VG II.– Dynamical zeta function of the β -shift - Renyi- Parry Dynamics of algebraic numbers

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Let f ∈ C[X ] be a nonzero polynomial with complex coefficients ofdegree d :

f = a0X d + a1X d−1 + . . .ad = a0

d

∏i=1

(X −αi).

...(i) the Mahler measure of f :

M(f ) = |a0|d

∏i=1

max{1, |αi |},

Theorem (Smyth, ’71)

Let f ∈ Z[X ], f (0) 6= 0 and f assumed not reciprocal. Then

M(f )≥Θ = 1.3247 . . . .

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Theorem (Dobrowolski, ’79)For any reciprocal algebraic integer α of degree d,

M(α) > 1 + (1− ε)

(LogLog d

Log d

)3

, d > d1(ε),

where 1− ε can be replaced by 1/1200 in an effective version (n ≥ 2).

Conjecture (Schinzel - Zassenhaus, ’65)There exists a constant C > 0 such that, for any algebraic integerα 6= 0, deg[α) = d, not being a root of unity, such that the houseα := max{|α(i)|} of α satisfies :

α > 1 +Cd.

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Conjecture (Lehmer’s Problem, ’33))There exists c > 0 such that

M(α)≥ 1 + c

for any reciprocal algebraic number α 6= 0 which is not a root of unity.

+ex.4 / 47

1.– The Renyi–Parry dynamical system

In 1957 Renyi introduced new representations of a real number x ,using a positive function y = f (x) and infinite iterations of it, in the formof an “f -expansion”, as

x = ε0 + f (ε1 + f (ε2 + . . .))

with “digits” εi in some alphabet and remainders f (εn + f (εn+1 + . . .)).

This approach considerably enlarged the usual decimal numerationsystem, and numeration systems in integer basis, by allowingarbitrary real bases of numeration (Fraenkel, Lothaire) : let β > 1 notan integer and consider f (x) = x/β if 0≤ x ≤ β , and f (x) = 1 if β < x .Then the f -expansion of x is the representation of x in base β as

x = ε0 +ε1

β+

ε2

β 2 + . . .+εn

β n + . . . .

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In terms of dynamical systems, in 1960, Parry has reconsidered andstudied the ergodic properties of such representations of real numbersin base β , in particular the conditions of faithfullness of the map :

x ←→ (εi)i

and the complete set of admissible sequences (ε0,ε1,ε2, . . .) for all realnumbers. This complete set is called the language in base β .

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DefinitionA representation in base β (or a β -representation ; or a β -aryrepresentation if β is an integer) of x is an infinite word (xi)i≥1 of A N

β,

where Aβ = {0,1, . . . ,dβ −1e}, such that

x = ∑i≥1

xiβ−i .

The main difference with the case where β is an integer is that x mayhave several representations. A particular β -representation, called theβ -expansion, or the greedy β -expansion, and denoted by

dβ (x),

of x can be computed either by the greedy algorithm, or equivalentlyby the β -transformation

Tβ : x 7→ βx ( mod 1) = {βx},

the iterates of Tβ providing the successive digits xi of dβ (x).7 / 47

Denoting T 0β

:= Id,T 1β

:= Tβ ,T iβ

:= Tβ (T i−1β

) for all i ≥ 1, we have :

dβ (x) = (xi)i≥1 if and only if xi = bβT i−1β

(x)c

and we write the β -expansion of x as

x = ·x1x2x3 . . . instead of x =x1

β+

x2

β 2 +x3

β 3 + . . . .

The digits are x1 = bβxc, x2 = bβ{βx}c, x3 = bβ{β{βx}}c, . . . , dependupon β .

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Theorem (Renyi, ’57)

Let β > 1. For any g(x) ∈ L1(0,1) we have

limn→∞

1n

n−1

∑k=0

g(T kβ

(x)) = R(g) for almost all x ,

where the constant R(g) does not depend upon x. There exists a measure ν ,equivalent to the Lebesgue measure, which is invariant with respect to theβ -transformation Tβ , such that

ν(E) =∫

Eh(x)dx for any measurable subset E ⊂ (0,1),

where h(x) is a measurable function, satisfying

1− 1β≤ h(x)≤ 1

1− 1β

,

for which R(g) =∫ 1

0g(x)h(x)dx .

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Theorem (Parry, ’60)For each β > 1, the density function h(x) of the invariant measure νβ is equalto the pure jump function

hβ (x) = ∑x<T n

β(1)

1β n .

It is a step function with a finite number of steps if and only if the β -expansiondβ (β ) of β (in base β ) is ultimately periodic.

The dynamical system ([0,1],Tβ ,ν = νβ ) is called the Renyi-Parry numerationsystem in base β . Tβ is an ergodic transformation sending [0,1) onto itself,i.e. ν(E) = ν(T−1

β(E)).

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The set A Nβ

is endowed with the lexicographical order (not usual innumber theory), and the product topology. The one-sided shiftσ : (xi)i≥1 7→ (xi+1)i≥1 leaves invariant the subset Dβ of theβ -expansions of real numbers in [0,1). The closure of Dβ in A N

βis

called the β -shift, and is denoted by Sβ . The β -shift is a subshift ofA N

β, for which

dβ ◦Tβ = σ ◦dβ

holds on the interval [0,1]. In other terms, Sβ is such that

x ∈ [0,1] ←→ (xi)i≥1 ∈ Sβ

is bijective. This one-to-one correspondence between the totallyordered interval [0,1] and the totally lexicographically ordered β -shiftSβ is fundamental. Parry has shown that only one sequence of digitsentirely controls the β -shift Sβ , and that the ordering is preservedwhen dealing with the greedy β -expansions.

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The usual inequality “<” on the real line is transformed into theinequality “<lex ”, meaning “lexicographically smaller with all its shifts”.

The greatest element of Sβ : it comes from the Renyi β -expansion of 1denoted by

dβ (1) = 0.t1t2t3 . . . and uniquely corresponds to 1 =+∞

∑i=1

tiβ−i ,

where

t1 = bβc, t2 = bβ{β}c= bβTβ (1)c, t3 = bβ{β{β}}c= bβT 2β

(1)c, . . .

The sequence (ti)i≥1 is given by the orbit of one (T jβ

(1))j≥0 by

T 0β

(1) = 1, T jβ

(1) = βj − t1β

j−1− t2βj−2− . . .− tj ∈ Z[β ]∩ [0,1]

for all j ≥ 1. The digits ti belong to Aβ .

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From (ti)i≥1 ∈A Nβ

is built (ci)i≥1 ∈A Nβ

, defined by

c1c2c3 . . . :=

{t1t2t3 . . . if dβ (1) = 0.t1t2 . . . is infinite,(t1t2 . . . tq−1(tq−1))ω if dβ (1) is finite, = 0.t1t2 . . . tq,

where ()ω means that the word within () is indefinitely repeated. Thesequence (ci)i≥1 is the unique element of A N

βwhich allows to obtain

all the admissible β -expansions of all the elements of [0,1).

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Conditions of Parry

DefinitionA sequence (yi)i≥0 of elements of Aβ (finite or not) is said admissibleif and only if

σj(y0,y1,y2, . . .) = (yj ,yj+1,yj+2, . . .) <lex (c1, c2, c3, . . .) for all j ≥ 0,

where <lex means lexicographically smaller.

Any admissible representation (xi)i≥1 ∈A Nβ

corresponds to a realnumber x ∈ [0,1) and conversely the greedy β -expansion of x is (xi)i≥1itself. For an infinite admissible sequence (yi)i≥0 of elements of Aβ the(strict) lexicographical inequalities constitute an infinite number ofinequalities, unusual in number theory, which are called ”Conditionsof Parry”.

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Any admissible representation (xi)i≥1 ∈A Nβ

corresponds to a realnumber x ∈ [0,1) and conversely the greedy β -expansion of x is (xi)i≥1itself.

For an infinite admissible sequence (yi)i≥0 of elements of Aβ the(strict) lexicographical inequalities constitute an infinite number ofinequalities, unusual in number theory.

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In number theory, inequalities are often associated to collections ofhalf-spaces in euclidean or adelic Geometry of Numbers (Minkowski’sTheorem, etc).

The conditions of Parry are of totally different nature since they refer toa reasonable control, order-preserving, of the gappiness (lacunarity)of the coefficient vectors of the power series Parry Upperfunctions, which are the generalized Fredholm determinants of thetransfer operators of the β -transformations :

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In the correspondence [0,1]←→ Sβ , the element x = 1 admits dβ (1)as counterpart. The uniqueness of the β -expansion dβ (1) and itsself-admissibility property characterize the base of numeration β asfollows.

Proposition (Parry, ’60)

Let (a0,a1,a2, . . .) be a sequence of non-negative integers wherea0 ≥ 1 and an ≤ a0 for all n ≥ 0. The unique solution β > 1 of

1 =a0

x+

a1

x2 +a2

x3 + . . .

is such that dβ (1) = 0.a0a1a2 . . . if and only if

σn(a0,a1,a2, . . .) = (an,an+1,an+2, . . .)<lex (a0,a1,a2, . . .) for all n≥1.

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Let 1 < β < 2 be a real number, with dβ (1) = 0.t1t2t3 . . ..

If β is a simple Parry number, then there exists N ≥ 2, depending uponβ , such that tN 6= 0 and tj = 0, j ≥ n + 1.

Parry proved that the set of simple Parry numbers is dense in thehalf-line (1,+∞).

If β is a Parry number which is not simple, the sequence (ti)i≥1 iseventually periodic : there exists an integer m ≥ 1, the preperiodlength, and an integer p ≥ 1, the period length, such that

dβ (1) = 0.t1t2 . . . tm(tm+1tm+2 . . . tm+p)ω ,

m and p depending upon β , with at least one nonzero digit tj , withj ∈ {m + 1,m + 2, . . . ,m + p}.

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dβ (1) = 0.t1t2 . . . tm(tm+1tm+2 . . . tm+p)ω ,

The gaps of successive zeroes in (ti)i≥1 are those of the preperiod(t1, t2, . . . , tm) then those of the period (tm+1, tm+2, . . . , tm+p), then occurperiodically up till infinity. The length of such gaps of zeroes is at mostmax{m−2,p−1}. The asymptotic lacunarity is controlled by theperiodicity in this case.If 1 < β < 2 is an algebraic number which is not a Parry number, thesequences of gaps of zeroes in (ti)i≥1 remain asymptotically moderateand controlled by the Mahler measure M(β ) of β , as follows :

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Theorem (Verger-Gaugry, ’06)

Let β > 1 be an algebraic number such that dβ (1) is infinite and gappyin the sense that there exist two infinite sequences {mn}n≥1 and{sn}n≥0 such that

1 = s0 ≤m1 < s1 ≤m2 < s2 ≤ . . .≤mn < sn ≤mn+1 < sn+1 ≤ . . .

with (sn−mn)≥ 2, tmn 6= 0, tsn 6= 0 and ti = 0 if mn < i < sn for all n ≥ 1.Then

limsupn→+∞

sn

mn≤ Log(M(β ))

Logβ

Adamczewski and Bugeaud gave another proof using Schmidt’ssubspace Theorem.

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In Theorem 7 the quotient sn/mn,n ≥ 1, is called the n-th Ostrowskiquotient of the sequence (ti)i≥1.

For a given algebraic number β > 1, whether the upper bound isexactly the limsup of the sequence of the Ostrowski quotient of (ti)i≥1is unknown.

For Salem numbers, this equality always holds since M(β ) = β , andthe upper bound is 1.

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Varying the base of numeration β in the interval (1,2) :

for all β ∈ (1,2), being an algebraic number or a transcendentalnumber, the alphabet Aβ of the β -shift is always the same : {0,1}.

All the digits of all β -expansions dβ (1) are zeroes or ones. Parry hasproved that the relation of order 1 < α < β < 2 is preserved on thecorresponding greedy α- and β - expansions dα (1) and dβ (1) asfollows :

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Proposition (Parry, ’60)

Let α > 1 and β > 1. If the Renyi α-expansion of 1 is

dα (1) = 0.t ′1t ′2t ′3 . . . , i .e. 1 =t ′1α

+t ′2α2 +

t ′3α3 + . . .

and the Renyi β -expansion of 1 is

dβ (1) = 0.t1t2t3 . . . , i .e. 1 =t1β

+t2β 2 +

t3β 3 + . . . ,

then α < β if and only if (t ′1, t′2, t′3, . . .) <lex (t1, t2, t3, . . .).

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For any integer n ≥ 1 the sequence of digits 10n−11, with n−1 times“0” between the two ones, is self-admissible. By the first Proposition itdefines a unique solution β ∈ (1,2). Denote by θ

−1n+1 this solution.

From the second Proposition we deduce that the sequence (θ−1n )n≥2 is

(strictly) decreasing and tends to 1 when n tends to infinity.

The real number θ−12 is the unique root > 1 of the equation

1 = 1/x + 1/x2, that is of X 2−X −1. Therefore it is the Pisot number(golden mean) = 1+

√5

2 = 1.618 . . ..

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Being interested in bases β > 1 close to 1 tending to 1+, we will focuson the interval (1, 1+

√5

2 ] in the sequel. This interval is partitioned bythe decreasing sequence (θ

−1n )n≥2 as

(1,

1 +√

52

]=

∞⋃n=2

[θ−1n+1,θ

−1n

) ⋃ {θ−12

}. (1)

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The following theorem shows that the gappiness of (ti)i≥1 also admitssome uniform lower bound, for all gaps of zeroes. The condition ofminimality on the length of the gaps of zeroes in (ti)i≥1 is only afunction of the interval

[θ−1n+1,θ

−1n

)to which β belongs, when β tends

to 1.

Theorem

Let n ≥ 2. A real number β ∈ (1, 1+√

52 ] belongs to [θ−1

n+1,θ−1n ) if and

only if the Renyi β -expansion of unity is of the form

dβ (1) = 0.10n−110n110n210n3 . . . ,

with nk ≥ n−1 for all k ≥ 1.

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Pf.

Since dθ−1n+1

(1) = 0.10n−11 and dθ−1n

(1) = 0.10n−21, the secondProposition implies that the condition is sufficient.

It is also necessary : dβ (1) begins as 0.10n−11 for all β such thatθ−1n+1 ≤ β < θ

−1n . For such βs we write dβ (1) = 0.10n−11u with digits in

the alphabet Aβ = {0,1} common to all βs, that is

u = 1h00n11h10n21h2 . . .

and h0,n1,h1,n2,h2, . . . integers ≥ 0.

The self-admissibility lexicographic Conditions of Parry applied to thesequence (1,0n−1,11+h0 ,0n1 ,1h1 ,0n2 ,1h3 , . . .), which characterizesuniquely the base of numeration β , readily implies h0 = 0 and hk = 1and nk ≥ n−1 for all k ≥ 1.

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We say that dβ (1) is finite if it ends in infinitely many zeros.

DefinitionLet β > 1 be a real noninteger number and dβ (1) = 0.t1t2t3 . . . its Renyiβ -expansion of 1. The power series fβ (z) :=−1 + ∑i≥1 tiz i of thecomplex variable z is called the Parry Upper function at β .

with all tis in the alphabet {0,1,2, . . . ,bβc}.

DefinitionIf dβ (1) is finite or ultimately periodic (i.e. eventually periodic), then thereal number β > 1 is said to be a Parry number. In particular, a Parrynumber β is said to be simple if dβ (1) is finite.

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Let (X ,Σ,µ) be a σ -finite measure space and let T : X → X be anonsingular transformation, i.e. T is measurable and satisfies : for allA ∈ Σ, µ(A) = 0 =⇒ µ(T−1(A)) = 0. In ergodic theory, by theRadon-Nikodym theorem, the operator PT : L1(X ,Σ,µ)→ L1(X ,Σ,µ)defined by ∫

APT fdµ =

∫T−1(A)

fdµ

is called the Perron-Frobenius operator associated with T .

Let β ∈ (1,θ−12 ), X = [0,1], Σ the Borel σ -algebra and Tβ the

β -transformation. The Tβ -invariant probability measure µ = µβ of theβ -shift, on Σ, is unique, ergodic, maximal [Hofbauer] and absolutelycontinuous with respect to the Lebesgue measure dt , withRadon-Nikodym derivative :

hβ = C ∑n:x<T n

β(1)

1β n+1 , so that dµβ = hβ dt ,

for some constant C > 0.29 / 47

We denote by PTβthe Perron-Frobenius operator associated with Tβ . It

is not compact by nature. The theory of Fredholm and Fredholmdeterminants cannot be applied as such.

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For each function f : [0,1]→ C, let

var(f ) := sup{ n

∑i=1|f (ei)− f (ei−1)| | n ≥ 1,0≤ e1 ≤ e2 ≤ . . .≤ en ≤ 1

},

‖f‖BV := var(f ) + sup(|f |),

and denote by BV the Banach space of functions with boundedvariation :

BV := {f : [0,1]→ C | ‖f‖BV < ∞}.

For g ∈ BV , one can define the following transfer operator

Ltβ ,g : BV → BV , Ltβ ,g f (x) := ∑y ,Tβ (y)=x

g(y)f (y).

We will only consider the case g ≡ 1 in the sequel and put Ltβ := Ltβ ,1.

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DefinitionThe (Artin-Mazur) dynamical zeta function of the β -shift is defined by

ζβ (z) := exp

(∞

∑n=1

Pn

nzn

), Pn := #{x ∈ [0,1] | T n

β(x) = x}

counting the number of periodic points of period dividing n (isolatedpoints).

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Theorem

Let β ∈ (1,θ−12 ). Then,

(i) the Artin-Mazur dynamical zeta function ζβ (z) is nonzero andmeromorphic in {|z|< 1}, and such that 1/ζβ (z) is holomorphic in{|z|< 1},

(ii) suppose |z|< 1. Then z is a pole of ζβ (z) of multiplicity k if andonly if z−1 is an eigenvalue of Ltβ of multiplicity k.

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The β -transformation Tβ is one of the simplest transformations amongpiecewise monotone intervals maps [Baladi Ruelle, Milnor Thurston,Pollicott, Hofbauer Keller, Preston]

When β > 1 is an algebraic integer and tends to 1+, the multiplicity k isequal to 1 for the first pole 1/β of ζβ (z) and for a subcollection ofGalois conjugates of 1/β in a small angular sector containg 1.

The relations between the poles of the dynamical zeta function ζβ (z),the zeroes of the Parry Upper function fβ (z) and the eigenvalues of thetransfer operator Ltβ come from :

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Theorem

Let β > 1 be a real number. Then the Parry Upper function fβ (z)satisfies

(i) fβ (z) =− 1ζβ (z)

if β is not a simple Parry number, (2)

and

(ii) fβ (z) =−1−zN

ζβ (z)if β is a simple Parry number (3)

where N, which depends upon β , is the minimal positive integer suchthat T N

β(1) = 0. It is holomorphic in the open unit disk {|z|< 1}. It has

no zero in |z| ≤ 1/β except z = 1/β which is a simple zero.

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From ergodic theory [Takahashi, Ito Takahashi]

ζβ (z) =1−zN

(1−βz)(

∑∞

n=0 T nβ

(1)zn)

where “zN ” has to be replaced by “0” if β is not a simple Parry number.Since βT n

β(1) = bβT n

β(1)c+{βT n

β(1)}= tn+1 + T n+1

β(1) expanding the

power series of the denominator readily gives :

−1 + t1z + t2z2 + . . . = fβ (z) =−(1−βz)( ∞

∑n=0

T nβ

(1)zn).

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The Parry Upper function fβ (z) admits an Euler product and alogarithmic derivative in terms of a power series with coefficients in Z :

ζβ (z) = ∏x periodic

(1−z order(x)

)−1,

ζ ′β

(z)

ζβ (z)=

∞

∑n=0

Pn+1zn

and, with N defined inthere, we deduce, with “zN ” equal to 0 if β is nota simple Parry number,

fβ (z) =−(1−zN)× ∏x periodic

(1−zorder (x)

),

f ′β

(z)

fβ (z)=−NzN−1(1−zN)−1−

∞

∑n=0

Pn+1zn.

The growth of the sequence (Pn)n≥0 was investigated by Flatto andLagarias in terms of the lap-function of the β -shift as a function of thetwo zeroes of smallest modulus of fβ (z).

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Definition

If β is a simple Parry number, with dβ (1) = 0.t1t2 . . . tm, tm 6= 0, thepolynomial

Pβ ,P(X ) := X m− t1X m−1− t2X m−2− . . . tm

is called the Parry polynomial of β . If β is a Parry number which is notsimple, with dβ (1) = 0.t1t2 . . . tm(tm+1tm+2 . . . tm+p+1)ω and not purelyperiodic (m is 6= 0), then

Pβ ,P(X ) := X m+p+1−t1X m+p−t2X m+p−1−. . .−tm+pX−tm+p+1

−X m + t1X m−1 + t2X m−2 + . . .+ tm−1X + tm

is the Parry polynomial of β .

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DefinitionIf β is a nonsimple Parry number such that dβ (1) = 0.(t1t2 . . . tp+1)ω ispurely periodic (i.e. m = 0), then

Pβ ,P(X ) := X p+1− t1X p− t2X p−1− . . .− tpX − (1 + tp+1)

is the Parry polynomial of β .

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Ex. : Bassino’s convergent family of cubic Pisot numbers

Let k ≥ 3. Consider

fβk(z) =−1+kz +

k−1

∑i=2

((i−1)z i +(k− i +1)zk+i+1)+kzk +zk+1 +kz2k+2.

Observe that the coefficient vector is self-admissible for the Conditionsof Parry.

Then it defines a unique βk > 1 whose inverse β−1k is the unique root of

fβk(z) = 0. How to find it ? its minimal polynomial ?

Then factorize fβk(z), for instance for k = 30. It comes :

fβk(z) = φ2(X )φ3(X )φ5(X )φ6(X )φ10(X )φ15(X )φ30(X )φ31(X )

×(X 3− (k + 2)X 2 + 2kX −k).

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The cyclotomic polynomials count for nothing. The minimal polynomialis necessarily :

Pβk(X ) = X 3− (k + 2)X 2 + 2kX −k ,

which is Pisot.We have : k < βk < k + 1 and limk→+∞(βk −k) = 0.Naıve heights go to ∞ with k .

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Theorem (Schmidt - Bertrand- Mathis)Every Pisot number is a Parry number.

Theorem (Boyd)Every Salem number of degree 4 is a Parry number.

Salem numbers of higher degrees above 10 : Boyd developed aprobabilistic model, based on frequencies of digits in dβ (1) and ondiscriminants of number fields, giving rise to a dichotomy. Some Salemnumbers are Parry numbers, others not.

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Theorem

The real number β > 1 is a Parry number if and only if the Parry Upperfunction fβ (z) is a rational function, equivalently if and only if ζβ (z) is arational function.The set of Parry numbers, resp. of nonParry numbers, in (1,∞), is notempty. If β is not a Parry number, then |z|= 1 is the natural boundaryof fβ (z). If β is a Parry number, with Renyi β -expansion of 1 given by

dβ (1) = 0.t1t2 . . . tm(tm+1tm+2 . . . tm+p+1)ω , t1 = bβc, ti ∈Aβ , i ≥ 2,

the preperiod length being m ≥ 0 and the period length p + 1≥ 1, fβ (z)admits an analytic meromorphic extension over C, of the followingform :

fβ (z) =−P∗β ,P(z) if β is simple,

fβ (z) =−P∗

β ,P(z)

1−zp+1 if β is nonsimple,

where the Parry polynomial is given above.43 / 47

If β ∈ (1,2) is a Parry number, the (naıve) height H(Pβ ,P) of Pβ ,P isequal to 1 except when : β is nonsimple and that tp+1 = bβT p

β(1)c= 1,

in which case the Parry polynomial of β has naıve height H(Pβ ,P) = 2.

ex. : Lehmer’s number β = 1.176280 . . ., the smallest Salem numberknown, dominant root of the degree 10 Lehmer’s polynomial

X 10 + X 9−X 7−X 6−X 5−X 4−X 3 + X + 1

admits :

fβ (z) =−1 + z + z12 + z31 + z44 + z63 + z86 + z105 + z118 + . . .

which is sparse with gaps of length ≥ 10 = dyg(β )−2.The usual degree has disappeared. Appearing : dynamical degree 12+ moderate lacunarity.

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Theorem (Akiyama)

The dominant root γn > 1 of −1−z + zn, for n ≥ 2, is a Perron numberwhich is a Parry number if and only if n = 2,3. If n = 2,3, γ2 = θ

−12 and

γ3 = θ−15 = Θ are Pisot numbers which are simple Parry numbers.

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The Salem numbers found by Lehmer are Parry

deg(β ) β = M(β ) minimal pol. of β dβ (1)

4 1.722 . . . X 4−X 3−X 2−X + 1 0.1(100)ω

6 1.401 . . . X 6−X 4−X 3−X 2 + 1 0.1(02104)ω

8 1.2806 . . . X 8−X 5−X 4−X 3 + 1 0.1(05105107)ω

10 1.17628 . . . X 10 + X 9−X 7−X 6−X 5 0.1(0101018101210181012)ω

−X 4−X 3 + X + 1

The respective dynamical degrees dyg(β ) of the last two Salem numbers β

are 7 and 12, with Parry polynomials of respective degrees 20 and 75, forwhich fβ (z) =

−z20−z19−z13−z7−z + 11−z19 =−1 + z + z7 + z13 + . . . = G7(z) + z13 + . . . ,

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resp.

fβ (z) =−z75−z74−z63−z44−z31−z12−z + 11−z74

=−1 + z + z12 + z31 + . . . = G12(z) + z31 + . . . .

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