A periodicity property of iterated morphisms

41 DOI: 10.1051/ita:2007016

A PERIODICITY PROPERTY OF ITERATED MORPHISMS

Juha Honkala

Abstract. ^Supposef:X^∗−→X^∗is a morphism andu, v∈X^∗. For every nonnegative integern, letzn be the longest common prefix of fⁿ(u) andfⁿ(v), and letun, vn∈X^∗be words such thatfⁿ(u) =znun

andfⁿ(v) =znvn. We prove that there is a positive integerqsuch that for any positive integerp, the prefixes ofun(resp. vn) of lengthpform an ultimately periodic sequence having period q. Further, there is a value ofqwhich works for all wordsu, v∈X^∗.

Mathematics Subject Classification.68Q45, 68R15.

1. Introduction

In the theory of D0L systems many deep decidability results are proved by showing that certain D0L systems or pairs of D0L systems have some kind of regular behavior. In this paper we investigate a regularity property encountered when we compare the iteration of a morphism on two different words. More precisely, letf :X^∗−→X^∗be a morphism and letu, v∈X^∗ be words. For every nonnegative integern, letz_nbe the longest common prefix of the wordsfⁿ(u) and fⁿ(v), and let u_n and v_n be words such that fⁿ(u) = z_nu_n and fⁿ(v) = z_nv_n. Hence, the wordsu_nandv_ngive the wordsfⁿ(u) andfⁿ(v) from the first position where the latter words differ. In [4] it was shown that for any positive integerpthe prefixes of lengthpof u_n (resp. v_n) form an ultimately periodic sequence. This result plays a key role in the solution of the DF0L language equivalence problem in [4]. In this paper we study this periodicity result in more detail. In particular, we prove that there is a period which works simultaneously for prefixes of any length and does not depend on the wordsuandv.

Keywords and phrases.Iterated morphism, periodicity.

1Department of Mathematics, University of Turku, 20014 Turku, Finland;

juha.honkala@utu.fi

c EDP Sciences 2007

Article published by EDP Sciences and available at http://www.edpsciences.org/ita or http://dx.doi.org/10.1051/ita:2007016

We assume that the reader is familiar with the basics concerning iterated mor- phisms (see [5–7]). Notions and notations that are not defined are taken from these references.

2. Definitions and results

SupposeX ={x1, . . . , x_m}is an alphabet withm≥1 letters. Ifw∈X^∗, then

#_xi(w) is the number of occurrences of the letter x_i in w. The Parikh mapping ψ:X^∗−→N^mis defined by

ψ(w) = (#_x1(w), . . . ,#_xm(w)), w∈X^∗.

Thelengthof a wordw∈X^∗ is denoted by |w|. The length of the empty wordε equals zero. The first letter of a nonempty wordw∈X^∗ is denoted by first(w).

Ifu, v∈X^∗ are words and there is a wordw∈X^∗ such thatuw =v, we say thatuis aprefixofv. Ifu∈X^∗andpis a positive integer, we denote by pref_p(u) the prefix ofuhaving lengthp. If|u|< p, it is understood that pref_p(u) =u. This notation is extended in a natural way for pairs of words. Hence, if (u, v)∈X^∗×X^∗ we denote

pref_p(u, v) = (pref_p(u),pref_p(v)).

Two wordsu, v∈X^∗ are calledcomparable (with respect to the prefix order) ifu is a prefix ofv or vice versa. The longest common prefix ofuandvis denoted by u∧v. Further, we denote

u∗v=

(ε, ε) ifuandv are comparable,

((u∧v)⁻¹u,(u∧v)⁻¹v) otherwise.

Next, let (a_n)_n≥0 be a sequence and letq be a positive integer. We say thatqis aperiodof (a_n)_n≥0if there is an integern₀ such that

a_n+q =a_n whenevern≥n₀. A sequence is calledultimately periodicif it has a period.

The following theorem gives a basic result concerning periodicities in D0L se- quences (see [2, 3]).

Theorem 2.1. Suppose X is an alphabet, f : X^∗ −→ X^∗ is a morphism and u∈X^∗. There is a positive integerq such that for any positive integer p,q is a period of the sequence

(pref_p(fⁿ(u)))_n≥0.

In this paper we study the iteration of a given morphism on two different words.

Iff :X^∗ −→X^∗ is a morphism, u, v∈X^∗ are words andpis a positive integer, we consider the sequence (pref_p(fⁿ(u)∗fⁿ(v)))_n≥0. It was shown in [4] that these sequences are ultimately periodic. (In [4] it is assumed thatψ(u) =ψ(v).) In this paper we will study these sequences in more detail. In particular, we will prove the following result.

Theorem 2.2. Suppose X is an alphabet and f : X^∗ −→ X^∗ is a morphism.

There is a positive integer q such that whenever u, v∈ X^∗ are words and p is a positive integer, thenqis a period of the sequence

(pref_p(fⁿ(u)∗fⁿ(v)))_n≥0. (1) The proof of Theorem 2.2 is given in the following section. The proof uses two lemmas from [4], but otherwise we do not assume familiarity with [4].

Let nowf :X^∗−→X^∗ be a morphism and letu, v∈X^∗ be words. For every nonnegative integern, define z_n =fⁿ(u)∧fⁿ(v) and let u_n, v_n ∈ X^∗ be words such thatfⁿ(u) =z_nu_nandfⁿ(v) =z_nv_n. Then there is a positive integerqsuch that for any positive integerp,qis a period of the sequences

(pref_p(u_n))_n≥0 and (pref_p(v_n))_n≥0.

To deduce the existence ofq from Theorem 2.2, choose a new letter $, extendf byf($) = $ and consider the wordsu$ andv$.

3. Proofs

We will prove Theorem 2.2 in two steps. The first subsection contains the main part of the proof. Without loss of generality we assume thatX contains at least two letters.

3.1. Proof of Theorem 2.2 – a special case

Iff :X^∗−→X^∗is a morphism, a lettera∈X is calledbounded(with respect tof) if the length sequence (|fⁿ(a)|)n≥0is bounded above by a constant. Ifa∈X is not bounded, thena is calledgrowing. A word w∈X^∗ is calledboundedif no letter ofwis growing.

In this subsection we assume thatf :X^∗−→X^∗ is a morphism which has the following additional properties:

(i) There is a positive integerK such that ifx, y ∈X^∗ are nonempty words and first(x)= first(y) then|f(x)∧f(y)| ≤K.

(ii) If a∈X is a bounded letter, thenf(a) =a.

(iii) Ifa∈X is a growing letter, then|f(a)| ≥2.

In this subsection we prove Theorem 2.2 for morphisms which share properties (i)–(iii). To proceed we fix such a morphismf :X^∗−→X^∗.

Leta ∈X be a growing letter and let a_n be the first growing letter of fⁿ(a) forn≥0. Then the sequence (a_n)_n≥0is ultimately periodic. Hence we can find a positive integern₀such that ifa, b∈X are growing letters anda_n=b_n for some n≥0, thena_n0=b_n0. Denote

M = max

x∈X|fⁿ⁰(x)|.

Lemma 3.1. If a, b∈X are growing letters andz ∈X^∗ is a bounded word with

|z| ≥M thenfⁿ(a)∧fⁿ(zb)is a bounded word for all n≥0.

For the proof of Lemma 3.1 we refer to [4].

Next we define a family of mappings needed for the proof of Theorem 2.2. Let M be as above. Denote N = K(M +K) where K is as in (i). First, define the mappingτ₁ :X^∗ −→X^∗ as follows. If w∈X^∗ containsM +K consecutive bounded letters, thenτ₁(w) is the shortest prefix ofwending withM+Kbounded letters. Otherwise,τ₁(w) =w. Next, the mappingτ₂:X^∗−→X^∗ is defined by

τ₂(w) = pref_N(w), w∈X^∗.

Then, define τ : X^∗ −→ X^∗ as follows. If w ∈ X^∗ has a prefix of length M consisting of bounded letters,τ(w) = pref_M(w). Otherwiseτ(w) =τ₂τ₁(w).

Next, we consider pairs of words and extend the mapping τ in a natural way by setting

τ(u, v) = (τ(u), τ(v)), (u, v)∈X^∗×X^∗.

Finally, if (u, v)∈X^∗×X^∗, define the mappingρ_u,v:N−→X^∗×X^∗by ρ_u,v(n) =τ(fⁿ(u)∗fⁿ(v)), n≥0.

Denote

Λ_u,v={ρu,v(n)|n∈N}.

Define the set R ⊆ X^∗ ×X^∗ as follows. Let (u, v) ∈ X^∗ ×X^∗ and denote (u_n, v_n) = fⁿ(u)∗fⁿ(v) for n≥0. Then (u, v)∈ R if and only if ψ(u) = ψ(v) and for alln≥0 the wordu_n (resp. v_n) contains a growing letter and|un| ≥N (resp. |vn| ≥N).

Let (u, v)∈ R and let n≥0. Write ρ_u,v(n) = (w₁, w₂). Then the wordsw_i, i= 1,2, satisfy the following conditions:

(1) M ≤ |wi| ≤N;

(2) if|wi|=M, thenw_i is a bounded word;

(3) ifM <|w_i|< N, thenw_i ends withM+K bounded letters;

(4) if |w_i| =N, eitherw_i does not have a bounded factor of length M +K or the suffix ofw_i of length M +K is the only bounded factor of length M+K ofw_i.

A pair (u, v)∈X^∗×X^∗ is calledspecialif|u|=M or|v|=M.

Lemma 3.2. Suppose (u, v)∈ R and(x, y)∈Λ_u,v. Ifw₁, w₂∈X^∗ and(x, y)is not special, then

τ(f(x)∗f(y)) =τ(f(x)w₁∗f(y)w₂).

For the proof of Lemma 3.2 we again refer to [4].

Lemma 3.3. Suppose(u, v)∈ Rand writeρ_u,v(n) = (x_n, y_n)for n≥0. If none of the pairs(x_n, y_n),n≥0, is special, then

ρ_u,v(i+t) =τ(f^t(x_i)∗f^t(y_i)) (2)

for alli≥0 andt≥0.

Proof. If (w₁, w₂)∈X^∗×X^∗, we haveτ(w₁, w₂) =τ(τ(w₁, w₂)). Hence ρ_u,v(i) =τ(fⁱ(u)∗fⁱ(v)) =τ(τ(fⁱ(u)∗fⁱ(v))) =τ(x_i, y_i) for alli≥0. Consequently, (2) holds for alli≥0 ift= 0.

Suppose then that (2) holds for a fixed pairt≥0,i≥0. In other words (x_i+t, y_i+t) =τ(f^t(x_i)∗f^t(y_i)).

Hence there exist wordsα, β₁, β₂∈X^∗such that

f^t(x_i) =αx_i+tβ₁, f^t(y_i) =αy_i+tβ₂.

Becauseτ(f^i+t(u)∗f^i+t(v)) = (x_i+t, y_i+t), there are wordsγ, δ₁, δ₂∈X^∗such that f^i+t(u) =γx_i+tδ₁, f^i+t(v) =γy_i+tδ₂.

Now we get

ρ_u,v(i+t+ 1) = τ(f^i+t+1(u)∗f^i+t+1(v))

= τ(f(γx_i+tδ₁)∗f(γy_i+tδ₂))

= τ(f(x_i+t)∗f(y_i+t))

= τ(f(α)f(x_i+t)f(β₁)∗f(α)f(y_i+t)f(β₂))

= τ(f^t+1(x_i)∗f^t+1(y_i)).

Here the third and fourth equations follow by Lemma 3.2. Consequently, (2) holds

for alli≥0 andt≥0.

By Theorem 2.1 there is a positive integer q₁ such that q₁ is a period of the sequence

(pref_p(f^nt(a)))_n≥0

for allp≥1,a∈X andt <card(X)^2N+2. Fix such an integerq₁. Thenq₁is also a period of

(pref_p(f^nt(w)))_n≥0 for allp≥1,w∈X^∗ andt <card(X)^2N+2.

Lemma 3.4. Let (u, v) ∈ R and write ρ_u,v(n) = (x_n, y_n) for n ≥ 0. Suppose none of the pairs (x_n, y_n), n ≥0, is special. Then q₁ is a period of (1) for any positive integerp.

Proof. Because the set Λ_u,vhas less than card(X)^2N+2elements, there are positive integersiandt such thatt <card(X)^2N+2 and

ρ_u,v(i) =ρ_u,v(i+t).

By Lemma 3.3 we have

(x_i, y_i) = (x_i+t, y_i+t) =τ(f^t(x_i)∗f^t(y_i)).

Hence there exist wordsα, γ₁, γ₂∈X^∗ such that

f^t(x_i) =αx_iγ₁, f^t(y_i) =αy_iγ₂. Letw₁, w₂∈X^∗ be words such that

fⁱ(u)∗fⁱ(v) = (x_iw₁, y_iw₂).

Then

f^i+nt(u)∗f^i+nt(v) =f^nt(x_iw₁)∗f^nt(y_iw₂)

= (x_iγ₁f^t(γ₁). . . f^(n−1)t(γ₁)f^nt(w₁), y_iγ₂f^t(γ₂). . . f^(n−1)t(γ₂)f^nt(w₂)) forn≥1. This implies thatq₁is a period of

(pref_p(f^i+nt(u)∗f^i+nt(v)))_n≥0

for anyp≥1. Because the morphism f has property (i), it follows thatq₁ is also

a period of (1) for anyp≥1.

Next, define

q₂ = max{1,|β| | there are a growing letterc∈X, a positive integer s≤card(X)² and a wordγ∈X^∗ such thatf^s(c) =βcγ}

and define

q₃=q₂!q₁.

Lemma 3.5. Let(u, v)∈ Rand writeρ_u,v(n) = (x_n, y_n)forn≥0. If there is an integer n such that (x_n, y_n) is special, then q₃ is a period of (1) for any positive integerp.

Proof. Suppose that k is a nonnegative integer such that ρ_u,v(k) = (x_k, y_k) is special. Without loss of generality assume that|x_k|=M. Hencex_k is a bounded word. Let

f^k(u) =αβ₁aγ₁, f^k(v) =αβ₂bγ₂

whereβ₁, β₂∈X^∗ are bounded words,a, b∈X are growing letters andα, γ₁, γ₂∈ X^∗. Furthermore,x_k is a prefix ofβ₁ andy_k is a prefix ofβ₂bγ₂.

Ifβ₂ is not empty, then

f^k+n(u)∗f^k+n(v) =fⁿ(β₁aγ₁)∗fⁿ(β₂bγ₂) = (β₁fⁿ(aγ₁), β₂fⁿ(bγ₂)) forn≥0. Henceq₁ andq₃ are periods of (1).

To proceed assume thatβ₂ is the empty word. Then Lemma 3.1 implies that fⁿ(β₁a)∧fⁿ(b) is a bounded word for alln≥0. Now, there exist integersk₁≥0, k₂ ≥ 1, growing letters c, d ∈ X, bounded words β₃, β₄, β₅, β₆ ∈ X^∗ and words γ₃, γ₄, γ₅, γ₆∈X^∗ such thatk₂≤card(X)² and

f^k1(a) =β₃cγ₃, f^k1(b) =β₄dγ₄, f^k2(c) =β₅cγ₅, f^k2(d) =β₆dγ₆. (Here we may havec=d.) Hence

f^k+k1+nk2(u)∗f^k+k1+nk2(v) =

β₁β₃β₅ⁿcγ₅f^k2(γ₅). . . f^nk2−k2(γ₅)f^nk2(γ₃)f^k1+nk2(γ₁)

∗β₄β₆ⁿdγ₆f^k2(γ₆). . . f^nk2−k2(γ₆)f^nk2(γ₄)f^k1+nk2(γ₂)

for n ≥ 1. Furthermore, the words β₁β₃β₅ⁿc and β₄βⁿ₆d are not comparable.

Consequently the sequence

(pref_p(f^k+k1+nk2(u)∗f^k+k1+nk2(v)))_n≥0 (3) has periodq₃. In fact, if there is an integernsuch thatβ₁β₃βⁿ₅ andβ₄β₆ⁿ are not comparable, (3) has periodq₃. Otherwise, without loss of generality assume that β₁β₃βⁿ₅ is a prefix ofβ₄β₆ⁿ for all large n, say, ifn≥n₀. Then the sequence

(pref_p((β₁β₃β₅ⁿ)⁻¹β₄β₆ⁿ))_n≥n0 has periodq₂! for any p≥1. Because the sequences

(pref_p(cγ₅f^k2(γ₅). . . f^nk2−k2(γ₅)f^nk2(γ₃)f^k1+nk2(γ₁)))_n≥0 and

(pref_p(dγ₆f^k2(γ₆). . . f^nk2−k2(γ₆)f^nk2(γ₄)f^k1+nk2(γ₂)))_n≥0

have period q₁ for any p≥ 1, it follows that (3) has period q₃. Because f has property (i) it again follows that (1) has periodq₃. We have now shown that iff satisfies conditions (i)–(iii) thenq₃is a period of (1) for allp≥1 and for all (u, v)∈ R.

To proceed, let $ be a new letter. DefineX₁=X∪ {$}and let f₁:X₁^∗−→X₁^∗ be the extension off defined byf₁($) = $. Thenf₁satisfies (i)–(iii) and we know that there are positive integersqandN₁ such thatqis a period of

(pref_p(f₁ⁿ(u₁)∗f₁ⁿ(v₁)))_n≥0 (4) for allp≥1 and for allu₁, v₁∈X₁^∗such thatψ(u₁) =ψ(v₁) and each component off₁ⁿ(u₁)∗f₁ⁿ(v₁) contains a growing letter and has length at leastN₁for alln≥0.

Let now u, v ∈ X^∗. If there is an integer n such that fⁿ(u) and fⁿ(v) are comparable, then (1) has period one for allp≥1. Also, if neitherunorvcontains a growing letter, (1) has period one for allp≥1. Assume that fⁿ(u) and fⁿ(v)

are incomparable for alln≥0 and assume thatuor v contains a growing letter.

Now, define

u₁=u$v(uv)^N1, v₁=v$u(vu)^N1.

Thenqis a period of (4) for allp≥1. Consequently (1) has periodqfor allp≥1.

This concludes the proof of Theorem 2.2 for a morphismf which satisfies con- ditions (i)–(iii).

3.2. Proof of Theorem 2.2 – the general case

We assume that the reader is familiar with elementary morphisms (see [1, 5]).

In particular, recall that an elementary morphism permutes the set of bounded letters.

Letf :X^∗ −→X^∗ be a morphism. Then there exist a positive integerj₁, an alphabetY and morphismsg₁:X^∗−→Y^∗,g₂:Y^∗−→X^∗such that

f^j1 =g₂g₁

and the morphismsg₂ andg₁g₂ are elementary (see Th. III 2.2 in [5]). Consider the morphismg₁g₂:Y^∗−→Y^∗. There exists a positive integerj₂ such that

(g₁g₂)^j2(y) =y ify∈Y is a bounded letter with respect tog₁g₂ and

|(g1g₂)^j2(y)| ≥2

ify∈Y is a growing letter with respect tog₁g₂. Defineg= (g₁g₂)^j2 andj =j₁j₂. Then

f^nj+j1 =g₂gⁿg₁ forn≥0.

Because g satisfies conditions (i)-(iii) of the previous subsection, there is a positive integerq such thatqis a period of

(pref_p(gⁿg₁fⁱ(u)∗gⁿg₁fⁱ(v)))_n≥0

for allp≥1,i= 0,1, . . . , j−1 andu, v∈X^∗. Becauseg₂is elementary,qis also a period of

(pref_p(g₂gⁿg₁fⁱ(u)∗g₂gⁿg₁fⁱ(v)))_n≥0

for allp≥1,i= 0,1, . . . , j−1 andu, v∈X^∗. In other words,qis also a period of (pref_p(f^nj+j1+i(u)∗f^nj+j1+i(v)))_n≥0

for allp≥1,i= 0,1, . . . , j−1 andu, v∈X^∗. Hencejq is a period of (1) for all p≥1 andu, v∈X^∗.

This concludes the proof of Theorem 2.2 in the general case.

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(1981).

Communicated by C. Choffrut.

Received August 30, 2005. Accepted September 13, 2006.

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