Table Of Content

Orbit Equivalent Substitution Dynamical Systems 2 and Complexity 1 0 2 S. Bezuglyi and O. Karpel n Institute for Low Temperature Physics, a J 47 Lenin Avenue, 61103 Kharkov, Ukraine 8 (e-mail: [email protected], [email protected]) ] S D Abstract . h For any primitive proper substitution σ, we give explicit construc- t a tionsofcountablymanypairwisenon-isomorphicsubstitutiondynami- m calsystems{(Xζn,Tζn)}∞n=1 suchthattheyallare(strong)orbitequiv- [ alentto(Xσ,Tσ). Weshowthatthecomplexityofthesubstitutiondy- 1 namicalsystems{(Xζn,Tζn)}isessentiallydifferentthatpreventsthem from being isomorphic. Given a primitive (not necessarily proper) v 2 substitution τ, we find a stationary simple properly ordered Bratteli 2 diagram with the least possible number of vertices such that the cor- 6 responding Bratteli-Vershik system is orbit equivalent to (X ,T ). τ τ 1 . 1 1 Introduction 0 2 1 The seminal paper [10] answers, among other outstanding results, the ques- : v tion of orbit equivalence of uniquely ergodic minimal homeomorphisms of a i X Cantor set. Itwas provedthattwosuchminimalsystems, (X,T)and(Y,S), r are orbit equivalent if and only if the clopen values sets S(µ) = {µ(E) : a E clopen in X} and S(ν)= {ν(F) :F clopen in Y} coincide where µ and ν are the unique invariant measures with respect to T and S, respectively. It is well known now that Bratteli diagrams play an extremely important role in the study of homeomorphisms of Cantor sets because any minimal (and even aperiodic) homeomorphism of a Cantor set is conjugate to the Vershik map acting on the path space of a Bratteli diagram [10], [11], [13]. This realization turns out to be useful in many cases, in particular, for the study of substitution dynamical systems because the corresponding Bratteli dia- gramsareofthesimplestform. Itwasprovedin[6]thattheclass ofminimal substitution dynamical systems coincides with Bratteli-Vershik systems of 1 stationary simple Bratteli diagrams. Later on, it was shown in [3] that a similar result is true for aperiodic dynamical systems. These facts allow us to findeasily the clopen values set S(µ) for a substitution dynamical system in terms of the matrix of substitution (see [4] and subsection 2.2). In order to construct a minimal substitution dynamical system which is orbit equivalent to a given one, (X ,T ), (in other words, a simple stationary Bratteli σ σ diagram B ) one has to find another stationary simple Bratteli diagram B σ such that the clopen values set S(µ) is kept unchanged where µ is a unique T -invariant measure. Moreover, if one wants to have a substitution dynam- σ ical system which is strongly orbit equivalent to (X ,T ), then additionally σ σ the dimension group of the diagram B must be unchanged. Of course, we σ are not interesting in the case when powers of σ are considered since it leads trivially to conjugate substitution systems. We focus here on the study of orbit equivalence of minimal substitution dynamical system because aperiodicnon-minimal substitution systems were considered before in [2]. We note that the simplest case when the invariant measure µ has rational S(µ) and λ is an integer was studied in [16]. The main results of the present paper are as follows. Let (X ,T ) be σ σ a minimal substitution dynamical system and let B be a stationary sim- σ ple Bratteli diagram corresponding to (X ,T ). We give an explicit con- σ σ struction of countably many substitutions {ζ }∞ defined on the Bratteli n n=1 diagrams {B }∞ , obtained by telescoping of B , such that the systems n n=1 σ {(X ,T )}∞ are strong orbit equivalent to (X ,T ) and pairwise non- ζn ζn n=1 σ σ isomorphic. In the other construction, we build pairwise non-isomorphic orbit equivalent minimal substitution dynamical systems by using alphabets of different cardinality. In both constructions, we use the complexity function n 7→ p (n) to dis- σ tinguish non-isomorphic systems. Recall that the function p (n) counts the σ number of words of length n in the infinite sequence invariant with respect to σ. In the first construction, the incidence matrices of built substitution systems are the powers of A. In the case of fixed alphabet, the complexity function can be made increasing by enlarging the length of substitution and an appropriate permutation of letters. Using this method, we produce a countable family of pairwise non-isomorphic strong orbit equivalent substitution systems. In the second construction, the complexity of the systems is growing by increasing the number of letters in the alphabet. In other words, for a propersubstitution σ definedon thealphabet A, we findcount- ably many proper substitutions {ζ }∞ on the alphabets A of different n n=1 n cardinality such that (X ,T ) is orbit equivalent to (X ,T ), but the set σ σ ζn ζn {(X ,T )}∞ consists of pairwise non-isomorphic substitution dynamical ζn ζn n=1 2 systems. GiventheBratteli-Vershiksystemonasimplestationarydiagram(B,≤), we findan orbitequivalent stationary Bratteli-Vershik system with the least possible number of vertices. This number is the degree of the algebraic integer λ, the Perron-Frobenius eigenvalue of the transpose A to the incidence matrix of B. 2 Preliminaries 2.1 Minimal Cantor systems A minimal Cantor system is a pair (X,T) where X is a Cantor space and T: X → X is a minimal homeomorphism, i.e. for every x ∈ X the set Orb (x) = {Tn(x) |n ∈ Z} is dense in X. T Given a minimal Cantor system (X,T) and a clopen A ⊂X, let r (x) = A min{n ≥ 1 : Tn(x) ∈ A} be a continuous integer-valued map defined on A. Then TA(x) = TrA(x) is a homeomorphism of A, and a Cantor minimal system (A,T ) is called induced from (X,T). A There are several notions of equivalence for minimal Cantor systems: Definition 2.1. Let (X,T) and (Y,S) be two minimal Cantor systems. Then (1) (X,T) and (Y,S) are conjugate (or isomorphic) if there exists a homeomorphism h: X → Y such that h◦T = S ◦h. (2) (X,T) and (Y,S) are orbit equivalent if there exists a homeomorphism h: X → Y such that h(Orb (x)) = Orb (h(x)) for every x ∈ X. In T S other words, there exist functions n,m: X → Z such that for all x ∈ X, h◦T(x) = Sn(x) ◦h(x) and h◦Tm(x) = S ◦h(x). The functions n,m are called orbit cocycles associated to h. (3) (X,T) and (Y,S) are strong orbit equivalent if they are orbit equivalent and each of the corresponding orbit cocycles has at most one point of discontinuity. (4) (X,T) and (Y,S) are Kakutani equivalent if they have conjugate induced systems. (5) (X,T) and (Y,S) are Kakutani orbit equivalent if they have orbit equivalent induced systems. A Cantor system is called uniquely ergodic if it has a unique invariant probability measure. For a full non-atomic Borel measure µ on a Cantor space X, define the clopen values set S(µ) = {µ(U) : U clopen in X}. Let (X ,T ) and (X ,T ) be two uniquely ergodic minimal Cantor systems and 1 1 2 2 3 µ and µ be the unique probability invariant measures for T and T , re- 1 2 1 2 spectively. Among other results on orbit equivalence, it is proved in [10] that (X ,T ) and (X ,T ) are orbit equivalent if and only if S(µ ) = S(µ ). 1 1 2 2 1 2 2.2 Bratteli diagrams Definition2.2. ABrattelidiagram isaninfinitegraphB = (V,E)suchthat the vertex set V = V and the edge set E = E are partitioned i≥t0 i i≥1 i into disjoint subsets V and E such that i i S S (i) V = {v } is a single point; 0 0 (ii) V and E are finite sets; i i (iii) there exist a range map r and a source map s from E to V such that r(E ) = V , s(E ) = V , and s−1(v) 6= 0, r−1(v′) 6= 0 for all v ∈ V i i i i−1 and v′ ∈ V \V . 0 The pair (V ,E ) or just V is called the i-th level of the diagram B. A i i i sequence of edges (e : e ∈ E ) such that r(e ) = s(e ) is called a path. i i i i i+1 We denote by X the set of all infinite paths starting at the vertex v . This B 0 set is endowed with the standard topology turning X into a Cantor set. B Given a Bratteli diagram B = (V,E), define a sequence of incidence (n) (n) matrices F = (f ) of B: f = |{e ∈ E : r(e) = v,s(e) = w}|, where n vw vw n+1 v ∈ V and w ∈ V . Here and thereafter |V| denotes the cardinality of the n+1 n set V. A Bratteli diagram is called stationary if F = F for every n ≥ 2. n 1 A Bratteli diagram B′ = (V′,E′) is called the telescoping of a Bratteli diagram B = (V,E) to a sequence 0 = m < m < ... if V′ = V and E′ 0 1 n mn n is the set of all paths from V to V , i.e. E′ = E ◦... ◦E = mn−1 mn n mn−1 mn {(e ,...,e ): e ∈ E ,r(e )= s(e )}. mn−1 mn i i i i+1 Observe that every vertex v ∈ V is connected to v by a finite path, 0 and the set E(v ,v) of all such paths is finite. A Bratteli diagram is called 0 simple if for any n > 0 there exists m > n such that any two vertices v ∈ V n and w ∈ V are connected by a finite path. m A Bratteli diagram B = (V,E) is called ordered if every set r−1(v), v ∈ V , is linearly ordered. Given an ordered Bratteli diagram (B,≤ n≥1 n ) = (V,E,≤), any two paths from E(v ,v) are comparable with respect 0 S to the lexicographical order [11]. We call a finite or infinite path e = (e ) i maximal (minimal) ifevery e is maximal(minimal)amongsttheedges from i r−1(r(e )). A simple ordered Bratteli diagram (B,≤) is properly ordered if i thereareuniquemaximalandminimalinfinitepaths. Anysimplestationary Bratteli diagramcanbeproperlyordered. ABrattelidiagramB =(V,E,≤) 4 is called stationary ordered if it is stationary and the partial linear order on E does not depend on n. n Let (B,≤) = (V,E,≤) be a simple properly ordered stationary Bratteli diagram. Define a minimal homeomorphism φ : X → X as follows. B B B Let φ (x ) = x . If x = (x ,x ,...) 6= x , let k be the smallest B max min 1 2 max number so that x is not a maximal edge. Let y be the successor of x k k k (hence r(x ) = r(y )). Set φ (x) = (y ,...,y ,y ,x ,x ,...), where k k B 1 k−1 k k+1 k+2 (y ,...,y ) is the minimal path in E(v ,s(y )). The resulting minimal 1 k−1 0 k Cantor system (X ,φ ) is called a Bratteli-Vershik system. If (B′,≤′) is B B a telescoping of (B,≤) which preserves the lexicographical order then the Bratteli-Vershik systems (XB,φB) and (XB′,φB′) are isomorphic. Definition 2.3. Let B = (V,E) be a Bratteli diagram. Two infinite paths x = (x ) and y = (y ) from X are called tail equivalent if there exists i i i B 0 such that x = y for all i ≥ i . Denote by R the tail equivalence relation i i 0 on X . B ABratteli diagramissimpleifthetailequivalencerelation Risminimal. (n) Denote X (e) := {x = (x ) ∈ X : x = e ,i = 1,...,n}, where e = w i B i i (e ,...,e ) ∈ E(v ,w), n ≥ 1. A measure µ on X is called R-invariant 1 n 0 B if for any two paths e and e′ from E(v ,w) and any vertex w, one has 0 µ(X(n)(e)) = µ(X(n)(e′)). The measure invariant for a stationary Bratteli- w w Vershik system is R-invariant. In the paper, we will consider only simple stationary Bratteli diagrams. Let A = FT be the matrix transpose to the incidence matrix of a diagram B. Let λ be a Perron-Frobenius eigenvalue of A and let x = (x ,...,x )T be 1 K the corresponding positive eigenvector such that K x = 1. Suppose B i=1 i has no multiple edges between levels 0 and 1. Then the ergodic probability P measure µ defined by λ and x satisfies the relation: x (n) i µ(X (e)) = , i λn−1 where i ∈ V and e is a finite path with s(e) = i. Therefore, the clopen n values set for µ has has the form: K x (n) i (n) (n) S(µ)= k : 0 ≤ k ≤ h ; n= 1,2,... , i λn−1 i i ( ) i=1 X where h(n) = |E(v ,v)|, v ∈ V . Let H(x) be an additive subgroup of R v 0 n generated by x ,...,x . Since the Bratteli-Vershik system is minimal and µ 1 n 5 is a uniqueinvariant measure, µ is good and S(µ) = ∞ 1 H(x) ∩[0,1] N=0 λN (see [1, 2]). It is easy to see that λH(x) ⊂ H(x) and λm ∈ H(x) for any (cid:0)S (cid:1) m ∈ N (see [2]). 2.3 Substitution dynamical systems Let A = {a ,...,a } be a finite alphabet. Let A∗ be the collection of finite 1 s non-empty words over A. Denote by Ω = AZ the set of all two-sided infinite sequences on A. A substitution σ is a map σ: A → A∗. It extends to maps σ: A∗ → A∗ and σ: Ω → Ω by concatenation. Denote by T the shift on Ω: T(...x .x x ...) = ...x x .x .... −1 0 1 −1 0 1 Let A = (a )s be the incidence matrix associated to σ where a is σ ij i,j=1 ij the number of occurrences of ai in σ(aj). Clearly, Aσn = (Aσ)n for every n ≥ 0. A substitution σ is called primitive if there is n such that for each a ,a ∈ A, a appears in σn(a ). Note that σ is primitive if and only if A i j j i σ is a primitive matrix. If it happens that |σ(a)| = q for any a ∈ A, then the substitution σ is called of constant length q. For x ∈ Ω, let L (x) be the n set of all words of length n occurring in x. Set L(x) = L (x). The n∈N n language of σ is the set L of all finite words occurring in σn(a) for some σ S n ≥ 0, a ∈A. Set X = {x ∈ Ω : L(x)⊂ L }. σ σ Throughout this paper we will consider only primitive substitutions σ such that X is a Cantor set. The dynamical system (X ,T ), where T σ σ σ σ is the restriction of T to the T-invariant set X , is called the substitution σ dynamical system associated to σ. It is well known (see [15]) that every primitive substitution generates a minimal and uniquely ergodic dynamical system. The following statements can be found in [15]. For every integer p > 0 the substitution σp defines the same language as σ, hence the systems (Xσ,Tσ) and (Xσp,Tσp) are isomorphic. Substituting σp for σ if needed, we can assume that there exist two letters r,l ∈ A such that r is the last letter of σ(r), l is the first letter of σ(l) and rl ∈ L . The sequence ω = σ lim σn(r.l) ∈ X is a fixed point of σ (that is σ(ω) = ω) and ω = r, n→∞ σ −1 ω = l. Then X = Orb (ω). 0 σ T The complexity of u ∈ Ω is the function p (n) which associates to each u integer n ≥ 1 the cardinality of L (u). It is easy to see that n p (k+1)−p (k) = (Card {a ∈ A :wa ∈ L (u)}−1). (2.1) u u k+1 w∈XLk(u) The sequence u is called minimal if every word occurring in u occurs in 6 an infinitenumberof places with boundedgaps. Afixed pointof aprimitive substitutionisalwaysminimal(see[7,15]). LetX bethesetofallsequences u x ∈ΩsuchthatL (x) = L (u)foreveryn ∈ N. Foraprimitivesubstitution n n σ with thefixedpoint uwehave X = X = Orb (u). Hence p (n)= p (n) σ u T x u for every n and every x ∈ X . Sometimes we will denote p by p to stress u u σ that the complexity function is defined by σ. The following results can be found in [5, 7, 15]. Theorem 2.4. (1) If the symbolic systems (X ,T) and (X ,T) associated u v to minimal sequences u and v are topologically conjugate, then there exists a constant c such that, for all n > c, p (n−c) ≤ p (n)≤ p (n+c). u v u Hence a relation p (n) ≤ ank+o¯(nk) when n → ∞ is preserved by conjugacy u (isomorphism). (2) Let ζ be a primitive substitution and p the complexity function of a fixed sequence u = ζ(u). Then, there exists a constant C > 0 such that p(n) ≤ Cn for every n ≥ 1. (3) Let u∈ Ω and p be the complexity function of u. Suppose that there exists a > 0 such that p(n)≤ an for all n ≥ n . Then 0 p(n+1)−p(n)≤ Ksa3 for all n≥ n , where K does not depend on u. 0 Definition 2.5. A substitution σ on an alphabet A is called proper if there exists an integer n > 0 and two letters a,b ∈ A such that for every c∈ A, a is the first letter and b is the last letter of σn(c). For every primitive substitution ζ, there exists a proper substitution σ such that the substitution systems (X ,T ) and (X ,T ) are isomorphic. ζ ζ σ σ The substitution σ is built using the method of return words (see [6]). The following theorem establishes the link between incidence matrices of ζ and σ: Theorem 2.6. [16] Let ζ be a non-proper primitive substitution and let λ be the Perron-Frobenius eigenvalue of its incidence matrix A . Let σ be the ζ corresponding proper substitution built by means of return words. Then the Perron-Frobenius eigenvalue of A is λk for some k ∈ N. σ Stationary Bratteli diagrams are naturally related to substitution dynamical systems (primitive substitutions are considered in [6], [9], and the 7 non-primitive case is studied in [3]). More precisely, let (B,≤) = (V,E,≤) be a stationary ordered Bratteli diagram with no multiple edges between levels 0 and 1. Choose a stationary labeling of V by an alphabet A: n V = {v (a) : a ∈ A}, n > 0. For a ∈ A consider the ordered set (e ,...,e ) n n 1 s of edges that range at v (a), n ≥ 2. Let (a ,...,a ) be the corresponding n 1 s orderedsetof thelabels ofthesourcesof theseedges. Themapa 7→ a ···a 1 s from A to A∗ does not depend on n and determines a substitution called the substitution read on (B,≤). Conversely, for any substitution dynamical system we can build the corresponding ordered stationary Bratteli diagram. Thefollowing theorem, proved in [6], shows thelink between simpleBratteli diagrams and primitive substitution dynamical systems. Theorem 2.7. Let (B,≤) be a stationary, properly ordered Bratteli diagram with only simple edges between the top vertex and the first level. Let σ be the substitution read on (B,≤). (i) If σ is aperiodic, then the Bratteli-Vershik system (X ,φ ) is iso- B B morphic to the substitution dynamical system (X ,T ). σ σ (ii) If σ is periodic, the Bratteli-Vershik system (X ,φ ) is isomorphic B B to a stationary odometer. In [16], the following result was proved: Theorem 2.8. Let σ be a primitive substitution whose incidence matrix has natural Perron-Frobenius eigenvalue. Then (X ,T ) is orbit equivalent to a σ σ stationary odometer system. We will need the next result (see [6, 9]): Theorem 2.9. Let (B,≤) be a stationary properly ordered Bratteli diagram. Then there exists a stationary properly ordered Bratteli diagram (B′,≤′) such that B′ has no multiple edges between levels 0 and 1 and the systems (XB,φB), (XB′,φB′) are isomorphic. 3 Orbit equivalence class for a primitive substitution Given a primitive proper substitution σ, we build countably many pairwise non-isomorphic substitution dynamical systems {(X ,T )}∞ in the or- ζn ζn n=1 bit equivalence class of (X ,T ). Two essentially different constructions σ σ are elaborated. In the first one, we obtain countably many strong orbit equivalent substitution systems defined on the same alphabet. The second 8 constructionproducescountablymanyorbitequivalentsubstitutionsystems with increasing cardinality of alphabets. Finally, given a primitive (not necessarily proper)substitution τ, we find a stationary simple properly ordered Bratteli diagram with the least possible number of vertices such that the corresponding Bratteli-Vershik system is orbit equivalent to (X ,T ). τ τ Theorem 3.1. Let (B,≤) be a stationary properly ordered simple Bratteli diagram. Let σ be the substitution read on (B,≤). Then there exist countably many telescopings B of B with proper orders ≤ and corresponding n n substitutions ζ read on B such that the substitution dynamical systems n n {(X ,T )}∞ are pairwise non-isomorphic and strong orbit equivalent to ζn ζn n=1 (X ,T ). σ σ Proof. Let A = {a ,...,a } be the alphabet for σ. Fix a number l ∈ N. Let 1 s {ω }sl denote the set of all possible words of length l over the alphabet A. r r=1 Take arbitrary N ∈ N(N willbechosen below) andconsider thetelescoping B of B with incidence matrix AN. In our construction, we will define a N proper substitution ζ = ζ(l) that is read on the Bratteli diagram B whose N incidence matrix is A = AN. This means that the number of occurrences ζ of any letter a in ζ(a ) is known but we are free to choose any order of i j letters in the word ζ(a ). In other words, we will change the lexicographical j order ≤ , that obviously determines σN, in order to define ζ. We take N lex sufficiently large to guarantee that ζ satisfies the following conditions: (1) for all 1 ≤ j ≤ s the word ζ(a ) starts with the word a a and ends j 1 j with the letter a ; 1 (2) the word ζ(a ) contains as subwords all words {ω a } for 1 ≤ i ≤ sl 1 i j and 1 ≤ j ≤ s. Obviously, itfollows thatζ isapropersubstitutionandthetwosubstitu- tion dynamical systems, (X ,T ) and (X ,T ), are strongly orbit equivalent ζ ζ σ σ because the dimension groups associated to these minimal Cantor systems (that is to the diagrams B and B ) are order isomorphic by a map preserv- N ing the distinguished order unit (see [10]). We need to show that for an appropriate choice of l the substitution ζ = ζ(l) is such that the systems (X ,T ) and (X ,T ) are not isomorphic. ζ(l) ζ(l) σ σ We see that ζ∞(a .a )= lim ζn(a .a ) is a fixed point. The parameter 1 1 n→∞ 1 1 l should be now chosen in such a way that the complexity function p of ζ ζ∞(a .a ) grows essentially faster then the complexity function associated 1 1 to (X ,T ). σ σ To clarify the idea of the proof, we first prove the theorem in the case when σ is a substitution of constant length q. Then ζ is a substitution of length qN. 9 By definition of ζ, we have p (1) = s, p (2) = s2 and p (k) = sk for ζ ζ ζ 1 ≤ k ≤ l+1. The word ζ2(a ) contains the words ζ(ω )ζ(a ) for 1 ≤ i ≤sl 1 i j and 1 ≤ j ≤ s. Recall that ζ(a ) starts with a a . Thus, ζ2(a ) contains j 1 j 1 sl different words {ζ(ω )a }sl and each word can be followed by any letter i 1 i=1 fromA. Sincelq+1= |ζ(ω )|+|a |,weobtainp (lq+2)−p (lq+1)≥ sl(s−1) i 1 ζ ζ by (2.1). Consider ζ3(a ). Then apply the previous arguments with ζ(a ) instead 1 i of a . Since |ζ2(ω )|+|ζ(a )|+|a | = lq2 +q+1, we get p (lq2 +q +2)− i i 1 1 ζ p (lq2 + q + 1) ≥ sl(s − 1). Thus, we conclude by induction that for all ζ m ∈ N m−1 m−1 p (lqm+ qi+1)−p (lqm+ qi) ≥ sl(s−1). ζ ζ i=0 i=0 X X Taking l large enough, we can make the difference p (k +1)−p (k) arbi- ζ ζ trary large for infinite number of values of k. Now if we assumed that the substitution systems (X ,T ) and (X ,T ) are isomorphic, then we would ζ ζ σ σ have the following relation that follows from Theorem 2.4: (i) there exists C > 0 such that p (n) ≤ Cn for all n ≥ 1 and p (n) ≤ (C +1)n for suffi- σ ζ ciently large n; (ii) p (n+1)−p (n)≤ Ks(C +1)3 for all sufficiently large ζ ζ n, where K is a “universal” constant. Clearly, these statements contradict to the proved above fact that the values of p (n+1)−p (n) are unbounded. ζ ζ To prove the theorem in the general case, denote q = |ζm(a )| for m 1 m ∈ N. Let D = max |ζm(a )| and d = min |ζm(a )|. Since |ω | = l, we m i m i i 1≤i≤s 1≤i≤s have ld ≤ |ζm(ω )|≤ lD for any 1 ≤ i≤ sl and m ≥ 1. The matrix A is m i m ζ strictly positive, hence there exist positive constants M , M such that for 1 2 every m ≥ 1 we have M λm ≤ d ≤ D ≤ M λm, where λ is the Perron- 1 m m 2 Frobenius eigenvalue of A (see [15]). Hence Dm ≤ M2 for all m ≥ 1. For ζ dm M1 r > 0, set l = l(r) = M2 +1 r. Then l ≥ rM2 ≥ rDm and ld ≥ rD M1 M1 dm m m for all m ≥ 1. Thus,(cid:16){hζm(iω )}s(cid:17)l contains at least sr different suffices of i i=1 length D r and each suffix can be followed by any word from {ζm(a )}s . m j j=1 By the same argument as in the case of substitution of constant length, we conclude that m−1 m−1 p (D r+ q +1)−p (D r+ q ) ≥sr(s−1). ζ m i ζ m i i=0 i=0 X X By (2.1), we obtain the needed result. Thus, p (n) ≤ Cn for all n ≥ 1 but for any N ∈ N there exists M > N σ such that p (M) > (C + 1)M. Set ζ = ζ. There exists C > 0 such ζ 1 1 10