δ 1 • • • δ (i-1)q+r+q+1-d(r) and so u r+iq+1-d(r) = Pal(v i-1 xa). Moreover v i-1 xay = δ 1 • • • δ (i-1)q+q+r and so h r+iq = L v i-1 xay (a). Thus lim i→∞ |Pal(v i xa)| |L v i xay (a)| = lim i→∞ |u r+iq+1-d(r) | |h r+iq | and consequently b 4.25 ≤ b 6.1 .Let r be an integer in L 6.1 . We consider the word ∆[1 . . . r+ q] = δ 1 • • • δ r a ℓ δ r+ℓ+1 • • • δ r+q . Since q is a period of ∆, δ r+q+1 • • • δ r+q+ℓ =a ℓ and so by definition of ℓ we have δ r+q = a. Let y be the word such that a ∈ alph(y) and ay is a suffix of aδ r+ℓ+1 • • • δ r+q and let x be the word such that δ 1 • • • δ r+q = xay. Observe that by construction |v| ≤ |xay| < |v 2 | and so (x, a, y) ∈ L 4.25 . Hence for i ≥ 1, u r+iq+1-d(r) = Pal(v i-1 xa) and h r+iq = L v i-1 xay (a) showing that lim i→∞ |u r+iq+1-d(r) | |h r+iq | = lim i→∞ |Pal(v i xa)| |L v i xay (a)| : b 6.1 ≤ b 4.25 . This ends the proof of the equivalence between Theorems 4.25 and 6.1.

  Proposition 4.14 is a direct corollary of Proposition 4.12 and Lemma 4.13. Proposition 4.14. s E,a,b is not ultimately periodic. Another way to prove Proposition 4.14 uses the following proposition.Proposition 4.15. Let ϑ be an involutory antimorphism over an alphabet A. An infinite word obtained by the Pal ϑ operator is ultimately periodic if and only if its directive word has the form uα ω , with u ∈ A * and α ∈ A. Proof. Assume t = Pal ϑ (w) is ultimately periodic. By Theorem 4.11, t = µ ϑ (Pal(w)). Thus Proposition 4.15 appears as a direct corollary of Lemmas 4.1 and 4.13. Proposition 4.15 is interesting by itself, since it generalizes a well-known useful result of Droubay, Justin and Pirillo to pseudostandard words (see Theorem 3 in [13]).We now consider the following corollary of Proposition 4.14.

Corollary 4.

[START_REF] Glen | Quasiperiodic and Lyndon episturmian words[END_REF]

. w E is not a fixed point of some nontrivial morphism.

Proof. By Theorem 4.3, the continued fraction expansion of the slope of w E is ultimately periodic if and only if its directive word, which is s E,a,b by definition, is ultimately periodic. Hence by Proposition 4.14, the continued fraction expansion of the slope of w E is not ultimately periodic which implies by Theorem 4.4 that w E is not a fixed point of a nontrivial morphism.

  and from injectivity of µ E over the set of infinite words, we deduce that w E,a,b is the fixed point of η • µ E , contradicting Corollary 4.16. Lemma 4.18. The length of the blocks of s E,a,b are 1 and 2 for both letters.Proof. By Theorem 4.11, s E,a,b = µ E (w E). Thus, s E,a,b ∈ {ab, ba} ω and aa, bb, bab, aba are all factors of s E,a,b , which implies that the maximal length of a block is 2.

Proposition 4.

[START_REF] Justin | Episturmian words and episturmian morphisms[END_REF]

. w E contains 4-th powers, but is 5-th power-free.

Proof. By Theorem 4.3, the partial quotients of the continued fraction expansion of the slope of w E correspond to the blocks' lengths of s E,a,b , and so by Lemma 4.18 they have value 1 or 2. The statement is then a direct corollary of Lemma 4.7.

  3. By Theorem 4.11, s H,n,a,b = µ H (w H,n ). Here µ H is defined by µ H (a) = a, µ H (b) = bc, µ H (c) = cb. We let the reader verify that, for any integer k ≥ 2, a word w (finite or infinite) contains a k-th power if and only if µ H (w) contains a k-th power. Then this item follows from Corollary 4.27. 4. This is once again a direct consequence of s H,n,a,b,c = µ H (w H,n ), since µ H is a morphism such that for every letter α, |µ H

Appendix: More on Theorem 4.25

We have already mentioned that Theorem 4.25 is Theorem 5.2 in [START_REF] Justin | Episturmian words and episturmian morphisms[END_REF]. Nevertheless our formulation is slightly different from the original one which is: Theorem 6.1 ([19], Theorem 5.2). Let s be an A-strict standard episturmian word generated by a morphism and let q be the period of its directive word ∆ = (δ i ) i≥1 (with each δ i a letter). Let ℓ ∈ N be maximal such that y ℓ ∈ F (∆) for some letter y.

Moreover for any letter u in s the limit above can be obtained as a rational function with rational coefficients of the frequency α u of this letter.

To understand this statement, it is useful to recall that for n ≥ 0:

Note also that in this statement, A must contain at least two letters to allow the definition of ℓ, and that in this statement and from now on let δ i denote the i-th letter of ∆ instead of ∆[i].

We now explain the equivalence between the statements of Theorems 4.25 and 6.1. First note that symbols ∆, s and ℓ denote the same objects, and q = |v|. In the next formulas, the subscripts 4.25 and 6.1 refer to the statements of Theorem 4.25 and Theorem 6.1 respectively. Let us define :


PhD Tim Takken
email: t.takken@umcutrecht.nl

Samantha Stephens

MSc Astrid Balemans

PhD Mark S Tremblay

Dale W Esliger

PhD Jane Schneiderman

W Douglas Biggar

Patricia Longmuir

PhD Virginia Wright

MD Brian W Mccrindle

Samantha Stephens Msc

Dale W Esliger Msc

MD Douglas Biggar

Patricia Longmuir Msc

Moniek Hendricks Msc

Audrey Abad Msc

PhD Janjaap Van Der Net

MD Brian M Feldman

Validation of the Actiheart Activity Monitor for Measurement of Activity Energy Expenditure in Children and Adolescents with Chronic Disease

Keywords: physical activity, measurement, indirect calorimetry, exercise, heart rate

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Children with chronic disease are often sedentary and this behavior pattern can have various etiologies. For example in children with congenital heart disease (CHD) and hemophilia, hypoactivity might be caused by overprotection by parents and educators (Bar-Or 1987, Bergman andStamm 1967), while in juvenile arthritis children might be hypoactive because of painful joints [START_REF] Takken | Physical activity and health related physical fitness in children with juvenile idiopathic arthritis[END_REF].

There is an increasing body of literature showing that children with a chronic disease or condition are able to improve their physical fitness levels through an exercise training program [START_REF] Edouard | Training programs for children: literature review[END_REF][START_REF] Johnson | The Benefits of Physical Activity for Youth With Developmental Disabilities: A Systematic Review[END_REF][START_REF] Klijn | Effects of anaerobic training in children with cystic fibrosis: a randomized controlled study[END_REF], Takken et al 2007[START_REF] Van Brussel | Physical training in children with osteogenesis imperfecta[END_REF][START_REF] Verschuren | Exercise training program in children and adolescents with cerebral palsy: a randomized controlled trial[END_REF]. However, a frequently encountered problem is the deconditioning after the cessation of the structured exercise training program [START_REF] Van Brussel | Physical training in children with osteogenesis imperfecta[END_REF][START_REF] Verschuren | Exercise training program in children and adolescents with cerebral palsy: a randomized controlled trial[END_REF]. This may be caused by the fact that children with a chronic condition encounter many barriers in performing physical activities in their daily life. Many sport clubs are not equipped to have these children participate in their programs. Moreover, there may be perceived barriers to regular physical activity because of restriction in physical activity by parents, caregivers, health-care professionals, trainers and teachers as well as the child's own perception of the consequences of activity on their condition [START_REF] Edouard | Training programs for children: literature review[END_REF][START_REF] Murphy | Promoting the Participation of Children With Disabilities in Sports, Recreation, and Physical Activities[END_REF].

Increasing daily physical activity levels for children with a chronic disease or condition might be an effective method to maintain their physical fitness levels [START_REF] Morris | Physical activity recommandations for children and adolescents with chronic disease[END_REF]. However, research in this area is limited because of a lack of robust direct measurement methods for assessing free-living physical activity in children with chronic disease. Activity monitoring seems to be a promising method to objectively assess and profile physical activity (de Graauw et al 2010, Esliger andTremblay 2007, Hoos et al 2004), and seems to be more valid than indirect assessments (e.g. questionnaires and activity logs) [START_REF] Adamo | A comparison of indirect versus direct measures for assessing physical activity in the pediatric population: a systematic review[END_REF].

A recent report has called into question to use of methodologies established for children without chronic disease when predicting energy expenditure (EE) in children with chronic disease [START_REF] Stephens | Validation of Accelerometry as a Measure of Physical Activity Energy Expenditure in Children with Chronic Disease[END_REF]. Stephens and colleagues (2009) assessed the validity of prediction equations based on healthy subjects for measuring EE via accelerometry [START_REF] Freedson | Calibration of the Computer Science and Applications, Inc. accelerometer[END_REF][START_REF] Puyau | Validation and calibration of physical activity monitors in children[END_REF] for children with chronic disease [START_REF] Stephens | Validation of Accelerometry as a Measure of Physical Activity Energy Expenditure in Children with Chronic Disease[END_REF]. In this study, EE was measured by indirect calorimetry during seven activity tasks of variable intensities [START_REF] Stephens | Validation of Accelerometry as a Measure of Physical Activity Energy Expenditure in Children with Chronic Disease[END_REF]. Results showed significant differences between predicted EE and measured EE in children with chronic disease.

Therefore, the purpose of this study was to develop an AEE prediction equation for the Actiheart activity monitor for children with chronic disease and to assess the validity of the Actiheart to determine AEE in this population.

Methods

Participants

Sixty-three participants with chronic disease were recruited from several clinics of the Wilhelmina Children's Hospital, University Medical Center Utrecht. Diagnostic distribution and descriptive characteristics of participants can be found in Table 1. Participants included children and adolescents in that ages ranged from 8-18 years. All participants' parents gave written informed consent and participants aged 12 years and older signed their own assent form. This study was approved by the Medical Ethics Committee of the University Medical Center Utrecht, the Netherlands. Participants were not included in the study if they used medications that interfered with heart rate response to exercise or if they were unable to ambulate and/or cooperate. One patient with a Fontan circulation had a rate responsive pacemaker. This cohort was a subsample of a larger multi-center study on accelerometry, using different activity monitors.

Anthropometry

Weight was determined using an electronic scale (Seca, Hamburg, Germany), and a wallmounted stadiometer was used to measure height. Body mass index (BMI) was calculated by dividing body mass (in kg) by height squared (in meters).

Testing protocol

Participants participated in one testing session with a duration of ~1.5 hr, which consisted of three protocols:

Resting Protocol: Expired gas was collected using indirect calorimetry (IC) to determine resting metabolism over 20 minutes with the participant supine in a quiet room. Recordings were taken at 1-min intervals for the duration of the last 10 minutes of rest. The average values for heart rate (HR) and EE over the last 10 minutes was taken as resting HR (RHR) and resting EE (REE) since this provided steady-state values for RHR and REE.

IC measurements were made with the same equipment as the Activities protocol.

Activities Protocol:

The participants completed a questionnaire on the computer while seated.

In addition they performed a 6-minute cleaning task (sweeping), a 6-minute hallway walking task (at a self-selected comfortable walking pace) and a 6-minute stair stepping routine at a metronome-guided pace of 70 steps/min. A 6-minute duration was chosen to allow for a steady state oxygen uptake within the first 4 minutes of each activity. Furthermore, participants were asked to walk or jog on a treadmill at three different speeds.

The treadmill protocol selected depended on the walking speed chosen during the hallway walking task. For participants who walked with a self-selected walking speed of >3.5 km/h, a high function treadmill protocol was used (5 minutes at each of 3.5, 5 and 6.5 km/h, respectively). When the walking speed was <3.5km/h, a low function protocol was used (5 minutes each of 2.0, 3.5 and 5.0 km/h, respectively).

Actiheart

The Actiheart (AH2, CamnTech, Cambridge, UK) is a chest-mounted, combined-unit, synchronized heart rate monitor and accelerometer capable of storing time-sequenced data.

Acceleration is measured by a piezoelectric element within the unit with a frequency range of 1-7 Hz. The Actiheart unit was clipped to two ECG electrodes (3M™ Red Dot™ 2271 Monitoring Electrodes, 3M Nederland BV, Zoeterwoude, the Netherlands) adhered to the child's chest. The Actiheart was positioned at the level of the third intercostal space [START_REF] Brage | Effect of combined movement and heart rate monitor placement on physical activity estimates during treadmill locomotion and free-living[END_REF]. Acceleration movement counts and heart rate from Actiheart were sampled every 15 seconds. The participants wore the Actiheart unit during the entire testing session. The existing AEE prediction equation developed on healthy children for the Actiheart used heart rate above sleep (HRaS) as a variable [START_REF] Corder | Comparison of two methods to assess PAEE during six activities in children[END_REF]. Accordingly, we converted resting HR to HRaS using the following formula from healthy children (Corder et

Indirect calorimetry (IC)

During all tasks expired gas was collected through a face mask in order to measure oxygen uptake (VO 2 , L/min) and carbon dioxide exhalation (VCO 2 , L/min) breath-by-breath, to determine energy expenditure. Data were averaged over a 15-s epoch. A portable respiratory gas analysis system was used during all activities (Cortex Metamax, CORTEX Biophysik GmbH, Leipzig Germany). The Cortex Metamax was calibrated against ambient air and a standard calibration gas mixture provided by the manufacturer (15% 2 O , 4% 2 CO ), and volume calibrated prior to every exercise test session. VO 2 and VCO 2 data were converted to EE using the de Weir Equation (de Weir 1949):

EE (J/kg/min) = (3.9 × VO 2 + 1.1 × VCO 2 ) × 4184
Resting energy expenditure (REE) was determined from the average of the last 10 minutes of the resting protocol. AEE of the activities protocol were derived from the measured total EE of each task minus REE. The metabolic and activity data of the last two minutes of each task were averaged and used for analysis of actual AEE.

Statistical analysis

In order to best predict activity energy expenditure from accelerometry counts and heart rate we used mixed linear modeling techniques using SAS 9.1 (SAS Institute, Cary, N.C. 2009).

Mixed linear regression modeling is similar to simple linear regression only it allows for one to account for correlated data as well as non-constant variability. This difference in modeling techniques means that each subjects data contributes to the overall model as a random effect which allows for more accurate inferences to be made about the overall population effect rather than just being confined to making inferences about the existing data. Activity energy expenditure determined by indirect calorimetry was entered into the model as the main outcome. Heart rate above rest, accelerometer counts and gender were entered into the model as fixed effects. We also included random effects for heart rate above sleep and counts in order to account for the variation in each individuals response over the 7 different physical tasks. Further, as each subject was tested using 7 different tasks a repeated term was used to account for the within subject correlation [START_REF] Singer | Using SAS PROC MIXED to Fit Multilevel Models, Hierarchical Models, and Individual Growth Models[END_REF].

The difference between estimated AEE and actual AEE was analyzed using Bland-Altman plots and limits of agreement [START_REF] Bland | Statistical methods for assessing agreement between two methods of clinical measurement[END_REF]. Moreover, Lin's concordance coefficient (LCC) and the corresponding 95% confidence interval were also used to determine the agreement between the two methods [START_REF] Lin | A concordance correlation coefficient to evaluate reproducibility[END_REF]). The LCC measures both precision (Pearson correlation) and accuracy (bias correction factor) to determine whether the observed data deviate significantly from the line of perfect concordance, which occurs at 45 degrees;

that is, it assesses the linear relationship. The Bland-Altman plots and LCC were calculated using MedCalc 10.0. (Medcalc BVBA, Mariakerke, Belgium). A P-value of < 0.05 was considered as statistically significant.

Results

The descriptive characteristics of the participants are provided in Table 1. The BMI Z-scores were significantly increased in the patients with JDM and Haemophilia and was significantly decreased in the patients with CF.

Five participants (1 CF, 1 JDM, 2 JIA and 1 HMSN) were not able to finish fast treadmill walking because of joint pain or exhaustion. Two participants with JDM stopped already at the start of slow treadmill walking because of exhaustion. The mean estimated AEE of the prediction equation of Corder et al. [START_REF] Corder | Comparison of two methods to assess PAEE during six activities in children[END_REF] and the currently developed prediction equation for each task are shown in Table 3 andFigure

Discussion

The results of this study suggest that although activity monitors are regarded as an objective measure of physical activity, it remains a quite crude measure for estimating AEE in children with chronic disease. The derived prediction equation using both HR and activity counts accounted for 65% of the variance in actual AEE. The large range of the 95% CI in estimated AEE indicates that prediction equation seem to be more accurate for groups (e.g. research), than for individuals (e.g. clinical practice). The observed limits of agreement from the Bland-Altman plots were comparable to those reported in healthy adults using the Actiheart [START_REF] Crouter | Accuracy of the Actiheart for the assessment of energy expenditure in adults[END_REF]. The diversity of chronic conditions included in these analyses may have contributed to the individual variability. However, the derived prediction equation performs significantly better in children with chronic disease than a previously developed equation for healthy children [START_REF] Corder | Comparison of two methods to assess PAEE during six activities in children[END_REF].

There In our study a large group of children with different types of chronic diseases was tested to assess the validity of the Actiheart to determine AEE in this population. A strong overall correlation was found between estimated AEE from Actiheart with actual AEE for the complete exercise testing session. This suggests high overall criterion validity in measurement. The Actiheart showed no significant correlation with IC for working at the computer and a moderate to strong correlation for other activities. The correlations between predicted and measured AEE increased with higher intensity activities. From the current data it seems that the Actiheart becomes accurate starting from activities with an intensity of ~ 2.5 metabolic equivalents (METs).

The lack of correlation between Actiheart AEE estimation for working at the computer and IC can be explained by the fact that the activity monitor measures almost no acceleration of the chest during sedentary activities and low heart rates [START_REF] Crouter | Accuracy of the Actiheart for the assessment of energy expenditure in adults[END_REF]. In general, the Actiheart, and activity monitors, are better suited to predict EE during activity, rather than during sedentary activities [START_REF] Crouter | Accuracy of the Actiheart for the assessment of energy expenditure in adults[END_REF][START_REF] Bock | Combining Accelerometry and Heart Rate for Assessing Preschoolers' Physical Activity[END_REF]..

Since HRaS was a significant predictor of AEE in the mixed linear regression modeling (Table 2), AEE estimation in participants with a blunted HR response to exercise (e.g. chronotropic incompetence or beta-blocking medication usage [START_REF] Thoren | Effects of beta-adrenergic blockade on heart rate and blood lactate in children during maximal and submaximal exercise[END_REF]) might impede the validity of this combined HR and activity counts prediction equation.

Our results indicate that other approaches should be explored to improve the estimation of AEE for individual subjects. A study with adults with acquired brain injury suggested an individual calibration of the activity monitor output (e.g. activity counts) with actual AEE [START_REF] Tweedy | Validity of Accelerometry for Measurement of Activity in People with Brain Injury[END_REF]. Moreover, our Actiheart prediction equation was developed using the mean HR response in a group of subjects; which compromises the accuracy at the individual level [START_REF] Brage | Reliability and validity of the combined heart rate and movement sensor Actiheart[END_REF]. Future studies should examine approaches that take into account individual HR response at known exercise intensities to improve the accuracy of the prediction of AEE.

An other approach to improve accuracy might be more advanced mathematical modeling.

Zakeri et al have recently used multivariate adaptive regression splines (MARS) modeling in healthy children and adolescents to predict EE using the Actiheart activity monitor [START_REF] Zakeri | Multivariate Adaptive Regression Splines (MARS) Models for the Prediction of Energy Expenditure in Children and Adolescents[END_REF].

This study has a number of strengths. It is the first study to validate an objective measure of physical activity (Actiheart) in children and adolescents with chronic disease using actual energy expenditure (indirect calorimetry) as a criterion measure. The use of several different activities with both self-selected exercise intensities as well as a standardized treadmill protocol, allows our findings to be generalizable to free living activities.

This study had a few important limitations. We defined chronic disease using a noncategorical approach that considers chronically-ill young people as one population rather than specific disease classes [START_REF] Stein | A noncategorical approach to chronic childhood illness[END_REF]. The current sample size prohibited a subgroup analysis in specifying the disease classes, however a secondary analysis revealed no significant effects between groups. Furthermore, it is unclear whether this equation is also valid for other pediatric patient groups including younger children.

LEGEND TO

  al 2007): HRaS = 0.4195 × RHR + 27.4. Actiheart accelerometry counts and HRaS were converted to AEE using the established equation of Corder et al (Corder et al 2007) developed in healthy children: AEE (J/kg/min) = (5.17 × HRaS) + (0.61 × (gender [0=girls, 1=boys] × HRaS)) + (0.07 × activity counts) -(0.6 × gender) -74

1 .

 1 The Corder et al. equation significantly over-predicted the AEE compared to the currently developed AEE prediction equation. In addition, the Bland-Altman plot analyzing the differences between actual AEE and estimated AEE is shown in Figure 2. The current equation gives a mean difference of 2.1 J/kg/min (Limits of Agreement: -144.2 to 148.4) for the overall estimation AEE compared with actual AEE, instead of a mean difference of 49.1 J/kg/min (Limits of Agreemeent; -89.3 to 187.5; P<0.05) using the model of Corder et al.developed in healthy children[START_REF] Corder | Comparison of two methods to assess PAEE during six activities in children[END_REF]. The LCC for the Corder et al. equation with IC-AEE was 0.81 (95% CI 0.78-0.84), r = 0.87, bias correction factor: 0.93. The LCC for the current AEE prediction with actual AEE was 0.81 (95%CI: 0.77 to 0.84), r = 0.82, bias correction factor 0.99.

  are several potential explanations for why the equation of Corder et al. might be inaccurate for use in children with chronic disease. These include: delayed gross motor skill development among children with chronic disease; inefficiency of movement from disease complications; increased resting and activity energy expenditure associated with heart and lung function (Groen et al 2010, Hjeltnes et al 1984, Knops et al 1999); higher HR for a given VO 2 because of deconditioning (Bar-Or and Rowland 2004); and altered movement patterns to reduce pain (Broström et al 2007). Each of these would increase HR and EE for a given activity which may explain why the equation of Corder et al. consistently over-estimated AEE (figure 1).

FIGURESFigure 1 .

 1 Figure 1. Scatter plot of actual AEE as measured by indirect calorimetry and estimated by two Actiheart prediction equations (according to Corder et al (Corder et al 2007) and the current study).

Figure 2 .

 2 Figure 2. Bland-Altman plot of actual AEE (in J/kg/min) as measured by indirect calorimetry and AEE estimated using the Actiheart (current prediction equation).

Table 2

 2 reports the regression coefficients of the prediction of AEE from activity counts,

	HRaS, and gender. The new regression equation for prediction of AEE was:
	AEE (J/kg/min) = -93.7 + (4.8 × HRaS) + (0.04 × Accelerometer Counts) + (39.8 × Gender
	[girls=0, boys=1])

Table 1 .

 1 TABLES Characteristics of the participants (mean ± SD).

	Group	Age (years)	Height (cm)	Weight (kg)	BMI (kg/m 2 )	BMI for age (Z-score)	HR rest (bpm)	REE (J/kg/min)
	CF (N=9)	13.3±3.53	155.7±15.7	43.3±12.8	17.5±2.03	-.55±.55 *	74.1±14.4	173.9±44.8
	JDM (N=9)	13.9±1.96	159.0±9.0	55.1±10.5	21.6±2.42	.89±.97 *	74.9±14.4	113.6±20.9
	JIA (N=10)	11.7±2.58	152.5±11.5	41.2±12.1	16.2±6.19	-.43±1.55	78.8±8.77	153.7±43.8
	Hemophilia	12.5±3.31	166.0±17.6	59.3±22.5	20.7±4.73	.94±.98 *	70.5±5.15	129.3±33.5
	(N=10)
	CHD (N=8)	14.4±3.20	159.1±13.2	49.8±10.8	19.7±2.62	-.02±1.68	71.2±10.3	128.0±29.9
	NMD (N=17) 13.3±3.41	159.9±15.4	48.8±13.0	20.1±6.34	-.12±1.25	75.4±8.99	137.2±36.0

Abbreviations: CF: Cystic Fibrosis, JDM, Juvenile Dermatomyositis, JIA: Juvenile Idiopathic Arthritis, CHD: Congenital Heart Disease, NMD: Neuromuscular Disorders, REE: Resting Energy Expenditure, HR rest : Resting Heart Rate, BMI: Body Mass Index; *=significantly different from Dutch growth charts.

Table 3 .

 3 Comparison of the actual AEE with the estimated AEE from Actiheart using the prediction equation of Corder et al and the currently developed prediction equation.

	Task	Actual AEE	Estimated AEE	Actual AEE -	Estimated AEE	Actual AEE -current AH
		(J/kg/min)	Corder prediction	AEE Corder prediction	current AH prediction	prediction
			(J/kg/min)	(correlation, r)	(J/kg/min)	(correlation, r)
	Computer	19.2 ± 43.8	63.3 ± 40.0	0.07	48.3 ± 39.5	0.11
	Sweeping	139.6 ± 42.9	182.8 ± 63.0	0.25*	150.2 ± 55.0	0.27*
	Hallway	249.7 ± 78.6	293.3 ± 99.0	0.71*	232.9 ± 81.3	0.31*
	walking
	Steps	202.9 ± 50.8	254.7 ± 82.1	0.50*	209.6 ± 71.7	0.51*
	Slow TW	178.9 ± 36.3	231.5 ± 68.4	0.32*	187.3 ± 61.6	0.42*
	Moderate TW	263.7 ± 47.6	318.6 ± 76.9	0.47*	257.9 ± 66.3	0.47*
	Fast TW	424.5 ± 72.9	487.2 ± 125.0	0.69*	393.7 ± 102.8	0.69*
	All activities	206.0±125	257.83±143.2	0.87*	208.4 ± 118.2	0.81*

Abbreviations: AEE: Activity Energy Expenditure, TW: Treadmill walking, AH: Actiheart, * Significant (P<0.05).

Acknowledgements

This study was funded by a grant from the Canadian Institutes of Health Research (# 167391/CIHR). We would like to thank Fypro BV, Amsterdam, the Netherlands for their unconditional support with the Actiheart. We want to thank Preeya Govan, and Danial Ingas for their assistance in this project.

Funding source: This study was funded by a grant from the Canadian Institutes of Health Research (# 167391/CIHR).

In addition, the HRaS was calculated using an equation developed in healthy 12 year old children [START_REF] Corder | Comparison of two methods to assess PAEE during six activities in children[END_REF]. Unfortunately, we do not have overnight HR data of the participants to confirm this equation. Lastly, the derived prediction equation should be confirmed in a cross-validation study using similar and different activities; we only measured AEE during structured activities and did not measure AEE during free play activities of the participants.

A limitation of the practical use of the current prediction equation is the necessity of the determination of RHR. However, if overnight data are obtained, HRaS can be directly obtained from the data when the monitors are worn during sleep.

In conclusion, these results demonstrate that the Actiheart is valid for determining AEE when using the new prediction equation for groups of children with chronic disease. However, the prediction error limits the use of the equation in individual subjects.

Future research should determine the need for disease-specific AEE prediction equations and the applicability of the new prediction equation in other patient groups. Moreover, additional approaches should be explored to improve the prediction of AEE for individual subjects.