Periodic $p$-adic Gibbs measures of $q$-states Potts model on Cayley tree: The chaos implies the vastness of $p$-adic Gibbs measures

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Periodic p-adic Gibbs measures of q-states Potts model on Cayley tree: The chaos implies the vastness of p-adic

Gibbs measures

Mohd Ali Khameini Ahmad, Lingmin Liao, Mansoor Saburov

To cite this version:

Mohd Ali Khameini Ahmad, Lingmin Liao, Mansoor Saburov. Periodic

�hal-01572080�

PERIODICp-ADIC GIBBS MEASURES OFq-STATES POTTS MODEL ON CAYLEY TREE: THE CHAOS IMPLIES THE VASTNESS OFp-ADIC GIBBS

MEASURES

MOHD ALI KHAMEINI AHMAD, LINGMIN LIAO, AND MANSOOR SABUROV

ABSTRACT. We study the set ofp-adic Gibbs measures of theq-states Potts model on the Cayley tree of order three. We prove the vastness of the periodicp-adic Gibbs measures for such model by showing the chaotic behavior of the correspondence Potts–Bethe mapping overQpforp ≡ 1 (mod 3). In fact, for0 < |θ−1|p < |q|²_p < 1, there exists a subsystem that isometrically conjugate to the full shift on three symbols. Meanwhile, for 0<|q|²_p≤ |θ−1|p<|q|p<1, there exists a subsystem that isometrically conjugate to a subshift of finite type onrsymbols wherer ≥ 4. However, these subshifts onr symbols are all topologically conjugate to the full shift on three symbols. Thep-adic Gibbs measures of the same model for the casesp= 2,3and the corresponding Potts–

Bethe mapping are also discussed.

Furthermore, for0<|θ−1|p<|q|p<1,we remark that the Potts–Bethe mapping is not chaotic whenp= 2, p= 3andp≡2 (mod 3)and we could not conclude the vastness of the periodicp-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case0<|q|_p≤ |θ−1|_p<1for allp.

1. INTRODUCTION

Ap-adic valued theory of probability (a non–Kolmogorov model in which probabilities take values in the fieldQpofp-adic numbers) was proposed in a series of papers [17–22, 24–26] in order to resolve the problem of the statistical interpretation ofp-adic valued wave functions in non-Archimedean quantum physics [7,57,58]. On the other hand, in order to formalize the measure-theoretic approach forp-adic probability theory, in the papers [11–14] the authors developed severalp-adic probability logics which are sound, complete and decidable extensions of the classical propositional logic. This general theory ofp- adic probability was applicable to the problem of the probability interpretation of quantum theories with non-Archimedean valued wave functions [5,23,28]. The applications of p-adic functional and harmonic analysis have also shown up in theoretical physics and quantum mechanics [1–6].

Gibbs measure which plays the central role in statistical mechanics is a branch of prob- ability theory that takes its origin from Boltzmann and Gibbs who introduced a statistical approach to thermodynamics to deduce collective macroscopic behaviors from individual microscopic information. Gibbs measure associated with the Hamiltonian of a physical system (a model) generalizes the notion of a canonical ensemble (see [46]). In the classical case (where the mathematical model was prescribed over the real numbers), the physical

2010Mathematics Subject Classification. Primary 46S10, 82B26; Secondary 60K35.

Key words and phrases.p-adic number,p-adic Potts model,p-adic Gibbs measure.

The first author (M.A.K.A) is grateful to Embassy of France in Malaysia and Labex B´ezout for the financial support to pursue his Ph.D at LAMA, Universit´e Paris–Est Cr´eteil, France. The third author (M.S.) thanks the Junior Associate Scheme, Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, for the invitation and hospitality. He was partially supported by the MOHE grant FRGS14-141-0382.

1

phenomenon of phase transition should be reflected in a mathematical model by the non- uniqueness of the Gibbs measures or the size of the set of Gibbs measures for a prescribed model. Due to the convex structure of the set of Gibbs measures over the real numbers field, in order to describe the size of the set of Gibbs measures, it was sufficient to study the number of its extreme elements. Hence, in the classical case, to predict a phase transi- tion, the main attention was paid to finding all possible extreme Gibbs measures. We refer to the book [10] for more details.

The rigorous mathematical foundation of the theory of Gibbs measures on Cayley trees was presented in the books [46,47]. Thep-adic counterpart of the theory of Gibbs measures on Cayley trees has also been initiated [9,42,43]. The existence ofp-adic Gibbs measures as well as the phase transition for some lattice models were established in [34,35,41,45].

Recently, in [44,51,52], all translation-invariantp-adic Gibbs measures of thep-adic Potts model on the Cayley tree of order two and three were described by studying allocation of roots of quadratic and cubic equations over some domains of theQp.

In thep-adic case, due to lack of convex structure of the set ofp-adic (quasi) Gibbs measures, it is quite difficult to constitute a phase transition with some features of the set ofp-adic (quasi) Gibbs measures. Moreover, unlike the real case [30], the set ofp- adic Gibbs measures of lattice models on the Cayley tree has a complex structure in a sense that it is strongly tied up with a Diophantine problem (i.e. to find all solutions of a system of polynomial equations or to give a bound for the number of solutions) over theQp. In general, the same Diophantine problem may have different solutions from the field ofp-adic numbers to the field of real numbers because of the different topological structures. On the other hand, the rise of the order of the Cayley tree makes difficult to study the corresponding Diophantine problem over theQ^p. In this aspect, the question arises as to whether a root of a polynomial equation belongs to the domains Z^∗p, Zp \ Z^∗p, Zp, Qp \Z^∗p, Qp \ Zp\Z^∗p

, Qp\Zp, Qp or not. Recently, this problem was fully studied for monomial equations [40], quadratic equations [51,53], depressed cubic equations for primesp > 3in [38,39,54]. Meanwhile, in [48–50], the depressed cubic equations for primesp= 2,3are discussed.

This paper can be considered as a continuation of the papers [44,51,52]. We are going to study the set ofp-adic Gibbs measures of theq-states Potts model on the Cayley tree of order three. For such study, the conditions |θ−1|p < 1 and|q|p ≤ 1 are natural (see Subsections 2.1 and 2.5). We remark that the case0 < |θ−1|p < |q|p = 1has recently been considered in [37] and the regularity of the the dynamics of the Potts–Bethe mapping for primesp≥5, p≡2 (mod 3)with the condition0 <|θ−1|p <|q|p <1 has been studied in [55]. In this paper, we prove the vastness of (periodic)p-adic Gibbs measures by showing the chaotic behavior of the Potts–Bethe mapping overQpfor primes p≡1 (mod 3)with the condition0<|θ−1|_p <|q|_p <1. To complete our work, in a forthcoming paper with the same title, we will study the Potts–Bethe mapping overQ^pfor the case0<|q|p≤ |θ−1|p<1.

We organize our paper as follows. In Section 2, we provide the preliminaries of the paper. In Section 3, we discuss thep-adic Gibbs measure of the Potts model associated with anm-boundary function. Such a measure is associated to the cycle of lengthmof the corresponding Potts–Bethe mapping. In Section 4, the dynamical behavior of the Potts–

Bethe mapping is studied forp ≡ 1 (mod 3). We find a subsystem that isometrically conjugate to a shift dynamics i.e., the full shift on three symbols or a subshift of finite type onr≥ 4symbols. All these subsystems are proven to be topologically conjugate to the full shift over three symbols. Finally, in Section 5, we discuss the dynamics of Potts–Bethe

mapping overQ2andQ3. We prove that except for a fixed point and the inverse images of the singular point, all points converge to an attracting fixed point.

2. PRELIMINARIES

2.1. p-adic numbers. For a fixed primep, the fieldQpofp-adic numbers is a completion of the setQof rational numbers with respect to the non-Archimedean norm| · |_p:Q→R given by

|x|p=

p^−k, x6= 0, 0, x= 0,

wherex = p^{k m}_n withk, m ∈ Z, n ∈ N,p6 | mandp6 | n. The numberkis called the p-orderofxand is denoted byordp(x).Note thatordp(x) =−log_p|x|p=k.Moreover, this norm is non-Archimedean because it satisfies the strong triangle inequality:|x+y|p≤ max (|x|p,|y|p).The metric onQpinduced by this norm,d(x, y) =|x−y|p,satisfies the ultrametric property: for allx, y, z∈Qp, d(x, y)≤max (d(x, z), d(z, y)).

We respectively denote the set of allp-adic integersandp-adic unitsofQp byZp = {x∈Qp:|x|p≤1}andZ^∗p ={x∈Qp:|x|p= 1}.Anyp-adic numberx∈Qpcan be uniquely written in the canonical formx=p^ordp(x) x₀+x₁·p+x₂·p²+. . .

where x0∈ {1,2, . . . , p−1}andxi ∈ {0,1,2, . . . , p−1}fori∈N:={1,2, . . .}. Therefore, x=_|x|^x∗

p, for somex^∗∈Z^∗p.

We denote byBr(a) ={x∈Qp:|x−a|p < r}andSr(a) ={x∈Qp:|x−a|p=r}

the open ball and the sphere with centera∈Qpand radiusr > 0respectively. Note that, by non-Archimedean property, an open ball is also closed. Thep-adic logarithm function lnp(·) :B1(1)→B1(0)is defined by

lnp(x) = lnp(1 + (x−1)) =

∞

X

n=1

(−1)ⁿ⁺¹(x−1)ⁿ

n .

Thep-adic exponential functionexp_p(·) :Bp^−1/(p−1)(0)→B1(1)is defined by exp_p(x) =

∞

X

n=0

xⁿ n!.

Lemma 2.1(see [15,57]). Letx∈Bp^−1/(p−1)(0).Then we have

|exp_p(x)|p= 1, |exp_p(x)−1|p=|x|p<1, |ln_p(1 +x)|p=|x|p< p^−1/(p−1), lnp(exp_p(x)) =x, exp_p(lnp(1 +x)) = 1 +x.

LetEp={x∈Qp:|x−1|p< p^−1/(p−1)}.Obviously, ifp≥3thenEp={x∈Z^∗p :

|x−1|p<1}=B1(1).Due to Lemma 2.1, we have the following result.

Lemma 2.2(see [15,57]). The setEphas the following properties:

(i) Epis a group under multiplication;

(ii) one has|a−b|_p<1for alla, b∈ E_p; (iii) ifa, b∈ Ep, then one has|a+b|p=

1

2, if p= 2 1, if p6= 2;

(iv) ifa∈ Ep, then there existsh∈Bp^−1/(p−1)(0)such thata= exp_p(h).

Note that a compact-open set in Qp is a finite union of balls. The following is the definition of local scaling on a compact-open setX ⊂Qp.

Definition 2.3(Definition 4.1 of [29]). LetX =

n

S

i=1

Br_i(ai)⊂Qpbe a compact-open set.

We say that a mappingf :X →Xis locally scaling forr_i∈ {pⁿ, n∈Z}if there exists a functionS:X →R≥1such that for anyx, y∈Bri(a_i),S(x) =S(y) =S(a_i)and

|f(x)−f(y)|_p=S(a_i)|x−y|_p.

The functionSis called a scaling function. The following lemma will be useful.

Lemma 2.4(Theorem 5.1 of [29]). LetX =

n

S

i=1

Br_i(ai)⊂ Qp be a compact-open and f :X →X be a scaling forr_i∈ {pⁿ, n∈Z}.LetS:X →R≥1be the correspondence scaling function. Then for allr⁰≤ri, the restricted map

f :B^r0(ai)→Br⁰S(ai)(f(ai)) is a bijection.

The following result is a direct consequence of Lemma 2.4.

Corollary 2.5. Keep the same assumption as in Lemma 2.4, we have for allr⁰ ≤ri, the restricted map

f :Sr⁰(a_i)→Sr⁰S(a_i)(f(a_i)) is a bijection.

Proof. RecallSr⁰(ai) ={x∈Qp:|x−ai|p=r⁰}.ThenSr⁰(ai) =Bpr⁰(ai)\Br⁰(ai)is a difference of consecutive balls. By Lemma 2.4,f are bijections from balls to balls. Thus

when restricted to the spheresf is also bijective.

2.2. p-adic subshift. In this paper, we will apply a theorem in [32]. We take the same notation as in [32]. Letf :X→Qpbe a mapping from a compact open setX⊂Qpinto Qp. We assume that (i)f⁻¹(X) ⊂X and (ii)X =S

j∈IBp^−τ(a_j)can be written as a finite disjoint union of balls of centersa_j and of the same radiusp^−τ, τ ∈Zsuch that for eachj∈Ithere is an integerτ_j∈Zsuch that for anyx, y∈Br(a_j)

|f(x)−f(y)|_p=p^τj|x−y|_p. (2.1) For such a mapf, we define its Julia set by

Jf =

∞

\

n=0

f⁻ⁿ(X). (2.2)

It is clear thatf⁻¹(Jf) =Jfandf(Jf)⊂Jf.

The triple(X, Jf, f)is calledap-adic weak repellerif allτjin (2.1) are nonnegative, but at least one is positive. It is calledap-adic repellerif allτjin (2.1) are positive. For anyi∈I,we let

Ii:={j ∈I:Br(aj)∩f(Br(ai))6=∅}={j∈I:Br(aj)⊂f(Br(ai))}

(the second equality holds because of the expansiveness and the ultrametric property). We then define a so-calledincidence matrixA= (a_ij)_I×I as follows

a_ij=

(1 if j∈I_i, 0 if j6∈Ii.

IfAis irreducible, we say that(X, Jf, f)istransitive. Recall thatAis irreducible if for any pair(i, j)∈I×Ithere is a positive integermsuch thata^(m)_ij >0, wherea^(m)_ij is the

entry of the matrixA^m. For an index setIand an irreducible incidence matrixAgiven above, we denote by

Σ_A=

(x_k)_k≥0:x_k∈I, A_xk,xk+1 = 1, k≥0

the subshift space. We equipΣAwith a metricdf depending on the dynamics which is defined as follows. Fori, j ∈ I, i 6= j letκ(i, j)be the integer such that|ai−aj|p = p^−κ(i,j). It is clear thatκ(i, j)< τ. By the ultrametric inequality, we have

|x−y|p=|ai−aj|p, i6=j, ∀x∈Br(ai), ∀y∈Br(aj).

Forx= (x0, x1,· · · , xn,· · ·)∈ΣAandy= (y0, y1,· · · , yn,· · ·)∈ΣA, we define df(x, y) =

(p^{−τx0−τx1−···−τxn−1−κ(xn,yn)} if n6= 0,

p^−κ(x0,y0) if n= 0

wheren = n(x, y) = min{k ≥ 0 : xk 6= yk}. It is clear that df defines the same topology as the classical metric defined byd(x, y) = p^−n(x,y). Letσbe the (left) shift transformation onΣA. Then(ΣA, σ, df)becomes a dynamical system.

Theorem 2.6( [32]). Let(X, Jf, f)be a transitivep-adic weak repeller with incidence matrixA. Then the dynamics(Jf, f,| · |p)is isometrically conjugate to the shift dynamics (ΣA, σ, df).

The shift dynamics(Σ_A, σ)is called a subshift of finite type determined by matrixA.

WhenAis an×nmatrix whose coefficients are all equal to1,(Σ_A, σ)is called a full shift over an alphabet ofnsymbols, and is usually denoted by(Σ_n, σ).

2.3. p-adic measure. Let(X,B)be a measurable space, whereBis an algebra of subsets X. A functionµ:B →Q^pis said to be ap-adic measureif for anyA1, . . . , An∈ Bsuch thatAi∩Aj =∅(i6=j) the following equality holds

µ ⁿ

[

j=1

A_j

=

n

X

j=1

µ(A_j).

Ap-adic measure is called aprobability measureifµ(X) = 1. Ap-adic probability measureµis called boundedifsup{|µ(A)|p : A ∈ B} < ∞. We are interested in this important class ofp-adic measures in which the boundedness condition itself provides a fruitful integration theory. The reader may refer to [25,26,28] for more detailed informa- tion aboutp-adic measures.

2.4. Cayley tree. LetΓ^k₊= (V, L)be a semi-infinite Cayley tree of orderk≥1with the rootx⁰ (each vertex has exactlyk+ 1edges except for the rootx⁰which haskedges).

LetV be a set of vertices andLbe a set of edges. The verticesxandyare callednearest neighborsif there exists an edgel ∈ L connecting them. This edge is also denoted by l=hx, yi.A collection of the pairshx, x1i,· · ·,hx_d−1, yiis calleda pathbetween vertices xandy. The distanced(x, y)betweenx, y ∈ V on the Cayley tree is the length of the shortest path betweenxandy. Let

W_n=

x∈V :d(x, x⁰) =n , V_n=

n

[

m=1

W_m, L_n={< x, y >∈L:x, y∈V_n}. The set of direct successors ofxis defined by

∀x∈W_n, S(x) ={y∈W_n+1:d(x, y) = 1}.

We now introduce a coordinate structure inV. Every vertexx6=x⁰has the coordinate (i1,· · · , in)whereim∈ {1, . . . , k}, 1≤m≤nand the vertexx⁰has the coordinate(∅).

More precisely, the symbol(∅)constitutes level0 and the coordinates(i1,· · ·, in)form levelnofV (from the rootx⁰). In this case, for anyx∈V, x= (i₁,· · · , i_n)we have

S(x) ={(x, i) : 1≤i≤k},

where(x, i)means(i1,· · ·, in, i). Let us define a binary operation◦ : V ×V → V as follows: for any two elementsx= (i1,· · · , in)andy= (j1,· · · , jm)we define

x◦y= (i1,· · ·, in)◦(j1,· · · , jm) = (i1,· · ·, in, j1,· · ·, jm) and

y◦x= (j1,· · ·, jm)◦(i1,· · ·, in) = (j1,· · ·, jm, i1,· · ·, in).

Then(V,◦)is a noncommutative semigroup with the unitx⁰= (∅). Now, we can define a translationτ_g :V →V forg ∈V as followsτ_g(x) =g◦x.Consequently, by means of τg,we define a translationτ˜g:L→Lasτ˜g(hx, yi) =hτg(x), τg(y)i.

LetG⊂V be a sub-semigroup ofV andh:L→Qpbe a function. We say thathis G-periodicifh(˜τg(l)) =h(l)for allg∈Gandl∈L. AnyV-periodic function is called translation-invariant.

2.5. p-adicq-states Potts model. LetΦ ={1,2,· · · , q}be a finite set. A configuration (resp. a finite volume configuration, a boundary configuration) is a functionσ: V →Φ (resp. σn : Vn → Φ, σ⁽ⁿ⁾ : Wn → Φ). We denote byΩ(resp. ΩV_n,ΩW_n) the set of all configurations (resp. all finite volume configurations, all boundary configurations). For given configurationsσn−1∈ΩVn−1 andσ⁽ⁿ⁾∈ΩW_n, we define their concatenation to be a finite volume configurationσ_n−1∨σ⁽ⁿ⁾∈Ω_Vnsuch that

σn−1∨σ⁽ⁿ⁾ (v) =

(σ_n−1(v) if v∈V_n−1 σ⁽ⁿ⁾(v) if v∈Wn

.

The Hamiltonian ofq-states Potts modelwith the spin value setΦ ={1,2,· · ·q}on the finite volume configuration is defined as follows

Hn(σn) =J X

hx,yi∈Ln

δ_σn(x)σn(y),

for allσn ∈ΩV_nandn∈NwhereJ ∈Bp^−1/(p−1)(0)is a coupling constant,hx, yistands for nearest neighbor vertices, andδis Kronecker’s symbol.

2.6. p-adic Gibbs measure. Let us present a construction of ap-adic Gibbs measure of thep-adicq-states Potts model. We define ap-adic measureµ⁽ⁿ⁾_˜

h : Ω_Vn→Qpassociated with a boundary functionh˜:V →Q^qp,h(x) =˜

˜h⁽¹⁾x , . . . ,˜h^(q)x

, x∈V as follows µ⁽ⁿ⁾_˜

h (σ_n) = 1 Z_˜⁽ⁿ⁾

h

exp_p (

H_n(σ_n) + X

x∈Wn

˜h^(σ_x^n(x)) )

(2.3) for allσn ∈ΩV_n, n∈Nwhereexp_p(·) :Bp^−1/(p−1)(0)→B1(1)is thep-adic exponential function andZ_˜⁽ⁿ⁾

h isthe partition functiondefined by Z⁽ⁿ⁾_˜

h = X

σ_n∈Ω_Vn

exp_p (

Hn(σn) + X

x∈W_n

˜h^(σ_x^n(x)) )

for alln∈N.Throughout this paper, we considerh(x)˜ ∈ Bp^−1/(p−1)(0)q

for anyx∈V.

Thep-adic measures (2.3) are calledcompatibleif one has that X

σ⁽ⁿ⁾∈Ω_Wn

µ⁽ⁿ⁾_˜

h (σ_n−1∨σ⁽ⁿ⁾) =µ⁽ⁿ⁻¹⁾_˜

h (σ_n−1) (2.4)

for allσ_n−1∈Ω_Vn−1andn∈N.

Due to Kolmogorov’s extension theorem of thep-adic measures (2.3) (see [9,27,31]), there exists a uniquep-adic measureµ˜h: Ω→Qpsuch that

µ_˜h({σ|Vn=σ_n}) =µ⁽ⁿ⁾_˜

h (σ_n), ∀σ_n ∈Ω_Vn, ∀n∈N.

This uniquely extended measureµ˜h : Ω → Qp is calledap-adic Gibbs measure. The following theorem describes the condition on the boundary functionh˜:V →Q^qpin which the compatibility condition (2.4) is satisfied.

Theorem 2.7( [42,43]). Leth˜:V →Q^qp,˜h(x) =

h˜⁽¹⁾x ,· · ·,˜h^(q)x

be a given boundary function andh:V →Q^q−1p ,h(x) =

h⁽¹⁾x ,· · · , h^(q−1)x

be a function defined ash⁽ⁱ⁾x =

˜h⁽ⁱ⁾x −˜h^(q)x for alli= 1, q−1.Then thep-adic probability distributionsn µ⁽ⁿ⁾_˜

h

o

n∈Nare compatible if and only if

h(x) = X

y∈S(x)

F(h(y)), ∀x∈V \ {x⁰}, (2.5) where S(x) is the set of direct successors ofxand the function F : Q^q−1p → Q^q−1p , F(h) = (F1,· · ·, Fq−1)forh= (h1,· · · , hq−1)is defined as follows

F_i= ln_p (θ−1) exp_p(hi) +Pq−1

j=1exp_p(hj) + 1 θ+Pq−1

j=1exp_p(hj)

!

, θ= exp_p(J).

2.7. Translation-invariantp-adic Gibbs measures. Ap-adic Gibbs measure is translation- invariant (in short TIpGM) if and only if the boundary function˜h:V →Q^qpis constant, i.e.,˜h(x) =h˜forx∈ V.In this case, the condition (2.5) takes the formh =kF(h)or equivalently

h_i = ln_p (θ−1) exp_p(hi) +Pq−1

j=1exp_p(hj) + 1 θ+Pq−1

j=1exp_p(hj)

!^k

, 1≤i≤q−1.

Letz= (z₁,· · · , z_q−1)∈ E_p^q−1such thatz_i = exp_p(h_i)for any1 ≤i≤q−1.In what follows, we writez= exp_p(h).Then we have the following result.

Theorem 2.8 ( [42,43]). Let ˜h : V → Q^qp be a boundary function, ˜h(x) = ˜h = (˜h1,· · ·,˜hq). There exists a translation-invariantp-adic Gibbs measureµh˜ : Ω → Q^p associated with a boundary function ˜hfor anyx ∈ V and h = (h₁,· · ·, h_q−1) such thathi = ˜hi−˜hq for1 ≤i ≤ q−1if and only if(z1,· · · , z_q−1) = z = exp_p(h) = (exp_p(h₁),· · · , exp_p(h_q−1))∈ E_p^q−1is a solution of the following system of equations

z_i= (θ−1)zi+Pq−1 j=1zj+ 1 θ+Pq−1

j=1zj

!^k

, 1≤i≤q−1. (2.6)

LetIq−1 ={1,· · ·, q−1}. Forj ∈Iq−1, letej = (δ1j, δ2j,· · ·δq−1j)∈Q^q−1p and for α ⊂ Iq−1, let eα = P

j∈α

ej. For anyz = (z1,· · ·, zq−1) ∈ Q^q−1p , let {αj(z)}^d_j=1 be a disjoint partition of the index setIq−1, i.e.,

d

S

j=1

αj(z) =Iq−1, αj₁(z)∩αj₂(z) = ∅ forj1 6=j2such thatzi₁ =zi₂ for alli1, i2 ∈αj(z)andzi₁ 6=zi₂ for alli1 ∈ αj₁(z), i2∈αj₂(z). Then

z=

d

X

j=1

z_j^◦e_αj(z) (2.7)

wherezi=z_j^◦for anyi∈αj(z)and1≤j≤d.

Theorem 2.9(See [52]). Ifz∈ E_p^q−1is a solution of the system(2.6)then1≤d≤k.

The descriptions of all TIpGMs for the casesk= 2,3were given in [44,52].

Theorem 2.10(Descriptions of TIpGMs fork= 2,[44]). There exists a TIpGMµ_h˜: Ω→ Qpassociated with a boundary function˜h= (˜h1,· · · ,˜hq)if and only if˜hj = lnp(hzj) for all1 ≤j ≤q−1andh˜_q = ln_p(h)whereh∈ Epandz= (z₁,· · ·z_q−1)∈ E_p^q−1is defined as either one of the following form

(A) z= (1,· · ·,1) ;

(B) z= (z,· · ·, z)wherez∈ Ep\ {1}is a root of the following quadratic equation (q−1)²z²+ 2(q−1)−(θ−1)²

z+ 1 = 0;

(C) z=eα1+zeα2with|αi|=mi, m1+m2=q−1such thatz ∈ Ep\ {1}is a root of the following quadratic equation

m²₂z²+

2m2(m1+ 1)−(θ−1)²

z+ (m1+ 1)²= 0.

Theorem 2.11(Descriptions of TIpGMs fork= 3,[52]). There exists a TIpGMµh˜: Ω→ Qpassociated with a boundary function˜h= (˜h₁,· · · ,˜h_q)if and only if˜h_j = ln_p(hz_j) for all1 ≤ j ≤ q−1and ˜hq = lnp(h)whereh ∈ Ep is anyp-adic number andz = (z1,· · ·z_q−1)∈ E_p^q−1is defined either one of the following form

(A) z= (1,· · ·,1) ;

(B) z= (z,· · ·, z)wherez∈ Ep\ {1}is a root of the following cubic equation (q−1)³z³+ 3(q−1)²−(θ−1)²(θ+ 3(q−1)−1)

z² + 3(q−1)−(θ−1)²(θ+ 2)

z+ 1 = 0;

(C) z=eα₁+zeα₂with|αi|=mi, m1+m2=q−1such thatz ∈ Ep\ {1}is a root of the following cubic equation

m³₂z³+

3m²₂(m1+ 1)−(θ−1)²(θ+ 3m2−1) z² +

3m2(m1+ 1)²−(θ−1)²(θ+ 3(m1+ 1)−1)

z+ (m1+ 1)³= 0;

(D) z=z1eα₁+z2eα₂ with|αi|=mi, m1+m2=q−1such that (i) Ifm₁6= ^1−θ₃ thenz₁=−^(θ−1+3m_(θ−1+3m^2)z2+θ+2

1) ∈ E_p\ {1}andz₂∈ E_p\ {1}is a root of the following cubic equation

[(m₁−m₂)z+ (m₁−1)]³

+ (θ−1 + 3m₁)²

(θ−1 + 3m₂)z²+ (θ+ 2)z

= 0;

(ii) Ifm26= ^1−θ₃ thenz2=−^(θ−1+3m_(θ−1+3m^1)z1+θ+2

2) ∈ Ep\ {1}andz1∈ Ep\ {1}is a root of the following cubic equation

[(m₂−m₁)z+ (m₂−1)]³

+ (θ−1 + 3m₂)²

(θ−1 + 3m₁)z²+ (θ+ 2)z

= 0;

(E) z=z1eα₁+z2eα₂+eα₃with|αi|=mi, m1+m2+m3=q−1such that (i) If m₁ 6= ^1−θ₃ then z₁ = −^(θ−1+3m_(θ−1+3m^{2)z2+3m3+θ+2}

1) ∈ E_p \ {1} and z₂ ∈ Ep\ {1}is a root of the following cubic equation

[(m1−m2)z+ (m1−m3−1)]³+ (θ−1 + 3m1)²(θ−1 + 3m2)z² + (θ−1 + 3m1)²(3m3+θ+ 2)z= 0;

(ii) If m₂ 6= ^1−θ₃ then z₂ = −^(θ−1+3m_(θ−1+3m^{1)z1+3m3+θ+2}

2) ∈ Ep \ {1} and z₁ ∈ Ep\ {1}is a root of the following cubic equation

[(m₂−m₁)z+ (m₂−m₃−1)]³+ (θ−1 + 3m₂)²(θ−1 + 3m₁)z² + (θ−1 + 3m₂)²(3m₃+θ+ 2)z= 0;

(iii) Ifθ= 1−qandm1=m2=m3+1then eitherz1∈ Ep\{1}orz2∈ Ep\{1}

is anyp-adic number so that the second one is a root of the cubic equation (z1+z2+ 1)³= 27z1z2.

3. H_m-PERIODICp-ADICGIBBS MEASURES

Letmbe some fixed natural number. A boundary function˜h: V →Q^qpis calledm- periodicifh(x) =˜ ˜h^(j)wheneverd(x, x⁽⁰⁾) ≡j (modm)with0 ≤j < m.Ap-adic Gibbs measure associated with anm-periodic boundary function is calledHm-periodic.

In this case, the condition (2.5) takes the formh^(j)=kF(h^(j+1))for all0≤j < mwith h^(m):=h⁽⁰⁾, or equivalently

h^(j)_i = lnp

(θ−1) exp_p(h^(j+1)_i ) +Pq−1

l=1exp_p(h^(j+1)_l ) + 1 θ+Pq−1

l=1exp_p(h^(j+1)_l )

!^k

, 1≤i≤q−1.

Letz^(j)= (z₁^(j),· · · , z_q−1^(j) )∈ E_p^q−1such thatz^(j)_i = exp_p(h^(j)_i )for any1≤i≤q−1.In what follows, we writez^(j)= exp_p(h^(j)).Hence, we have the following result.

Proposition 3.1. There exists anHm-periodicp-adic Gibbs measure if and only if the following system of equations

z_i^(j)= (θ−1)z^(j+1)_i +Pq−1

l=1 z^(j+1)_l + 1 θ+Pq−1

l=1 z^(j+1)_l

!k

, 1≤i≤q−1, 0≤j≤m−1 (3.1) has a solution{z^(j)}^m−1_j=0 ⊂ E_p^q−1in whichz^(m):=z⁽⁰⁾.

We are looking for solutions of the formz^(j)=z^(j)eα+eβwith|α|+|β|=q−1of the system of equations (3.1). In this case, the system of equations (3.1) takes the following form

z^(j)=

(θ+|α| −1)z^(j+1)+q− |α|

|α|z^(j+1)+θ+q− |α| −1 ^k

, 0≤j ≤m−1 (3.2)

withz^(m):=z⁽⁰⁾.This means that{z^(j)}^m−1_j=0 is anm-cycle ofthe Potts–Bethe mapping so-called

Gθ,α,q,k(z) =

(θ+|α| −1)z+q− |α|

|α|z+θ+q− |α| −1 ^k

. (3.3)

Ifθ = 1or θ = 1−qthen the Potts–Bethe mapping becomes a constant function.

Throughout this paper, we suppose that0 < |θ−1|p < |q|p < 1. This is one of the assumptions to ensure the non-unique translation-invariant p-adic Gibbs measures. We may assume that1 ≤ |α| ≤ q−1otherwise if|α|= 0then we obtain a trivial solution z^(j)= (1,1,· · · ,1)for all0≤j≤m−1of the system of equations (3.1). Thus, we have

Gθ,α,q,k(z) =



 θ−1

|α| + 1

z+_|α|^q −1 z+^θ−1+q_|α| −1





k

.

If we setΘ = ^θ−1_|α| + 1andQ=_|α|^q ,then we obtain

f_Θ,Q,k(z) =

Θz+Q−1 z+ Θ +Q−2

k

.

It is obvious that the conditionsθ 6= 1andθ 6= 1−qare equivalent to the conditions Θ6= 1andΘ 6= 1−Q.Moreover, the condition|θ−1|p <|q|p (resp. |θ−1|p = |q|p

or|θ−1|p >|q|p) is equivalent to the condition|Θ−1|p <|Q|p(resp.|Θ−1|p =|Q|p

or|Θ−1|p >|Q|p). Therefore, without loss of generality, we assume that|α| = 1and q∈Qp.We are now ready to formulate the main results of this paper.

Letfθ,q,k:Qp→Qpbe the Potts–Bethe mapping defined as

f_θ,q,k(x) =

θx+q−1 x+θ+q−2

^k

(3.4)

whereθ∈ Ep, q∈Qpsuch that0<|θ−1|p<|q|p<1.

Denote byDom{f}the domain of a mappingf :Qp→Qpand byFix{f}the set of its fixed points. ThenDom{fθ,q,k}=Qp\{x^(∞)}wherex^(∞)= 2−θ−q=x⁽⁰⁾−q−(θ−1) andx⁽⁰⁾ = 1∈Fix{fθ,q,k}is always a fixed point for anyk∈N.

The dynamics of the Potts–Bethe mapping (3.4) for the casesk= 1withθ, q∈Qpand k= 2with0<|θ−1|p <|q|²_p <1were studied in [33] and [36], respectively. Here, we study its dynamics in the casek = 3withp≡1 (mod 3)andp= 2,3. The casek = 3 withp≥5, p≡2 (mod 3)has been studied in [55].

Let us consider the following cubic equation

y³−(1 +θ+θ²)y²−(2θ+ 1)(1−θ−q)y−(1−θ−q)²= 0. (3.5) This cubic equation always has three rootsy⁽¹⁾,y⁽²⁾,y⁽³⁾ ∈Zp(see Proposition 4.1).

Letx⁽⁰⁾ = 1, x⁽ⁱ⁾ = x⁽⁰⁾−q+ (θ−1)(y⁽ⁱ⁾−1)for 1 ≤ i ≤ 3. We introduce the

following sets, A0 :=

x∈Qp:

x−x⁽⁰⁾ _p<|q|p

,

C1 :=

x∈Qp:

x−x⁽¹⁾ _p<

x−x^(∞)

_p=|θ−1|p

,

C2 :=

x∈Qp:

x−x⁽²⁾ _p<

x−x^(∞)

_p=|q|p|θ−1|p

,

C₃ :=

x∈Qp:

x−x⁽³⁾ _p<

x−x^(∞)

_p=|q|_p|θ−1|_p

.

The attracting basin of the fixed pointx⁽⁰⁾is defined as B(x⁽⁰⁾) :=

x∈Q^p: lim

n→+∞f_θ,q,3ⁿ (x) =x⁽⁰⁾

wheref_θ,q,3ⁿ (x)forn∈Nis then-th iteration offθ,q,3. The following theorem is our main theorem.

Theorem 3.2. Letk = 3, p ≡ 1 (mod 3), and 0 < |θ−1|_p < |q|_p < 1. Then the following statements hold.

(i) One hasFix{fθ,q,3} ={x⁽⁰⁾, x⁽¹⁾, x⁽²⁾, x⁽³⁾}in whichx⁽⁰⁾is attracting and x⁽¹⁾, x⁽²⁾andx⁽³⁾are repelling;

(ii) One has

Dom{fθ,q,3} \(C1∪ C2∪ C3)⊂B x⁽⁰⁾

=

+∞

[

n=0

f_θ,q,3⁻ⁿ (A0) ;

(iii) There existsJ ⊂ C1∪ C2∪ C3such that the dynamical systemf_θ,q,3:J → J is topologically conjugate to the full shift dynamics(Σ₃, σ)on three symbols;

(iv) One has

Qp=B x⁽⁰⁾

∪ J ∪

+∞

[

n=0

f_θ,q,3⁻ⁿ {x^(∞)}.

The proof of Theorem 3.2 will be given in Section 4. As an application of Theorem 3.2, we have the following theorem which shows the vastness of periodic p-adic Gibbs measures.

Theorem 3.3. Letk = 3, p ≡ 1 (mod 3). Suppose0 < |θ−1|p < |q|p < 1. Write q= _|q|^q∗

p withq^∗6 |p.Then there existH_m-periodicp-adic Gibbs measures associated with them-periodic boundary functions{˜h^(j)}^m−1_j=0 where

˜h^(j)= (ln_p(hz₁^(j)),· · ·,ln_p(hz^(j)_q−1),log_p(h))

withh∈ Ep, z^(j)= (z₁^(j),· · ·z^(j)_q−1)∈ E_p^q−1such thatz^(j)=z^(j)eα+eβ,|α|+|β|=q−1,

|α| 6∈ {_|q|¹

p,_|q|²

p,· · ·,^q_|q|^∗−1

p }and{z^(j)}^m−1_j=0 is anm-cycle of the Potts–Bethe mapping G_θ,α,q,3(z) =

(θ+|α| −1)z+q− |α|

|α|z+θ+q− |α| −1 ³

. (3.6)

Proof. As we have already mentioned (see Proposition 3.1) that there exists anHm-periodic p-adic Gibbs measure associated with anm-periodic boundary function{h˜^(j)}^m−1_j=0 ,˜h^(j)= (lnp(hz^(j)₁ ),· · · ,lnp(hz_q−1^(j) ),lnp(h))withh∈ Ep, z^(j) = (z^(j)₁ ,· · ·z^(j)_q−1)∈ E_p^q−1of the formz^(j)=z^(j)eα+eβ, |α|+|β|=q−1if and only if{z^(j)}^m−1_j=0 ⊂ Epis anm-cycle of the Potts–Bethe mapping (3.6) or equivalently anm-cycle of the following mapping

f_Θ,Q,3(z) =

Θz+Q−1 z+ Θ +Q−2

3

(3.7) whereΘ := ^θ−1_|α| + 1andQ:= _|α|^q .Due to Theorem 3.2(iii), if0<|Θ−1|p<|Q|p<1 then the Potts–Bethe mapping (3.7) has anm-cycle{z^(j)}^m−1_j=0 ⊂ C₁∪ C₂∪ C₃⊂ E_p.

We now describe all possible setsα ⊂ I_q−1 for which one has that0 <|Θ−1|p <

|Q|p < 1.Letq =p^l·q^∗ = _|q|^q∗

p such thatq^∗6 | p, l ∈ N.One has|Q|p = |_|α|^q |p < 1 if and only if|α| 6∈ {p^l,2p^l,· · · ,(q^∗−1)p^l} ={_|q|¹

p,_|q|²

p,· · ·,^q_|q|^∗−1

p }.Moreover, since

|θ−1|p <|q|pand|α| ∈N,we get|Θ−1|p <|Q|p. Remark3.4. LetHGM(m, q: 3)be the set ofHm-periodicp-adic Gibbs measures of the q-states Potts model on the Cayley tree of order three. It follows from Theorem 3.3 that

|HGM(m, q: 3)| ≥



2^q−1−

q^∗−1

X

i=1

q−1

|q|⁻¹p i



·3^m whereq=_|q|^q∗

p withq^∗6 |p.

4. DYNAMICS OF THEPOTTS–BETHE MAPPING FORk= 3ANDp≡1 (mod 3) We investigate the dynamics of the Potts–Bethe mappingfθ,q,3:Dom{fθ,q,3} →Q^p

fθ,q,3(x) =

θx+q−1 x+θ+q−2

³

(4.1) whereDom{fθ,q,3}=Qp\ {x^(∞)}andx^(∞)= 2−θ−q.

Throughout this section, we assume that0<|θ−1|p<|q|p<1andp≡1 (mod 3).

4.1. Fixed points. It is clear thatx⁽⁰⁾ = 1 ∈ Fix{fθ,q,3}. Moreover, it follows from fθ,q,3(x)−1 =x−1that

(x−1)(θ−1)(θx+q−1)²+ (θx+q−1)(x+θ+q−2) + (x+θ+q−2)² (x+θ+q−2)³

= (x−1).

Therefore, any other fixed pointx6=x⁽⁰⁾is a root of the following cubic equation (θ−1)(θx+q−1) ((θ+ 1)x+θ+ 2q−3) = (x+q−1)(x+θ+q−2)². (4.2) We introduce a new variabley := ^x−1+q_θ−1 + 1. Then the cubic equation (4.2) can be written with respect toyas follows

y³−(1 +θ+θ²)y²−(2θ+ 1)(1−θ−q)y−(1−θ−q)²= 0. (4.3) Let us find all possible roots of the cubic equation (4.3).

Proposition 4.1. Writeq=p^rq^∗for somer∈Nwithq^∗ =q0+q1p+· · · ∈Z^∗p.Then the cubic equation (4.3) always has three rootsy⁽¹⁾,y⁽²⁾,y⁽³⁾such that

(i) |y⁽¹⁾|p= 1>|1−θ−q|p=|y⁽²⁾|p=|y⁽³⁾|p;

(ii) y⁽¹⁾= 3 + (p−q0)p^r+· · · or equivalently|y⁽¹⁾−3 +q|p<|y⁽¹⁾−3|p=|q|p; (iii) y⁽²⁾ = _|q|¹

p

y₀⁽²⁾+y⁽²⁾₁ p+. . .

, y⁽³⁾ = _|q|¹

p

y₀⁽³⁾+y⁽³⁾₁ p+. . .

wherey₀⁽²⁾ andy⁽³⁾₀ are roots of the following congruent equation

3t²+ 3(1−θ−q)^∗t+ ((1−θ−q)^∗)²≡0 (mod p). (4.4) Remark4.2. The congruent equation (4.4) has discriminantD =−3 ((1−θ−q)^∗)². It is solvable if and onlyD (or equivalently−3) is a quadratic residue modulop. −3 is a quadratic residue modulopif and only ifp≡1 (mod 3).

Proof. We study the solvability of the cubic equation (4.3) overQ^p. We first show that if the cubic equation (4.3) has any root in thep-adic fieldQpthen it must lie in the setZp. Suppose the contrary, i.e. the cubic equation (4.3) has a rooty such that|y|p >1.Since p≡1 (mod3)andy³= (1 +θ+θ²)y²+ (2θ+ 1)(1−θ−q)y+ (1−θ−q)²,then

|y|³_p=|(1 +θ+θ²)y²+ (2θ+ 1)(1−θ−q)y+ (1−θ−q)²|p =|y|²_p. It is a contradiction. Therefore, any root of the cubic equation (4.3) must lie in the setZp. We refer to [52] for the detailed study of the general cubic equation overZp.

Leta=−(1+θ+θ²), b=−(2θ+1)(1−θ−q)andc=−(1−θ−q)². One can verify that|b|²_p=|c|_p=|1−θ−q|²_p<1 =|a|_pand|b|_p<|a|²_p, |c|_p<|a|³_p. Thus, by Theorem 5.1 of [52], the cubic equation (4.3) always has a rooty⁽¹⁾for which|y⁽¹⁾|p =|a|p = 1.

Moreover, it follows from |θ−1|p < |q|p < 1 andy⁽¹⁾ = y⁽¹⁾₀ +y₁⁽¹⁾p+. . . that

y₀⁽¹⁾3

−3 y⁽¹⁾₀ 2

≡0 (mod p).Thus,y⁽¹⁾₀ = 3andy⁽¹⁾= 3 +y₁⁽¹⁾p+. . ..

Let δ1 = b²−4ac. It is clear that δ1 = −3(1−θ−q)² and |b|²_p = |a|p|c|p =

|1−θ−q|²_p=|δ1|p. Sincep≡1 (mod3),there always exists√

δ1∈Qp(or equivalently there exists√

−3∈Qp). Therefore, by Theorem 5.1 of [52], the cubic equation (4.3) has two more rootsy⁽²⁾andy⁽³⁾such that|y⁽²⁾|p=|y⁽³⁾|p= ^|b|_|a|^p

p =|1−θ−q|p<1.

We know that|1−θ−q|p =|q|p=p^−rfor somer∈N. Then,y⁽ⁱ⁾=p^r(y⁽ⁱ⁾)^∗for i= 2,3and1−θ−q=p^r(1−θ−q)^∗. By pluggingy⁽ⁱ⁾into the cubic equation (4.3), we have

p^r((y⁽ⁱ⁾)^∗)³−(1 +θ+θ²)((y⁽ⁱ⁾)^∗)²−(2θ+ 1)(1−θ−q)^∗(y⁽ⁱ⁾)^∗−((1−θ−q)^∗)²= 0.

Then fori= 2,3

3(y⁽ⁱ⁾₀ )²+ 3(1−θ−q)^∗y₀⁽ⁱ⁾+ ((1−θ−q)^∗)²≡0 (modp).

Hence,y₀⁽²⁾andy⁽³⁾₀ are roots of the congruent equation

3t²+ 3(1−θ−q)^∗t+ ((1−θ−q)^∗)²≡0 (modp).

Now, to prove

y⁽¹⁾−3

p=|q|p >|y⁽¹⁾−3+q|p, we introduce a new variablez:=y−3.

Then the cubic equation (4.3) takes the following form with respect toz z³+

8−θ−θ²

z²+ [21−6θ(θ+ 1)−(2θ+ 1)(1−θ−q)]z +

q(4θ+ 5)−(θ−1)(4θ+ 14)−q²

= 0. (4.5)

LetA= (8−θ−θ²), B= 21−6θ(θ+ 1)−(2θ+ 1)(1−θ−q)andC=q(4θ+ 5)− (θ−1)(4θ+ 14)−q². One can check that|A|p = 1, |B|p= 1and|C|p=|q|p<1. Thus

|B|p =|A|²_pand|C|p<|A|³_pwhich imply that the cubic equation (4.5) always has a root z⁽¹⁾for which|z⁽¹⁾|_p = _|B|^|C|p

p =|q|_p(see the proof of Theorem 5.1, [52]). Therefore, we have|z⁽¹⁾|p=|y⁽¹⁾−3|p=|q|pwhich means thaty⁽¹⁾= 3 +y⁽¹⁾r p^r+· · · and we write the cubic equation (4.3) as follows

y²(y−3)−(θ+ 2)(θ−1)y²−(2θ+ 1)(1−θ−q)y−(1−θ−q)²= 0. (4.6) Since|θ−1|_p<|q|_p<1,we have

y⁽¹⁾²

≡ 9 + 6y⁽¹⁾_r p^r(modp^r+1), θ ≡ 1 (modp^r+1).

It follows from (4.6)

9y⁽¹⁾_r p^r+ 9p^rq0≡0 (modp^r+1), equivalently y_r⁽¹⁾+q0≡0 (modp).

Therefore,yr⁽¹⁾=p−q0≡ −q0(mod p),which implies|y⁽¹⁾−3 +q|p<|y⁽¹⁾−3|p. Proposition 4.3. Fix{fθ,q,3} ={x⁽⁰⁾, x⁽¹⁾, x⁽²⁾, x⁽³⁾}, wherex⁽⁰⁾ = 1, x⁽ⁱ⁾ = 1− q+ (θ−1)(y⁽ⁱ⁾−1)for1≤i≤3andy⁽¹⁾,y⁽²⁾,y⁽³⁾are the roots of the cubic equation (4.3).

Proof. The proof follows from Proposition 4.1.

4.2. Local behavior of the fixed points. Recall that a fixed pointxof the Potts–Bethe mapping is called attractingif 0 ≤ |λ|p < 1, indifferent if |λ|p = 1 andrepelling if

|λ|p>1whereλ=f_θ,q,3⁰ (x).

Proposition 4.4. The following statements hold:

(i) x⁽⁰⁾is an attracting;

(ii) x⁽¹⁾, x⁽²⁾andx⁽³⁾are repelling.

Proof. We have to show

f_θ,q,3⁰ (x⁽⁰⁾)

_p <1and

f_θ,q,3⁰ (x⁽ⁱ⁾)

_p >1for1 ≤i ≤3. It is easy to check that for0≤i≤3,

f_θ,q,3⁰ (x⁽ⁱ⁾) = 3(θ−1)(θ−1 +q)(θx⁽ⁱ⁾+q−1)² (x⁽ⁱ⁾+θ+q−2)⁴

= 3(θ−1)(θ−1 +q)x⁽ⁱ⁾

(θx⁽ⁱ⁾+q−1)(x⁽ⁱ⁾+θ+q−2). (4.7) Hence,

f_θ,q,3⁰ (x⁽⁰⁾)

_p = ^|θ−1|_|q| ^p

p < 1. Recallingx⁽ⁱ⁾ = 1−q+ (θ−1)(y⁽ⁱ⁾−1) for 1≤i≤3, we deduce from (4.7) that

f_θ,q,3⁰ (x⁽ⁱ⁾) = 3(θ−1 +q)x⁽ⁱ⁾

(θ−1)y⁽ⁱ⁾(θy⁽ⁱ⁾+ 1−θ−q). (4.8) Due to Proposition 4.1, we have

y⁽²⁾ _p =

y⁽³⁾

_p = |1−θ−q|_p <1 = y⁽¹⁾

_pand

|x⁽ⁱ⁾|p = 1for1≤i≤3. Therefore,

f_θ,q,3⁰ (x⁽¹⁾)

_p = _|θ−1|^|q|p

p >1. For the fixed points