Élise Janvresse
email: elise.janvresse@univ-rouen.fr

Beno Ît

Thierry De

L A Rue

Benoît Rittaud
email: rittaud@math.univ-paris13.fr

Dynamics of

Keywords: 2000 Mathematics Subject Classification. 11K16, 11J70, 37A45, 37B10 continued fractions, β-expansion. 1

Introduction

In all the paper, λ denotes a real number, 0 < λ < 2. A λ-continued fraction is an expression of the form [a 0 , . . . , a n , . . .] λ := a 0 λ + 1

a 1 λ + 1 . . . + 1 a n λ+ . . .
where (a n ) n≥0 is a finite or infinite sequence, with a n ∈ Z\ {0} for n ≥ 1. This kind of continued fractions has been studied by Rosen in [START_REF] Rosen | A class of continued fractions associated with certain properly discontinuous groups[END_REF], where specific properties are enlightened when λ = λ k := 2 cos(π/k) for some integer k ≥ 3. For 0 < λ < 2, any real number can be expanded in λ-continued fraction, even if the expansion is not unique in general. In this paper we study a transformation associated to a particular expansion in λ-continued fractions, in which we always have a 0 = 0, a 1 > 0 and the signs of the a n 's alternate.

The motivations for the present article stem from several works by the same authors [START_REF] Janvresse | How do random Fibonacci sequences grow?[END_REF][START_REF] Janvresse | Almost-sure growth rate of generalized random Fibonacci sequences[END_REF], where the exponential growth of random Fibonacci sequences with parameter λ is studied. In [START_REF] Janvresse | Almost-sure growth rate of generalized random Fibonacci sequences[END_REF], the case λ = λ k for some integer k ≥ 3 is solved and involves a probability distribution on R + invariant under some dynamics. This measure is defined inductively on generalized Stern-Brocot intervals, whose endpoints are described in terms of finite expansion in λ-continued fraction. A key fact proved in [START_REF] Janvresse | Almost-sure growth rate of generalized random Fibonacci sequences[END_REF] is that the sequence of partitions of R + given by generalized Stern-Brocot intervals for λ = λ k is isomorphic to the sequence of partitions of [0, 1] associated to the expansion of real numbers in base (k -1). In this paper we investigate the link between λ-continued fractions and expansions in non-integer basis, generalizing the correspondence observed in the case λ = λ k .

Roadmap. We introduce in Section 2 a transformation T λ naturally associated to expansion in λ-continued fraction, for which we also give a nice geometrical interpretation. The study of this transformation leads to a symbolic coding of the orbits. In Section 3, we prove that distinct points have different codings, which provides an algorithm to expand any positive real number in λ-continued fraction (see Section 3.3). In Section 4.1 we give a characterization of symbolic coding of orbits for T λ (Theorem 4.4), which enables to prove the conjugacy between T λ and some β-shift, β > 1 (Theorem 4.6). In Section 4.2, we present some properties of the map λ → β(λ): It is increasing and continuous from ]0, 2[ onto ]1, ∞[ but non-analytic (Theorem 4.8 and Corollary 4.22). We end up by raising some open questions on the subject.

Description of the dynamics

2.1. The transformation T λ . We start by defining the homographic functions h and h 0 by setting h(y) := 1 λy ; h 0 (y) := y λy + 1 .

Observe that, when y ranges over [0, ∞[, h 0 (y) increases from m λ 0 := 0 to m λ 1 := 1/λ = h(m λ 0 ). We recursively define the sequence (m λ i ) by setting, while m λ i < λ, m λ i+1 := h(m λ i ).

Lemma 2.1. The sequence (m λ i ) is increasing, and there exists i λ ≥ 1 such that m λ i λ ≥ λ.

Proof. Since h is increasing on [0, λ[ and m λ 0 < m λ 1 , an easy induction shows that (m λ i ) is increasing. If the sequence (m λ i ) were infinite, it would be bounded above by λ, therefore it would converge to a fixed point for h. But, as λ < 2, such a fixed point does not exist.

The sequence (m λ i ) is thus finite, with (i λ + 1) terms satisfying

0 = m λ 0 < m λ 1 < • • • < m λ i λ -1 < λ ≤ m λ i λ < ∞.
We now recursively define the homographic functions (h i ) 0≤i≤i λ by h i+1 (y) := h • h i (y).

Note that, for i < i λ ,

I λ i := h i ([0, ∞[) = [m λ i , m λ i+1
[, and that the function h i has no pole on [0, ∞[. If m λ i λ > λ, the last function h i λ has a pole ℓ λ := h -1 i λ -1 (λ) and

h i λ ([0, ℓ λ [) = [m λ i λ , ∞[. If m λ i λ = λ, h i λ maps [0, ∞[ onto [m λ i λ , ∞[ (
and is thus a degenerate homographic function: h i λ is affine in this case). We will consider h i λ to be defined only on [0, ℓ λ [, where ℓ λ = ∞ in the second case, so that

I λ i λ := h i λ ([0, ℓ λ [) = [m λ i λ , ∞[. From the above, it follows that {I λ i , 0 ≤ i ≤ i λ } is a partition of [0, ∞[. We then define the transformation T λ : [0, ∞[-→ [0, ∞[ by setting ∀x ∈ I λ i , T λ (x) := h -1 i (x).
(See Figure 1.)

h 1 (y) h i λ (y) . . . y h 0 (y) ℓ λ ℓ λ y = T λ (x) m λ 0 = 0 m λ 1 = 1/λ m λ i λ x . . . λ x m λ 1 . . . λ m λ 0 m λ i λ . . . I λ 1 I λ i λ I λ 0

2.2.

Coding of the orbits. The way T λ is defined naturally leads to a symbolic coding of the orbits under the action of T λ , on the alphabet {0, . . . , i λ }: For x ∈ [0, ∞[, we set ω λ (x) := x 0 x 1 x 2 . . . , where x ℓ is the unique element of {0, . . . , i λ } such that T ℓ λ (x) ∈ I λ x ℓ . It will be useful in the sequel to consider the lexicographic order on {0, . . . , i λ } Z+ : If ω = (ω j ) and ω ′ = (ω ′ j ) belong to {0, . . . , i λ } Z+ , we shall note ω ≺ ω ′ or ω ′ ≻ ω whenever there exists ℓ ≥ 0 such that ω ℓ < ω ′ ℓ and ω j = ω ′ j for each 0 ≤ j < ℓ. By

ω ω ′ (or ω ′ ω), we mean ω ≺ ω ′ or ω = ω ′ . Lemma 2.2. If 0 ≤ x < x ′ < ∞, ω λ (x) ω λ (x ′ ). Proof. If ω λ (x) = ω λ (x ′ ), let ℓ be the smallest integer such that x ℓ = x ′ ℓ . Since the functions h -1
i are increasing, it is easily proved by induction on j that for 0

≤ j ≤ ℓ, T j λ (x) < T j λ (x ′ ). Hence T ℓ λ (x) < T ℓ λ (x ′ ) and x ℓ < x ′ ℓ . In Section 3, we prove that in fact ω λ (x) = ω λ (x ′ ) if x = x ′ .
An object of particular interest in the study of T λ will be the limit, as x → ∞, of ω λ (x) (which exists by Lemma 2.2). We shall denote this limit

ω λ (∞) = ∞ 0 ∞ 1 ∞ 2 . . . := lim x→∞ ↑ ω λ (x).
For n ≥ 0 and a 0 , . . . , a n ∈ {0, . . . , i λ }, we define I λ a0...an := {x : x j = a j , 0 ≤ j ≤ n}. It is easily checked by induction that I λ a0...an is either empty or an interval of the form

h a0 • • • • • h an ([0, r a0...an [) where 0 < r a0...an ≤ ∞.
We denote by σ the shift on {0, . . . , i λ } Z+ , so that ω λ (T λ (x)) = σω λ (x).

Lemma 2.3. For each 0 ≤ x ≤ ∞, and each ℓ ≥ 0, σ ℓ ω λ (x) ω λ (∞).

Proof. This is an immediate consequence of the definition of ω λ (∞) and Lemma 2.2.

Remark 2.4.

There exist some links between ω λ (∞), ω λ (λ) and ω λ (ℓ λ ). Let We thus introduce the matrices

ω λ (λ) = a 0 a 1 a 2 . . . If T λ (λ) = 0, then ω λ (ℓ λ ) = σ(ω λ (λ)) = a 1 a 2 . . . If T j λ (λ) = 0 for any j ≥ 1, then ω λ (∞) = (a 0 + 1)ω λ (ℓ λ ) = (a 0 + 1)a 1 a 2 . . .
H := 0 1 -1 λ and H 0 := 1 0 λ 1
respectively associated to h and h 0 . For 0 ≤ i ≤ i λ , let H i := H i H 0 be the matrix associated to h i . An easy induction shows that H i is of the form

H i = P i+1 (λ) P i (λ) P i+2 (λ) P i+1 (λ)
,

where the sequence of polynomials (P i ) is defined by P 0 (X) := 0, P 1 (X) := 1, and P i+2 (X) := XP i+1 (X) -P i (X).

Lemma 2.5. For 1 ≤ i ≤ i λ + 1, P i (λ) > 0, and P i λ +2 (λ) ≤ 0.

Proof. Since, for i < i λ , the function h i has no pole on [0, ∞[, all the P i (λ) are nonnegative for 1 ≤ i ≤ i λ + 1. It follows that, for 1 ≤ i ≤ i λ , P i (λ) > 0 (for, if P i (λ) = 0 and P i-1 (λ) > 0, then P i+1 (λ) < 0). If we had P i λ +1 (λ) = 0, h i λ -1 would be an affine function with positive slope, and then m λ i λ would be ∞, which is not possible. Hence

P i λ +1 > 0. Finally, since [m λ i λ , ∞[= h i λ ([0, ℓ λ [), we have P i λ +2 ≤ 0 (otherwise h i λ would be bounded on [0, ℓ λ [).
2.4. Geometrical interpretation of P i (λ). The key observation is the following: Let θ ∈]0, π/2[ be such that λ = 2 cos θ. Fix two points M, M ′ on a circle centered at the origin O, such that the oriented angle (OM, OM ′ ) equals θ. Let M ′′ be the image of M ′ by the rotation of angle θ and center O. Then the respective abscissae t, t ′ and t ′′ of M , M ′ and M ′′ satisfy t ′′ = λt ′t.

Let us consider the circle centered at the origin with radius R = 1/ cos(θπ/2). We fix on the circle the point M 0 = (0, -R), and define the sequence of points (M i ) such that M i is the image of M i-1 by the rotation of angle θ and center O. Let t i be the abscissa of M i . Observe that t 0 = 0 = P 0 (λ), t 1 = 1 = P 1 (λ) by choice of R, and by induction t i = P i (λ) for all i ≥ 0. (See Figure 2.)

M 1 M 0 M 2 M i λ +1 θ M i λ +2 P 2 (λ) P 1 (λ) P 0 (λ) Figure 2
. Geometrical interpretation of the sequence P i (λ) as the successive abscissae of points on the circle centered at the origin with radius R = 1/ cos(π/2θ), where θ is such that λ = 2 cos θ. Proposition 2.6. Let us define the increasing sequence (λ k ) k≥2 by λ k := 2 cos(π/k).

Then i λ = k -2 for λ ∈]λ k-1 , λ k ] (∀k ≥ 3).
Proof. Recall that i λ is characterized by the fact that P i λ +2 (λ) ≤ 0 and P i (λ) > 0 for 1 ≤ i ≤ i λ + 1. Since for λ = 2 cos θ ∈]λ k-1 , λ k ], we have π/k ≤ θ < π/(k -1), the result of Proposition 2.6 is a direct consequence of our geometrical interpretation.

Besides, we see that the P i (λ)'s are bounded by R = 1/ cos(π/2θ), and that (1) P i (λ) > 1 for 2 ≤ i ≤ i λ , and 0 ≤ P i λ +1 (λ) ≤ 1.

2.5. Geometrical interpretation of T λ . By definition, T λ (x) can be obtained by the following recursive algorithm:

If 0 ≤ x < 1/λ, T λ (x) = h -1 0 (x) = x 1 -λx , else T λ (x) = T λ (h -1 (x)) = T λ λx -1 x .
Suppose that x is written x = t 1 /t 0 , with t 1 ≥ 0, t 0 > 0. By Lemma 6.4 in [START_REF] Janvresse | Almost-sure growth rate of generalized random Fibonacci sequences[END_REF], we can find a circle centered at the origin and two points M 0 and M 1 on this circle with respective abscissae t 0 and t 1 , such that the oriented angle (OM 0 , OM 1 ) equals θ. Let M 2 be the image of M 1 by the rotation of angle θ, and denote its abscissa by t 2 = λt 1t 0 .

If x < 1/λ, then t 2 < 0. We get

T λ (x) = h -1 0 (x) = t 1 t 0 -λt 1 = t 1 -t 2
, and

H 0 t 1 -t 2 = t 1 t 0 .
Else, we have t 2 ≥ 0. We obtain

T λ (x) = T λ (h -1 (x)) = T λ λt 1 -t 0 t 1 = T λ t 2 t 1 and H t 2 t 1 = t 1 t 0 .
We recursively define the points M j on the circle, where M j is the image of M j-1 by the rotation of angle θ. Denote by t j the abscissa of M j and let i(x) be such that t i(x)+1 is the first negative abscissa. Then x ∈ I λ i(x)-1 , and

(2)

T λ (x) = t i(x) -t i(x)+1 = T λ t j+1 t j ∀0 ≤ j < i(x) .
Moreover,

(3)

H i(x) t i(x) -t i(x)+1 = t 1 t 0 .
If we want to iterate T λ in this setting, we have to find a new circle centered at the origin and two points N 0 and N 1 on this circle with respective abscissae -t i(x)+1 and t i(x) , such that the oriented angle (ON 0 , ON 1 ) equals θ. Let us denote by R ′ the radius of the new circle, whereas R stands for the radius of the first circle. (See Figure 3.) Proposition 2.7. With the above notations, we always have R ′ ≤ R. Moreover, there exists K(λ) > 1 such that R ′ ≤ R/K(λ) whenever T λ (x) and T 2 λ (x) do not both belong to I λ 0 , and

T λ (x) / ∈ I λ i λ .
Proof. Let α be the argument of M i(x) , so that t i(x) = R cos α and

t i(x)+1 = R cos(α + θ). Observe that π 2 -θ < α ≤ π 2 because t i(x)+1 < 0 and t i(x) ≥ 0. Let τ be the argument of N 1 , so that t i(x) = R ′ cos τ and -t i(x)+1 = R ′ cos(τ -θ). Observe that, since -t i(x)+1 > 0, τ -θ > -π/2, hence (4) τ > -α. θ θ R ′ t 0 M 1 t i(x) N 0 t i(x)+1 N 1 M 0 t 1 M i(x) M i(x)+1 R 0 O Figure 3. Geometrical interpretation of T λ : The transformation maps x = t 1 /t 0 to t i(x) /(-t i(x)+1
). The figure also illustrates the evolution of the radius of the circle.

We want to estimate the ratio

R R ′ = cos τ cos α = cos(τ -θ) cos(α + θ + π) .
Write the last equality in the form

cos(τ -θ) cos τ = cos(α + θ + π) cos α .
Expanding the cosines yields

(5) tan τ = tan α -2 cos θ sin θ ≤ tan α -λ,
hence τ ≤ α (where equality only holds when tan α = ∞, that is α = π/2). Together with (4), this proves R/R ′ ≥ 1.

We are now going to prove that under the additional hypothesis of the Proposition, there exists δ(λ) > 0 such that

(6) π 2 -θ + δ(λ) < α ≤ π 2 -δ(λ).
Indeed, if T λ (x) and T 2 λ (x) do not both belong to I λ 0 , and

T λ (t i(x) /t i(x)-1 ) = T λ (x), we deduce that t i(x) /t i(x)-1 / ∈ I λ 000 . Therefore t i(x) t i(x)-1 = cos α cos(α -θ) ≥ sup I λ 000 > 0.
The inequality on the righthand side of ( 6) follows.

Assuming moreover that T λ (x) / ∈ I λ i λ , we get by (2)

t i(x) -t i(x)+1 = - cos α cos(α + θ) < min I λ i λ ,
which yields the inequality on the lefthand side of [START_REF] Janvresse | Almost-sure growth rate of generalized random Fibonacci sequences[END_REF].

Recalling that τ > θπ/2, we get by the lefthand side ( 6) that τ > -α + δ(λ). Moreover, (5) together with the righthand side of [START_REF] Janvresse | Almost-sure growth rate of generalized random Fibonacci sequences[END_REF] proves that there exists a constant C(λ) > 0 such that τ < α -C(λ). Therefore, there exists a constant

K(λ) > 1 such that R/R ′ = cos τ / cos α > K(λ).

Distinct points have different codings

The purpose of this section is to prove the following statement:

Theorem 3.1. For all 0 ≤ x < x ′ ≤ ∞, ω λ (x) = ω λ (x ′ )
The first step is an elementary particular case. Lemma 3.2. For any n ≥ 0, for any a 0 , . . . , a n-1 , there is at most one x such that ω λ (x) = a 0 a 1 . . . a n-1 0 . . . 0 . . .. Proof. We first prove that ω λ (x) = 0 . . . 0 . . . ⇒ x = 0. Note that for all 0 ≤ y < ∞, h 0 (y) ≤ y, with equality only when y = 0. If ω λ (x) = 0 . . . 0 . . ., for each n, x = h n 0 (T n λ x) and the sequence (T n λ x) is increasing, and bounded by m λ 1 = 1/λ. Therefore it converges to a fixed point for h 0 , that is 0. This is possible only if T n λ x = 0 for all n.

The result follows by an easy induction on n, using the fact that T λ restricted to an interval I λ i is one-to-one.

As a corollary, we get the following lemma which will be useful.

Lemma 3.3.

There exist infinitely many n such that ∞ n = 0.

Proof. If ω λ (∞) = ∞ 0 ∞ 1 . . . ∞ n 0 . . . 0 . . ., then by definition of ω λ (∞) we would have ω λ (x) = ∞ 0 ∞ 1 . . . ∞ n 0 . . . 0 .
. . for all large enough x. But this would contradict Lemma 3.2.

3.1.

Infinity is unreachable. The second step of the proof deals with the case of ∞.

Proposition 3.4. For x ∈ [0, ∞[, ω λ (x) ≺ ω λ (∞).
For a fixed n ≥ 0, the intervals I λ a0...an form a partition of R + associated to the coding of T λ up to time n. The object of interest is here the decreasing sequence of rightmost intervals (I λ ∞0...∞n ) n in the successive partitions. Observe that an equivalent statement to the above proposition is: For all x ∈ [0, ∞[, there exists n large enough such that x / ∈ I λ ∞0...∞n . Before turning to the proof of Proposition 3.4, we prove a few results about matrices associated to iterates of the homographic functions h i .

3.1.1. Matrices. Lemma 3.5. For each a 0 . . . a n such that I λ a0...an = ∅, the matrix

H a0 • • • H an associated to h a0 • • • • • h an is of the form α β γ δ with β ≥ 0 and δ > 0. Moreover, γ ≤ 0 if a 0 . . . a n = ∞ 0 . . . ∞ n .
Proof. We prove the result by induction on n. This is true for n = 0 by Lemma 2.5

(observe that ∞ 0 = i λ ). Assume the result is true up to n, and consider

α β γ δ = H a0 • • • H an+1 = H a0 H a1 • • • H an+1 .
Observe that Lemma 2.5 also ensures the positivity of the upper left coefficient of H a0 . Hence, by induction hypothesis,

β ≥ 0. Moreover, β δ = h a0 • • • • • h an+1 (0) ≥ 0, thus δ > 0.
If we have γ > 0, since δ > 0 the homographic function associated to the matrix is bounded on [0, ∞[. This is impossible for a 0 . . . a n = ∞ 0 . . . ∞ n , as I λ ∞0...∞n is the rightmost interval of the partition at order n + 1. [START_REF] Blanchard | β-expansions and symbolic dynamics[END_REF]. Since γ n ≤ 0 by Lemma 3.5, we obtain in this case δ n+1 ≤ δ n . On the other hand, if 0 < i < i λ , the pole -δ n /γ n of h ∞0 • • • •• h ∞n is smaller than or equal to the right endpoint P i+1 (λ)/P i+2 (λ) of I λ i . Since the matrix H i has determinant 1, we can write P i+1 (λ) 2 -P i (λ)P i+2 (λ) = 1, which is bounded above by P i+1 (λ) by [START_REF] Blanchard | β-expansions and symbolic dynamics[END_REF]. It follows that

Lemma 3.6. Let H (n) = α n β n γ n δ n := H ∞0 • • • H ∞n . Then δ n+1 ≤ δ n for all n ≥ 0. Proof. Observe that H (n+1) = H (n) H i , where i = ∞ n+1 is the largest index such that the pole -δ n /γ n of h ∞0 • • • • • h ∞n is larger than the left endpoint of I λ i , that is -δ n /γ n > h i (0) = P i (λ)/P i+1 (λ). (See Figure 4.) Considering the lower right coefficient of the product H (n) H i , we get δ n+1 = γ n P i (λ) + δ n P i+1(λ) . If i = i λ or if i = 0, 0 < P i+1 (λ) ≤ 1 by
δ n -γ n ≤ P i+1 (λ) P i+2 (λ) < P i (λ) P i+1 (λ) -1 ,
and δ n+1 ≤ δ n .

3.1.2. Proof of Proposition 3.4. The idea consists in proving that the slope of the upper branch h ∞0 • • • • • h ∞n is always larger than 1, so that each time ∞ n+1 = 0, the left endpoint of the rightmost interval increases by at least m λ 1 = 1/λ (see Figure 4).

Recall that the determinant of

H (n) is equal to 1. Therefore, the second deriva- tive of h ∞0 • • • • • h ∞n is -2γ n /(γ n x + δ n ) 3 , which is nonnegative on [0, -δ n /γ n [ by Lemma 3.5. This proves that h ∞0 • • • • • h ∞n is convex on [0, -δ n /γ n [. Moreover, by Lemma 3.6, we get that (h ∞0 • • • • • h ∞n ) ′ (0) = δ -2 n ≥ δ -2 0 = P i λ +1 (λ) -2 ≥ 1. Hence, (h ∞0 • • • • • h ∞n ) ′ ≥ 1 on [0, -δ n /γ n [. Observe that h ∞0 • • • • • h ∞n+1 (0) = h ∞0 • • • • • h ∞n (m λ ∞n+1 ). I λ ∞ n+1 -δn γn m λ ∞ n+1 -βn δn h∞ 0 • • • • • h∞ n -β n+1 δ n+1 I λ ∞ 0 •••∞n Figure 4.
The upper branch at order n + 1.

If ∞ n+1 ≥ 1, m λ ∞n+1 ≥ m λ 1 = 1/λ. Hence, for all n such that ∞ n+1 ≥ 1, we get that h ∞0 • • • • • h ∞n+1 (0) ≥ h ∞0 • • • • • h ∞n (0) + 1 λ .
But by Lemma 3.3, we know that there exist infinitely many such n's, thus the left endpoint of I λ ∞0...∞n satisfies

h ∞0 • • • • • h ∞n (0) ----→ n→∞ ∞.
This concludes the proof of Proposition 3.4.

3.2.

Proof of Theorem 3.1. It remains to prove that for any 0 ≤ x < ∞, the length of I λ x0...xn goes to 0, where x 0 . . . x n • • • = ω λ (x). We have already dealt with the case when x n = 0 for all n large enough (Lemma 3.2). From now on, we assume that there exist infinitely many n's such that x n = 0. Lemma 3.7. For all n ≥ 0, for all z ∈ I λ x0...xn , there exist real numbers u n (z) and v n (z) such that

(7) T n+1 λ z = u n (z) v n (z)
, where

H x0 • • • H xn u n (z) v n (z) = z 1 .
Moreover, there exists

(ρ n ) n depending only on ω λ (x) such that 0 ≤ u n (z) ≤ ρ n , 0 < v n (z) ≤ ρ n and ρ n goes to 0 as n → ∞.
Proof. Let n ≥ 0 and z ∈ I λ x0...xn . We obtain (7) by iteration of ( 2) and (3). Recalling the geometrical interpretation in Section 2.5, u n (z) and v n (z) can be seen as abscissae of points on a circle, whose radius R n (z) is non-increasing with n. Moreover the initial radius R 0 (z) is a continuous function of z. By taking n large enough so that

x 0 • • • x n ≺ ∞ 0 • • • ∞ n (application of Proposition 3.4), I λ x0...xn is bounded, hence R 0 (z) is bounded on I λ x0.
..xn . Let s n be the number of j ∈ {0, . . . , n-2} such that x j+1 = i λ and (x j+1 , x j+2 ) = (0, 0). Since we assumed that the orbit of x does not end with infinitely many 0's, and since it cannot end with infinitely many i λ 's (consequence of Proposition 3.4), we get s n ----→ n→∞ ∞. For each z ∈ I λ x0...xn , the number of times the hypothesis of Proposition 2.7 are fulfilled up to time n is s n , and whenever they are fulfilled, the radius is divided by at least K(λ) > 1. We thus get the announced result with

ρ n := sup z∈I λ x 0 ...xn R 0 (z) K(λ) -sn .
We say that a finite sequence a 0 , . . . , a n is a standard block if a i = ∞ i for all i < n and a n < ∞ n . By Proposition 3.4, ω λ (x) can be decomposed into standard blocks in a unique way. The interest of standard blocks is enlightened by the following result: Lemma 3.8. If a 0 , . . . , a n is a standard block, then the matrix

H a0 • • • H an associ- ated to h a0 • • • • • h an has nonnegative coefficients. Moreover, I λ a0...an = h a0 • • • • • h an ([0, ∞[).
Proof. We consider the homographic function

h a0 • • • • • h an-1 = h ∞0 • • • • • h ∞n-1
which is the upper branch at order n. Since its pole lies in I λ ∞n , this function is bounded on I λ an , which means that

h a0 • • • • • h an is bounded on [0, ∞[. Moreover, since a n < ∞ n ≤ i λ , I λ a0,...,an = h ∞0 • • • • • h ∞n-1 (I λ an ) = h ∞0 • • • • • h ∞n-1 h an ([0, ∞[) .
In particular, I λ a0,...,an = ∅, thus, by Lemma 3.5, we know that H a0

↑ αx + β γx + δ > 0, hence α > 0.
We are now ready to achieve the proof of Theorem 3.1. Consider n such that x 0 . . . x n is a concatenation of standard blocks. By the previous Lemma, the matrix

H x0 • • • H xn is of the form H x0 • • • H xn = α β γ δ ,
where αδγβ = 1, all the coefficients are nonnegative, and

I λ x0...xn = h x0 • • • • • h xn ([0, ∞[) = β δ , α γ .
Observe that the length of I λ x0...xn is 1/γδ. Our purpose is to prove that γδ is large if n is large enough.

By Lemma 3.7, we have 1 = γu n (z) + δv n (z) for any z ∈ I λ x0...xn . Choose z = β/δ, so that u n (z) = 0. Then δ = 1/v n (z) ≥ 1/ρ n .

Choose now z j = α/γǫ j with ǫ j → 0. By compacity, we can assume that (ǫ j ) is chosen such that u n (z j ) and v n (z j ) converge, respectively to u n and v n . Since lim j→∞ u n (z j )/v n (z j ) = ∞, we have v n = 0. Therefore, γ = 1/u n ≥ 1/ρ n . This concludes the proof of Theorem 3.1.

3.3.

Convergence of λ-continued fractions. The above results can be interpreted in terms of convergence of λ-continued fractions. Starting from x ∈ [0, ∞[, we define, via the coding of its orbit ω λ (x) = x 0 x 1 . . ., a sequence of real numbers having finite expansions in λ-continued fractions: For any j ≥ 0, we consider the left endpoint

m λ x0...xn := h x0 • h x1 • • • • • h xn (0) of the interval I λ x0,.
..,xn . We can recursively construct a finite expansion in λ-continued fractions of these endpoints by observing that, if

y = [0, b 1 , . . . , b ℓ ] λ , then h i (y) = [0, 1, -1, . . . , (-1) i-1 i terms , (-1) i (1 + b 1 ), (-1) i b 2 , . . . , (-1) i b ℓ ] λ .
If y = 0, and its expansion is [0] λ , the preceding formula is to be understood as

h i (0) = [0, 1, -1, . . . , (-1) i-1 ] λ .
We obtain in this way finite expansions in λ-continued fractions, where the first coefficient is zero, the second one is positive, and the signs of the following coefficients alternate. It will therefore be useful to introduce the following notation which corresponds to this particular form of λ-continued fractions: For positive integers (b i ) 1≤i≤ℓ , we set Lemma 3.9. Consider the prefix x 0 . . . x n of the orbit ω λ (x) and write this finite sequence as 0 e0 a 0 0 e1 a 1 . . . 0 e ℓ a ℓ 0 e ℓ+1 , where a i > 0 and e i ≥ 0. Proof. It is an immediate induction using [START_REF] Rényi | Representations for real numbers and their ergodic properties[END_REF].

b 1 , b 2 , . . . , b ℓ λ := [0, b 1 , -b 2 , . . . , (-1) ℓ-1 b ℓ ] λ = 1 b 1 λ - 1 . . . - 1
Note that the number of coefficients in the expansion of m λ x0...xn obtained in this way is 4.1.1. Classical results on β-shifts. For any β > 1, Renyi [START_REF] Rényi | Representations for real numbers and their ergodic properties[END_REF] introduced the βexpansion of a real number 0 ≤ t < 1 as the series

x 0 + • • • + x n ,
t = n≥0 t n β n+1 ,
where the coefficients t n are nonnegative integers defined as follows. We consider the transformation S β : [0, 1[→ [0, 1[, which sends t to βt mod 1; The natural partition of [0, 1[ associated to this transformation is composed of intervals of the form J β j := j β , j+1 β , for 0 ≤ j < β -1, together with J β ⌊β⌋ := ⌊β⌋ β , 1 , where ⌊β⌋ denotes the largest integer smaller than β; The coefficients t n are given by the orbit of t under the transformation S β :

t n = j if S n β (t) ∈ J β j . 0 1 0 1 1/β β -[β] J β 0 J β 1 2/β J β ⌊β⌋ Figure 5. Graph of S β for β = 9/4.
We denote by O β (t) the sequence (t n ) n≥0 coding the orbit of t ∈ [0, 1[ under the transformation S β . As in the case of T λ , the map t → O β (t) is increasing with respect to the lexicographic order of sequences. We denote by O β (1) the limit, as t → 1, of O β (t). Observe that for any n ≥ 0,

σ n O β (1) O β (1).
Parry characterized the sequences which code orbits for S β using the β-expansion of the fractional part β -[β] of β. For our purpose, it is useful to translate his result in terms of O β (1), which can be done by the following lemma.

σ n O β (β -[β]) = O β (1),
which would contradict the theorem.

4.1.2. Characterization of orbits of T λ . As in the case of β-expansions, we can characterize the sequences which code orbits for T λ .

Theorem 4.4. A sequence (x n ) n≥0 codes the orbit of a real number x under the action of T λ if and only if

(10) x n x n+1 • • • ≺ ω λ (∞), ∀n ≥ 0.
Proof. The condition is necessary by Proposition 3.4. Conversely, consider a sequence (x n ) n≥0 satisfying x n x n+1 • • • ≺ ω λ (∞) for any n ≥ 0. This means that this sequence can be decomposed into a concatenation of standard blocks

x 0 x 1 • • • = B 0 B 1 . . . . For any standard block B = b 0 . . . b ℓ , let h B := h b0 • • • • • h b ℓ . By Lemma 3.8, for each k ≥ 0, the interval [u k , v k [:= I λ B0...B k is equal to h B0 • . . . h B k ([0, ∞[). Therefore, v k = h B0 •. . . h B k (∞), and v k+1 = h B0 •. . . h B k (h B k+1 (∞)). Since h B k+1 (∞) < ∞, we get that v k+1 < v k . Hence n I λ x0...xn = k I λ B0...B k = k [u k , v k [= k [u k , v k ] = ∅.
Since the sets of sequences coding orbits for T λ and for S β have the same structure, we are led to find a correspondence between λ and β: In view of the statements of Theorems 4.2 and 4.4, given λ ∈]0, 2[, we want to find β > 1 such that O β (1) = ω λ (∞).

How to find β(λ).

The main tool to find β is the following result by Parry. It also follows from Parry that β is unique whenever it exists. By Lemma 2.3,

σ n ω λ (∞) ω λ (∞) for all n ≥ 1. If ω λ (∞) = ∞ 0 ∞ 1 .
. . is non periodic, then the previous inequality is strict. Therefore, there exists a unique β > 1 with integer part ∞ 0 such that the β-expansion of (β -[β]) is coded by (∞ i ) i≥1 . By Lemma 3.3 and Lemma 4.1, we conclude that O β (1) = ω λ (∞). It remains to study the case where ω λ (∞) is periodic. Let p be the smallest integer such that ω λ (∞) is a periodic repetition of the pattern ∞ 0 . . . ∞ p . We then define ω := ∞ 0 . . . ∞ p-1 (∞ p + 1)00 . . . . We let the reader check that σ n ω ≺ ω for all n ≥ 1. Therefore, there exists a unique

β > 1 with integer part ∞ 0 (∞ 0 +1 if p = 0) such that the β-expansion of (β -[β]) is coded by ∞ 1 . . . ∞ p-1 (∞ p + 1)00 . . . . By Lemma 4.1, we conclude that O β (1) = ω λ (∞).
Eventually, we proved that for any λ ∈]0, 2[, there exists a unique

β > 1 such that O β (1) = ω λ (∞).
By Theorem 4.4, x → ω λ (x) is a one-to-one map from [0, ∞[ onto the set of integer-valued sequences satisfying (10). In the same way, by Theorem 4.2, t → O β (t) is a one-to-one map from [0, 1[ onto the set of integer-valued sequences satisfying [START_REF] Rosen | A class of continued fractions associated with certain properly discontinuous groups[END_REF]. If β is such that O β (1) = ω λ (∞), we get by composition a one-toone map

φ λ sending x ∈ [0, ∞[ to t ∈ [0, 1[, where O β (t) = ω λ (x).
The following diagram commutes:

T λ S β x ∈ [0, ∞[ t ∈ [0, 1[ T λ x ∈ [0, ∞[ S β t ∈ [0, 1[ φ λ φ λ Figure 6.
Moreover, φ λ is increasing, and since it is onto, φ λ is continuous.

Theorem 4.6. For any λ ∈]0, 2[, there exists a unique β = β(λ) > 1 and a homeomorphism

φ λ : [0, ∞[→ [0, 1[ conjugating [0, ∞[, T λ and [0, 1[, S β .
We observed in [START_REF] Janvresse | How do random Fibonacci sequences grow?[END_REF] some connection, when λ = 1, with Minkowski's question mark function. Translated in the context of the present paper, this connection can be written as follows:

φ 1 (x) = 1 2 ?(x) if 0 ≤ x ≤ 1, 1 2 + 1 2 ?(1 -1/x) otherwise.
Since it is well known that the topological entropy of the β-shift is log β (see e.g. [START_REF] Ito | Markov subshifts and realization of β-expansions[END_REF]), we get the following corollary of Theorem 4.6: Corollary 4.7. For any λ ∈]0, 2[, the topological entropy of T λ is equal to log β(λ), where β(λ) is defined in Theorem 4.6.

4.2.

Properties of λ → β(λ). The purpose of this section is to prove the following theorem:

Theorem 4.8. The map λ → β(λ) is increasing and continuous from ]0, 2[ onto ]1, ∞[.

Variation of λ → β(λ).

Lemma 4.9. Let W be a finite sequence of nonnegative integers. The left endpoint m λ W of I λ W is a continuous decreasing function of λ on its interval of definition.

Proof. Lemma 3.9 gives the explicit expression of m λ W in terms of λ-continued fractions. It is easily checked by induction on ℓ that λ → a 1 , a 2 , . . . , a ℓ λ is a decreasing function of λ.

Lemma 4.10. Let 0 < λ < λ ′ < 2. If x ≥ 0 is such that there exists a 0 , . . . , a n-1 satisfying x ∈ I λ a0...an-1 ∩ I λ ′ a0...an-1 , then T n λ (x) < T n λ ′ (x).
Proof. For n = 1, we get the result by an easy induction on a 0 . We then make an induction on n to prove the lemma.

Lemma 4.11. For a fixed x ∈ [0, ∞[, if 0 < λ < λ ′ < 2, then ω λ (x) ≺ ω λ ′ (x).
Proof. Let ω λ (x) = a 0 . . . a n . . ., and

ω λ ′ (x) = a ′ 0 . . . a ′ n . . .. Assume that for some n ≥ 1, a 0 . . . a n-1 = a ′ 0 . . . a ′ n-1 . Then by Lemma 4.10, T n λ (x) < T n λ ′ (x). But T n λ (x) ∈ I λ an = [m λ an , m λ an+1 [, while T n λ ′ (x) ∈ I λ ′ a ′ n = [m λ ′ a ′ n , m λ ′ a ′ n +1 [, thus m λ an < m λ ′ a ′ n +1 . By Lemma 4.9, m λ ′ an ≤ m λ an , therefore m λ ′ an < m λ ′ a ′ n +1
. This proves that a n ≤ a ′ n . It follows that ω λ (x) ω λ ′ (x). It remains to show that the orbits are different. If a 0 = a ′ 0 by Lemma 4.10, we get T λ (x) < T λ ′ (x). Hence by Theorem 3.1, ω λ ′ (T λ (x)) ≺ ω λ ′ (T λ ′ (x)). But we already know that ω λ (T λ (x)) ω λ ′ (T λ (x)). Proposition 4.12. λ → β(λ) is increasing.

Proof. By Lemma 3 in [START_REF] Parry | On the β-expansions of real numbers[END_REF], it is enough to show that for λ < λ ′ , ω λ (∞) ≺ ω λ ′ (∞). Since ω λ (∞) and ω λ ′ (∞) respectively start with i λ and i λ ′ , we can assume that there exists k ≥ 1 such that λ k+1 < λ < λ ′ ≤ λ k+2 , so that i λ = i λ ′ = k (otherwise, we have i λ < i λ ′ which directly gives the result).

We first show that

ℓ λ < ℓ λ ′ . If λ ′ = λ k+2 , then ℓ λ < ∞ = ℓ λ ′ . Otherwise, ℓ λ ′ = T λ ′ (λ ′ )
. By Lemma 4.9, we have

m λ ′ k-1 < m λ k-1 < λ < λ ′ ≤ m λ ′ k < m λ k , hence λ ∈ I λ k-1 ∩ I λ ′ k-1 . Then by Lemma 4.10, ℓ λ ′ = T λ ′ (λ ′ ) > T λ ′ (λ) > T λ (λ) = ℓ λ . Observe now that, since ℓ λ ′ > ℓ λ , σω λ ′ (∞) = lim x↑ℓ λ ′ ↑ ω λ ′ (x) ≻ ω λ ′ (ℓ λ ),
which by Lemma 4.11 is lexicographically after ω λ (ℓ λ ). The conclusion follows by noting that ω λ (ℓ λ ) lim x↑ℓ λ ↑ ω λ (x) = σω λ (∞). 4.2.2. Surjectivity of λ → β(λ). We define the set of sequences which are lexicographically shift maximal (LSM) as:

LSM := W = x 0 x 1 • • • ∈ Z Z+ + : x 0 > 0 and ∀k ≥ 0, x k x k+1 • • • x 0 x 1 • • • . By Lemma 2.3, for any 0 < λ < 2, ω λ (∞) ∈ LSM.
In the same way, we define the set of words of length n + 1 which are lexicographically shift maximal as:

LSM n := W = x 0 • • • x n ∈ Z n+1 + : x 0 > 0 and ∀0 ≤ k ≤ n, x k • • • x n x 0 • • • x n . Lemma 4.13. Let W = x 0 • • • x n ∈ LSM n . Then succ(W ) := min{W ′ ∈ LSM n : W ′ ≻ W }
always exists and is obtained in the following way: Consider the longest strict suffix x k+1 . . . x n of W which is also a prefix of . Then

succ(W ) =          x 0 . . . x k-1 (x k + 1) 0 . . . 0 n-k terms if k = 0 (x 0 + 1) 0 . . . 0 n terms if k = 0. Proof. easy exercise !!! For any λ ∈]0, 2[, let MAX λ n be the prefix of length n + 1 of ω λ (∞): MAX λ n ∈ LSM n and by Lemma 4.11 λ → MAX λ n is non-decreasing. Lemma 4.14. Let W ∈ LSM n . If the set λ ∈]0, 2[: MAX λ n = W is nonempty, then it is an interval of the form λ min W , λ max W .
Proof. Since λ → MAX λ n is non-decreasing, the above set is an interval. Let λ be such that MAX λ n = W and consider x > m λ W . By Lemma 4.9, for λ ′ < λ close enough to λ, x > m λ ′ W , thus MAX λ ′ n W . On the other hand, λ ′ → MAX λ ′ n is non-decreasing. We conclude that MAX λ ′ n = W for all λ ′ < λ close enough to λ.

We denote by I the interior of I. If j = 0, x 0 = i λ and λ = m λ i λ . In this case, we have lim x→∞ T λ (x) = ∞, thus ω λ (∞) = i λ i λ . . . and W = i λ . . . i λ .

Assume now that j ≥ 1. Since λ ∈ Iλ x0...xj-1 , Lemma 4.15 proves that, for all λ ′ close enough to λ, MAX λ ′ j-1 = (x 0 + 1)x 1 . . . x j-1 . On the other hand, T j λ (λ) = b λ xj , x j = 0 and x j+1 = • • • = x n = 0. We have in this case ℓ λ = T λ (λ). Thus,

lim x↑ℓ λ ↑ T j-1 λ (x) = b λ xj and lim x↑ℓ λ ↑ T j λ (x) = ∞. Since lim x→∞ ↑ T λ (x) = ℓ λ , we get lim x↑∞ ↑ T j λ (x) = b λ xj and lim x↑∞ ↑ T j+1 λ (x) = ∞.
This means that the (j + 1)-th term of ω λ (∞) is x j -1 and ω λ (∞) is periodic of period j + 1. This proves that W is the prefix of length n + 1 of the periodic repetition of the pattern (x 0 + 1)x 1 . . . x j-1 (x j -1). Moreover, j + 1 is the smallest period for ω λ (∞). Indeed, if we had a smallest period r, we would have lim x→∞ T r λ (x) = ∞, which would imply by a similar argument that T r-1 λ (λ) = b λ xr-1 . This would contradict the definition of j.

It remains to prove that for any λ ′ > λ close enough to λ, MAX λ ′ n = succ(W ), that is (x 0 + 1)x 1 . . . x j-1 x j 0 . . . 0 when j ≥ 1 and (x 0 + 1)0 . . . 0 when j = 0. Observe that the prefix of length n + 1 of ω λ ′ (λ ′ ) is a right-continuous function of λ ′ by Lemmas 4.9 and 4.10. Therefore, for any λ ′ > λ close enough to λ, ω λ ′ (λ ′ ) starts with x 0 x 1 . . . x j 0 . . . 0 and λ ′ ∈ Iλ ′ x0...xj 0...0 . Applying Lemma 4.15 we conclude the proof. Proposition 4.17. For all n ≥ 0 and all W ∈ LSM n , there exists λ ∈]0, 2[ such that W is a prefix of ω λ (∞).

Proof. Fix n ≥ 0. The smallest sequence in LSM n is W = 10 . . . 0. We first prove that W = MAX λ n for λ small enough. For any λ < 1, i λ = 1. Thus, ℓ λ is the pole of h 1 : ℓ λ = λ(1λ 2 ) -1 which tends to zero as λ ↓ 0. By Lemmas 4.9 and 4.10, for λ small enough, we can ensure that W = MAX λ n . Any W ∈ LSM n is such that {W ′ ∈ LSM n : W ′ W } is finite. Hence, a repeated iteration of Lemma 4.16 gives the desired result. Proposition 4.18. For all W ∈ LSM, there exists λ ∈]0, 2[ such that W = ω λ (∞) if and only if W does not end with infinitely many zeros.

Proof. By Lemma 3.3, ω λ (∞) cannot end with infinitely many zeros. Conversely, let W ∈ LSM and for each n ≥ 0, let W n ∈ LSM n be the prefix of length n + 1 of W . By Proposition 4.17, ]λ min Wn , λ max Wn ] is nonempty. Thus, the decreasing sequence of these intervals has a nonempty intersection if λ min Wn+1 > λ min Wn infinitely often. Since λ → MAX λ n+1 is non-decreasing, the equality λ min Wn+1 = λ min Wn is equivalent to

W n+1 = W n 0.
Proof of Theorem 4.8. By Proposition 4.12, λ → β(λ) is increasing. By Proposition 4.18 and Lemma 4.

1, λ ∈]0, 2[ → β ∈]1, ∞[ is onto. By monotonicity, it is henceforth continuous. 4.2.
3. Particular values of λ → β(λ). We turn in this section to the determination of β corresponding to specific values of λ. The simplest case is when λ = λ k = 2 cos(π/k) for some integer k ≥ 3. We know by Proposition 2.6 that λ k is the largest λ for which i

λ = k -2. Hence ω λ k (∞) is the largest LSM sequence starting with (k -2), that is (k -2)(k -2)(k -2) . . . The corresponding β satisfies O β (1) = (k -2)(k -2)(k -2) . . ., thus β(2 cos(π/k)) = k -1.
Another family of λ's for which we determine the associated

β is λ = 1/ √ k for k ≥ 2. Lemma 4.19. For all k ≥ 1, if λ ≤ 1/ √ k the sequence ω λ (∞) starts with 1 0 . . . 0 k-1 .
Proof. We prove the result by induction on k.

If λ ≤ 1, Proposition 2.6 shows that i λ = 1 hence ∞ 0 = 1, which proves the result for k = 1.

Let k ≥ 1 such that the result holds for k. This means that for λ ≤ 1/ √ k, the upper branch at order k is h

1 • h k-1 0 . The associated matrix is (11) H 1 H k-1 0 = kλ 1 kλ 2 -1 λ .
Observe that the pole of

h 1 • h k-1 0 is λ/(1 -kλ 2 ) ≤ m λ 1 = 1/λ as soon as λ ≤ 1/ √ k + 1. Therefore ∞ λ k+1 = 0 if λ ≤ 1/ √ k + 1.

Note that the function h

1 • h k-1 0 is affine if λ = 1/ √ k by (11). We then have lim x↑∞ ↑ T k 1/ √ k (x) = ∞, hence ω 1/ √ k (∞) is a periodic repetition of the pat- tern 1 0 . . . 0 k-1 . It remains to find the associated β. By Lemma 4.1, O β (β -[β]) is 0 . . . 0 k-2 10 . . . 0 . . . Since β corresponds to λ = 1/ √ k < 1, we know that β < 2 = β(1), thus [β] = 1. Hence β satisfies β -1 = 1 β k-1 . Since β > 1, β(1/ √ k) is the largest real root of X k -X k-1 -1.
4.2.4. Asymptotic behaviour of τ → β(2 cos(π/τ )). As suggested by the particular values of β obtained for λ = 2 cos(π/k), k integer, k ≥ 3, we investigate here some properties of β seen as a function of τ , where τ > 2 is related to λ by the relation λ = 2 cos(π/τ ). By composition, the map τ → β is also increasing, continuous, and sends ]2, ∞[ onto ]1, ∞[. We present in Figure 7 a numerical plot of this map, on which an asymptotic phenomenon appears: As t grows to infinity, we see more and more abrupt steps passing from one integer to the next one, whereas β remains almost constant when τ ranges over an interval ]k -1/2, k + 1/2[. This staircase phenomenon is proved in the following proposition.

Proposition 4.20. For all 0 < ε < 1/2, we have

(12) sup τ ∈]k-1 2 +ε , k+ 1 2 -ε[ β(2 cos(π/τ )) -(k -1) ----→ k→∞ 0.
Lemma 4.21. Let β > 1 be such that O β (1) starts with k j, for some integer k ≥ 2 and some 0 ≤ j ≤ k. Then

β -k + j k ≤ 1 k . 3
β -k = j β + 1 β S β (β -[β]).
Since 0 ≤ S β (β -[β]) < 1, we get the announced result.

Proof of Proposition 4.20. The main tool here is the geometrical interpretation of the transformation T λ developped in Sections 2.4 and 2.5. Let us fix 0 < ε < 1/2, 0 < r < 1/2ε.

Let τ := k + r for some large integer k. Let θ := π/τ . We want to find the beginning of ω λ (∞) = ∞ 0 ∞ 1 . . . for λ := 2 cos θ. By Proposition 2.6, we already know that ∞ 0 = i λ = k -1. We introduce the points M j , 0 ≤ j ≤ k + 1 as defined on Figure 2: These points lie on a circle centered at the origin, M 0 and M 1 have respective abscissae 0 and 1, and M j+1 is the image of M j by the rotation of angle θ. Denote by t j the abscissa of M j : We have t k = R sin kθ > 0 and t k+1 = R sin(k + 1)θ < 0, where R is the radius of the circle. Note that (13) -t k+1t k t k ∼ k→∞ 1 -2r r

To estimate geometrically ∞ 1 , we have to introduce a new circle centered at the origin, and two points N 0 and N 1 lying on this circle so that their abscissae are respectively -t k+1 and t k , and the angle (ON 0 , ON 1 ) equals θ. We again define the sequence (N j ) of points on this circle by successive rotations of angle θ. Then ∞ 1 is the smallest j such that N j+2 has negative abscissa. Let ϕ be the argument of N 1 , so that the argument of N 0 is ϕθ. We have -t k+1t k t k = cos θ -1 + sin θ tan ϕ.

Hence, by (13), observing that θ ----→ k→∞ 0, we get lim k→∞ sin θ tan ϕ = 1 -2r r > 4ε.

When k → ∞, ϕ → π/2 uniformly with respect to r. This implies that ∞ 1 /k → 0 uniformly with respect to r. By Lemma 4.21, we conclude that

β 2 cos π τ -(k -1) ----→ k→∞ 0,
uniformly with respect to r ∈]0, 1/2ε[. It remains to study the case where τ := kr. We now have k -1 < τ < k, and Proposition 2.6 gives here ∞ 0 = k -2. The new points N 0 and N 1 have respective abscissae -t k = -R sin kθ and t k-1 = R sin(k -1)θ. The analog of the estimation (13) is

t k-1 + t k t k-1 ∼ k→∞ 1 -2r 1 -r ,
and the lefthand side is equal to 1cos θsin θ tan ϕ (where ϕ is the argument of N 1 ). We get now ϕ ----→ k→∞ -π/2 uniformly with respect to r. This implies that ∞ 1 /k → 1 uniformly with respect to r. We conclude by using Lemma 4.21.

Corollary 4.22. The map λ → β(λ) is not analytic.

Proof. We know that β(2 cos(π/k)) = k -1 for all integer k ≥ 3. If the map λ → β(λ) were analytic, we would then have β(2 cos(π/τ )) = τ -1 for all real τ > 2. This is clearly not the case by Proposition 4.20.

Open questions

5.1. Invariant measures. It can be shown that in the case λ = λ k , the transformation T λ admits a unique absolutely continuous invariant measure µ, whose density with respect to the Lebesgue measure is dµ dx (x) = 1/x. Can we describe absolutely continuous T λ -invariant measures for other λ's? 5.2. Algebraic properties of λ. Many works have been devoted to the connections between algebraic properties of β and the dynamical properties of the associated β-shift (see e.g. [START_REF] Blanchard | β-expansions and symbolic dynamics[END_REF]). Are these properties also connected to the algebraic properties of the corresponding λ? We can stress the fact that the particular values of λ studied in Section 4.2 are algebraic, and are always associated to an algebraic β. Does the correspondence between λ and β always associate algebraic numbers to algebraic numbers? 5.3. Lazy and random expansions in λ-continued fractions. For β > 1, there are generally many ways to expand a real number t ∈ [0, 1[ in the form t = n≥0 t n β n+1 , where the t n 's are taken from {0, 1, . . . , ⌊β⌋}. The expansion given by iteration of the map S β defined in Section 4.1.1 is sometimes refered to as the greedy expansion, since at each step it uses the largest possible digit. It therefore gives the maximal expansion of t with respect to the lexicographic order. Symmetrically, one can consider the minimal β-expansion of t, given by iteration of the so-called lazy map which at each step outputs the smallest possible digit (see [START_REF] Dajani | Measures of maximal entropy for random β-expansions[END_REF] and references therein). Between the lazy and the greedy expansions, one can find a continuum of possible β-expansions among which the random β-expansions studied in [START_REF] Dajani | Measures of maximal entropy for random β-expansions[END_REF][START_REF]Invariant densities for random β-expansions[END_REF]. Can we develop some expansions in λ-continued fractions corresponding to these notions of lazy and random β-expansions? 5.4. Regularity of λ → β(λ). We proved in Section 4.2 that the map λ → β(λ) is increasing and continuous but non-analytic. A careful look at the details in Figure 7 suggests some kind of self-similarity in the graph. We thus may expect the map to be not even differentiable.

Figure 1 .

 1 Figure 1. Top: Graphs of the homographic functions h i . Bottom: Graph of the transformation T λ . (Here λ = 1.5)

2. 3 .

 3 Matrices associated to the homographic functions h i . Recall that each homographic function can be written in the form x → ax + b cx + d where adbc = 1 and is associated to the matrix a b c d . Composition of homographic functions corresponds to matrix multiplication.

b ℓ λ

 λ With our new notation, we have (8) h i b 1 , b 2 , . . . , b ℓ λ = 1, . . . , 1 i terms , (1 + b 1 ), b 2 , . . . , b ℓ λ .

4 . 1 .

 41 and that the expansion of m λ x0...xn is a prefix of the expansion of m λ x0...xn+1 . Now, Theorem 3.1 can be interpreted as the convergence to x of m λ x0...xn . This allows to write x in the form of an infinite expansion in λ-continued fraction: x = e 0 + 1, 1, . . . , 1 a0 terms , e 1 + 2, 1, . . . , 1 a1 terms , . . . , e ℓ + 2, 1, . . . , 1 a ℓ terms , . . . λ . Moreover, we can deduce from the proof of Theorem 3.1 the speed of convergence of m λ x0...xn to x, which is essentially given by the combinatorics of the sequence ω λ (x) = x 0 x 1 . . . x n . . .: If there exists some n 0 such that x n = 0 for each n ≥ n 0 , then x = m λ x0...xn 0 ; Otherwise, the proof of Theorem 3.1 yields xm λ x0...xn ≤ I λ x0...xn ≤ ρ 2 n , where ρ n is defined in Lemma 3.7. Link with β-shifts 4.Conjugacy between T λ and some β-shift.

Lemma 4 . 1 .↑

 41 If β ∈ Z + , O β (1) = (β -1)(β -1) . . . . If β / ∈ Z + , let (b i ) i≥0 be the coding of the orbit of β -[β]. If (b i ) i contains infinitely many nonzero terms, then O β (1) = [β]b 0 b 1 . . . . Otherwise, O β (1) is a periodic repetition of the pattern [β]b 0 . . . b ℓ-1 (b ℓ -1), where b ℓ is the last nonzero term.Proof. The case where β is an integer is clear. We now assume that β / ∈ Z. Then lim t↑1 S β (t) = β -[β]. If (b i ) i contains infinitely many nonzero terms, the orbit of β -[β] never meets 0, hence S n β is continuous at β -[β]. We thus have lim t↑β-[β] ↑ O β (t) = O β (β -[β]). This proves that O β (1) = [β]b 0 b 1 . . . in this case. On the other hand, if there exists a smaller integer ℓ ≥ 0 such that b n = 0 for all n > ℓ, then S ℓ+1 β (β -[β]) = 0 and S ℓ β (β -[β]) = b ℓ /β, where b ℓ ≥ 1. Therefore, lim t↑β-[β] ↑ S ℓ+1 β (t) = 1, and lim t↑1 ↑ S ℓ+2 β (t) = 1. Then O β (1) is periodic, with period ℓ + 2, and it is easily checked that the periodic pattern is [β]b 0 . . . b ℓ-1 (b ℓ -1).

Proposition 4 . 5 (

 45 Parry [7], Corollary 1). Let b be a positive integer and let (b i ) i≥0 be a sequence of nonnegative integers. There exists β > 1 with integer part b such that the β-expansion of (β-[β]) is coded by (b i ) i≥0 if and only if b n b n+1 • • • ≺ b b 0 b 1 .. . , for all n ≥ 0.

Lemma 4 .

 4 15. Let λ be such that ω λ (λ) starts with x 0 . . . x n . If λ ∈ Iλ x0...xn , then MAX λ ′ n = (x 0 + 1)x 1 . . . x n for any λ ′ close enough to λ.Proof. Since λ ∈ Iλ x0 , we have x 0 = i λ -1, thus ω λ (∞) starts with i λ = x 0 + 1. Moreover, λ = m λ i λ implies that ℓ λ = T λ (λ). Hence, ℓ λ ∈ Iλ x1 , T λ (ℓ λ ) ∈ Iλ x2 , . . . , T n-1 λ (ℓ λ ) ∈ Iλ xn . It follows that MAX λ n = (x 0 + 1)x 1 . . . x n because lim x→∞ T λ (x) = ℓ λ .By continuity with respect to λ of the endpoints of the interval I λ x0...xn , any λ ′ close enough to λ also satisfies the hypothesis of the lemma and the claim follows.

Lemma 4 .

 4 16. Let W ∈ LSM n such that λ ∈]0, 2[: MAX λ n = W is nonempty. For any λ ′ > λ max W close enough to λ max W , MAX λ ′ n = succ(W ). Proof. Let λ := λ maxW and x 0 x 1 . . . := ω λ (λ). By definition, λ = λ max W does not satisfy the conclusion of Lemma 4.15. Therefore, λ / ∈ Iλ x0...xn . Consider the smallest j ≥ 0 such that λ / ∈ Iλ x0...xj .

Figure 7 .

 7 Figure 7. Graph of τ → β(2 cos(π/τ )). Detail of the graph for 10.5 ≤ τ ≤ 11. Proof. We use Lemma 4.1. If O β (1) = kkk . . . , then β = k + 1 = k + j/k. In all the other cases, observe that [β] = k. If O β (1) is a periodic repetition of the pattern kj with j < k, then O β (β -[β]) = (j + 1)000 . . . Hence we have βk = (j + 1)/β, which gives the result since βk ≤ 1. Otherwise, O β (β -[β]) starts with j, which yields


Oleg A Bessonov

Alexander I Fedoseyev

Efficiency

teaching and research institutions in France or abroad, or from public or private research centers.

Efficiency analysis and parallelization of the CNSPACK

implementation of a preconditioned CGS solver for modern multicore computer systems


O17 -Formal and Informal Sectors|Shadow Economy|Institutional Arrangements < O1 -Economic Development

The objective of this paper is to improve estimations of the size and scope of the underground economy by introducing a new approach that combines the advantages of the two most commonly used approaches, i.e., currency demand and MIMIC. The new approach is applied to Germany. Among other things, it is shown that the approach yields improved estimation results. Some policy perspectives are discussed in the concluding section.

I. INTRODUCTION

Estimates of the size of the underground economy by Giles (1999a,b), [START_REF] Schneider | Shadow Economies: Size, Causes and Consequences[END_REF], [START_REF] Schneider | Shadow economies around the world: What do we know?[END_REF], and others, indicate that the underground economy has risen over the last two decades in virtually all countries under investigation. Yet, in general, estimates of the size of the underground economy seem to be sensitive to the underlying approach that is used. Moreover, even if the relative size of the underground economy is virtually the same, the scope of the underground economy, i.e., the structural, social, geographical, etc., dimension of underground activities, may well differ with respect to the countries under consideration. To this extent, the purpose of this paper is to further improve estimates of the size and scope of the underground economies by introducing a new approach that combines the advantages of the two most commonly used approaches (currency demand and MIMIC). This new methodological aspect is applied to Germany.

The remainder of this paper is organized as follows. The next section introduces the methodology and the third section offers some estimation results.

Concluding remarks and some policy perspectives are provided in section four.

II. METHODOLOGY

There are several methods of estimating the size and scope of the underground economy. Surveys of the literature on measuring the underground economy are provided by Feige and Ott (1999) or [START_REF] Schneider | Shadow Economies: Size, Causes and Consequences[END_REF], among others.

In this paper, the size and scope of the underground economy are estimated by F o r P e e r R e v i e w 3 using macroeconomic data. In particular, the currency demand approach and the MIMIC approach are employed and a new combination of the latter two is introduced as a third option. All three approaches are discussed in some detail in the following subsections.

The currency demand approach

The currency demand approach rests on the assumption that cash is the only means of payment used in the underground economy, because cash leaves to traces for the authorities. In addition, it is assumed that taxes are the main reason why people get involved in the underground economy, because in this way they may at least partly avoid paying taxes. Hence, if the tax base is given, increasing (or decreasing) tax rates may induce more (or less) underground economy activities, which in turn may increase (or decrease) the demand for currency.

Thus, an equation for currency demand that includes a measure of fiscal pressure, which is composed of various tax items, and some standard variables such as income, prices and interest rates, is econometrically estimated over time.

Following [START_REF] Mauleón | Demanda de dinero y economía sumergida[END_REF] and Mauleón andSardá (1997, 2000), the demand for currency is specified in log-linear terms, that is:

log(E t ) = 0 + 1 .log(Y Lt ) + 2 .log(P t ) + 3 .c t + 4 .R t + u t , ( 1
)
where E is currency demand, Y L is observed legal income in terms of GDP, P is a price index, c is a measure of fiscal pressure composed of various tax system (3)

Equating ( 2) and (3) and rearranging yields:

Y Ut /Y Lt = exp( 3 .c t / 1 ) -1 3 .c t / 1 , (4)
where the size of the underground economy is explained as a function of fiscal pressure c, and the size of the underground economy, Y Ut , is measured as a fraction of the observed economy, Y Lt .

Note that this version of the currency demand approach neither requires a reference year in which the size of the underground economy is assumed to be zero nor does it require that the velocity of cash circulation is the same in both the legal and the underground sector of the economy. Likewise, if the modeling of currency demand includes additional variables, for example, financial innovations as in Hill and Kabir (2000: 184), it is not necessary to assume that these variables things, the currency demand approach is often criticized on the grounds that it focuses on just one cause of underground activities (taxation), and on just one indicator (fiscal pressure induced by the tax system) of underground activities (Schneider and Enste, 2000: 94-96). For this reason, the MIMIC approach is considered in the following subsection.

The MIMIC approach

In contrast to the currency demand approach, the MIMIC approach allows for several indicator variables and several causal variables in forming structural relationships to explain a latent variable, for example, the size of the underground economy.

Following Giles (1999a,b), Tedds and Giles (2000: 3-4) and others, a standard MIMIC model is formulated as follows. The size of the underground economy, , is defined as a scalar latent variable, with y' = (y 1 , y 2 , ..., y p ) and x' = (x 1 , x 2 , ..., x q )

representing vectors of indicators and causes of respectively. Likewise, and are (px1) and (qx1) vectors of parameters, is a (px1) random error, is a scalar random error and both error terms are assumed to be normal and mutually uncorrelated, with var( ) = , and cov( ) = . The MIMIC model is then specified as: 6) into (5) yields,

y t = t + t , ( 5
) t = ' x t + t . (6)
y t = x t + z t , (7)
where:

= ', (8)

z t = t + t , (9)
and:

cov(z) = ' + . ( 10
)
The rank of the regressor matrix, , of the p-equation multivariate regression model ( 7), is equal to 1 and the error covariance matrix, cov(z), is also constrained. Hence, a normalization of one of the elements of the vector to some pre-assigned value is required prior to the estimation of (7). Estimates of the elements of , and thus of and ', can then be obtained through a restricted Maximum Likelihood estimation. Next, ordinal values for the latent variable, , can be derived from equation ( 6), if the error term is set to zero. However, the ordinal values obtained for must be transformed into cardinal values and this inevitably requires a cardinal reference value for at a sample point, i.e., usually the base year of the period under consideration. But it is important to note that this cardinal reference value or benchmark must be obtained from an external source, for example, the currency demand approach.

Yet, despite its advantages, the MIMIC approach may be criticized for being too sensitive to the causes and indicators under consideration, the benchmark that

F o r P e e r R e v i e w
7 is used and the transformation process that is required. Given the criticism of both the currency demand approach and the MIMIC approach, in the next subsection, it is proposed to combine the two approaches in order to overcome some of the disadvantages.

The joint model

The new methodological aspect introduced here consists of estimating the currency demand model jointly with the MIMIC model. More precisely, expression (1) is considered as an additional equation of the p-equation multivariate regression model ( 7), which yields the (p+1)-equation multivariate regression model ( 11), hereafter referred to as the joint model:

y t = x t + z t ,
log(E t ) = 0 + 1 .log(Y Lt ) + 2 .log(P t ) + 3 .c t + 4 .R t + u t . (11)

Hence, the new aspect consists of adding information to the system of equations in (7) and this should improve the robustness of the estimation. In fact, as the following section shows, a SURE estimation procedure of the joint model ( 11) yields improved estimation results. In addition, the procedure addresses the calibration problem which is usually encountered with a MIMIC model, i.e., converting the resulting index into a series of levels.

III. ESTIMATION RESULTS

F

Underground economy calculations

With bearing the reservations mentioned in section II in mind and by applying the methodology discussed in section II, an estimation of the size of the underground economy in Germany can be obtained. In particular, applying the estimation results shown in Table 1 to equation ( 4), Table 2 (Model 2) to equation ( 5), and The results in Table 4 and Figure 1 confirm the findings of [START_REF] Schneider | Shadow Economies: Size, Causes and Consequences[END_REF] and [START_REF] Schneider | Shadow economies around the world: What do we know?[END_REF] with respect to the relative size of the German underground economy.

IV. CONCLUDING REMARKS AND POLICY PERSPECTIVES

According to the results presented in this paper the relative size of the underground economy in Germany increased over time until 1999 and then dropped back to its 1996 level in 2001. [START_REF] Giles | Causality between the measured and underground economies in New Zealand[END_REF] reports similar movements in the time-path of the New Zealand underground economy between 1987 and 1992, and finds evidence of Granger-causality from the measured to the underground economy. In addition, Giles (1999c) finds no evidence that an increase or decrease in real GDP causes asymmetric responses in the underground economy of New Zealand. Both findings have important implications for the use of fiscal policy as an instrument for controlling the size of the underground economy [START_REF] Giles | The Learning Path of the Hidden Economy: Tax and Growth Effects in New Zealand[END_REF][START_REF] Giles | The Canadian Underground and Measured Economies: Granger Causality Results[END_REF]. Moreover, according to Giles andCaragata (2001: 1863) the ratio of underground to measured GDP and the ratio of total tax revenue to GDP are cointegrated in the case of New Zealand, which implies that any exogenous shock to the system will be absorbed and the two variables never drift apart. As demonstrated in this paper, the exogenous shock "reunification" had a substantial impact on the ratio of underground to measured GDP in Germany. Thus, the German case seems to be particularly suitable for applying the methods of [START_REF] Giles | Causality between the measured and underground economies in New Zealand[END_REF]Giles ( , 1999c) ) and Giles and Caragata Regarding the scope of the underground economy, the results indicate that taxes explain more than a third of the German underground economy. Yet, more than 50 percent of the tax effect is explained by social security taxes. These payments, however, are directly linked to the labour market as they are economically born by both the employer and the employee (50 percent each), but transferred to the responsible social security body exclusively by the employer. As noted, another 10 percent of the underground economy in Germany is explained by other labour market variables. Hence, in the German case, labour market policy may be an important alternative or complement to fiscal policy as an instrument for controlling the size of the underground economy.

Finally, the findings in this paper suggest that both the size and the scope of a country's underground economy may serve as an indicator for a country specific social reform agenda, which in turn would allow for constructing a social reform index. In such an index the scope of the underground economy could well serve as a mirror imaged blueprint for the type of social reforms need in a country, while its relative size and its growth rate would hint at how urgent these reforms are.

Yet, the construction of such an index would certainly require searching for more variables that can be used to determine the scope of the underground economies in order to get a more comprehensive and complete image of the latter. Likewise, the conjecture that the joint model approach yields more efficient estimates, calls for applying the joint model approach to more countries. But these tasks delineate a future research agenda.

  of the economy in a similar way. However, among other

  Germany varies with the model under consideration. However, there are only a few minor differences among the three models. In particular, the MIMIC model yields slightly higher figures than the currency model throughout the entire sample period, whereas the figures of the joint model fall in between, except in 1991, but are usually closer to the figures of the currency model. Over the sample period 1980-2001, the size of the German underground economy peaks in 1999 and this result is true for all three models. The results in Table4also suggest that the reunification of Germany in 1990 had a substantial impact on the growth of the German underground economy. Figure1visualizes the evolution of the underground economy in Germany over time. ***Insert Figure 1 about here***

  . Any confirmation of their results would have important implications for controlling the size of the German underground economy via fiscal policy.

Fig. 1 .

 1 Fig.1. Size of the underground economy in Germany in percent of official GDP

  Given the limited availability of quarterly time series data for several variables, all models have been estimated with annual data from 1980 to 2001, using the EViews 4.1 software package. Data sources and definitions of all variables are provided in the appendix. For example, speculative cash holdings in the legal economy during periods of very low interest rates or other wealth related speculative motives.

		8 9 10
	We have used the natural logarithm of the GDP growth rate, the male labour force ***Insert Table 3 about here***
	participation rate, and the natural logarithm of cash as indicators. The causal
	variables for Germany are: inflation, unemployment, the number of full-time
	employees and various taxes. As noted, we need to constrain the coefficient of
	one variable to solve the models and for this reason we have constrained the male
	The currency demand model labour force participation rate to unity. As a consequence, all values obtained for
	the coefficients must be interpreted relative to this normalization. Also, we have
	o Regarding the currency demand model the estimations demonstrate reasonable statistical properties. Table 1 shows the results obtained from estimating equation F o considered several MIMIC model specifications. Tables 2 shows the results for Germany. F o
	(1).	r r r
		P e P e P e
		e e e
		r r r
		R e R e R e
		v i v i v i
	e w e w Germany. These three variables are taxes (34.4 percent), price indices (21.2 e w
	percent) and labour market variables (i.e. unemployment and fulltime) (10.4
	percent). Moreover, with respect to taxes model 4 suggests that 52.0 percent of the
	tax effect is explained by social security taxes (or payments), 30.8 percent by
	indirect taxes and just 17.2 percent by direct taxes.
	The MIMIC model Finally, Table 3 demonstrates that improved estimation results can be obtained
	with the joint model (11).

***Insert Table

1

about here*** Yet, the values of the income and price elasticity's differ from their usually expected value of unity, and the difference is particularly unexpected with regard to the income elasticity. But it may well be that these values capture those underground activities which are not affected by the evolution of the tax system, i.e., the fiscal pressure variable. An additional or alternative explanation might be that these values capture currency demand motives other than the transaction motive.

***Insert Table

2

about here*** In Table

2

the positive sign of the LogGDP variable in all model specifications indicates that there is a positive relation between the growth rate of the observed real GDP and the size of the underground economy. Based on the statistical properties, our preferred specification is model 2. According to model 2, only three variables explain about two thirds of the size of the underground economy in

Table 3

 3

to equation (11) yields Table 4. ***Insert Table 4 about here*** According to Table 4 the estimated relative size of the underground economy in

Table 1 .

 1 Currency demand model

	F o
	r
	P e
	e
	r
	R e
	v i
	e w

Table 2 .

 2 MIMIC results for Germany

	Model	1	2	3	4
	Indicators Log GDP 0.028 *	0.028 *	0.029 *	0.028 *
		(207)	(207)	(207)	(207)
	MR	1.000	1.000	1.000	1.000
	LogCash	(-) -	(-) -	(-) 0.069 *	(-) -
				(92.7)
	Causes LogP	0.206 *	0.220 *	0.213 *	0.240 *
	(79.4) TotalTax 0.350 *	(17.2) 0.357 *	(15.6) 0.345 *	(14.0) -
	(5.95) Unemp -0.280 *	(6.20) -0.353 *	(5.57) -0.348 *	-0.608 *
	Fulltime	(-6.31) -	(-10.3) -0.047 ***	(-4.02) -0.036	(-4.29) -0.163 *
	D1	0.059 *	(-1.06) 0.061 *	(-0.75) 0.056 *	(-2.74) 0.012 *
	SSTax	(9.21) -	(9.30) -	(7.84) -	(2.52) 0.588 *
					(3.90)
	TdTax	-	-	-	0.194
	TiTax	-	-	-	(0.75) 0.348 *
					(2.22)
	Chi 2	13.14	26.56	21.28	23.45
	p-value	0.86	0.40	0.78	0.42
	RMSEA	0.0	0.0	0.0	0.0
	RMR	0.02	0.05	0.02	0.06
	AGFI	0.85	0.87	0.82	0.87
	A *, ** or *** respectively indicates significance at .01, .05 or .10 for a two-sided
	t-test, t-statistics in parentheses.

Table 3 .

 3 Joint Model results for GermanyA *, ** or *** respectively indicates significance at .01, .05 or .10 for a two-sided t-test, t-statistics in parentheses.

			Submitted Manuscript
	MIMIC	Currency
	Indicators coefficient	Variables	coefficient
	Log GDP MR Causes LogP Unemp	0.028 * (207) 1.000 (-) 0.234 * (14.8) -0.592 *	Constant(U 0 ) LogGDP (U 1 ) LogP (U 2 ) TotalTax (U 3 ) Rc (U 4 ) D1	-17.111 * (-4.51) 0.790 * (5.03) 1.245 * (4.73) 2.678 * (3.74) -0.004 (-1.20) 0.285 *
	D1	(-4.51) 0.011 *	Deuro	(4.17) 0.130 *
	Fulltime	(2.41) -0.146 *		(6.46)
	SSTax	(-2.67) 0.578 *
		(4.33)
	TdTax	0.150
	TiTax	(1.20) 0.315 **
		(2.17)
	Chi 2	23.83
	p-value	0.43
	RMSEA	0.0
	RMR	0.06
	AGFI	0.94

Page 19 of 21

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 4 .

 4 Relative size of the underground economy inGermany (1980Germany ( -2001) )

		Currency	MIMIC (2)	Joint Model
	Year	(% of GDP)	(% of GDP)	(% of GDP)

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UKSubmitted Manuscript

ACKNOWLEDGEMENTS

The authors are indebted to Axel Dreher, Andrea Gebauer, Ignacio Mauleón, Michael Neugart, and an anonymous referee for valuable comments and suggestions. However, any shortcomings are the authors' sole responsibility.

Sardà is grateful to the DAAD for grant No 314/2002.

< O -Economic Development, Technological Change, and Growth, H26 -Tax Evasion < H2 -Taxation, Subsidies, and Revenue < H -Public Economics

DATA SOURCES

Germany

-Cash: Statistisches Jahrbuch 1960-2002, Bundesamt für Statistik; CPI, GDP, Fulltime, MR, Rc, Unemp: OECD Statistical Compendium 1/2002; SSTax, TdTax, TiTax: Statistisches Jahrbuch 1979-2002, Bundesamt für Statistik.


Johnnie Johnson

Alistair Bruce
email: alistair.bruce@nottingham.ac.uk

Jane Yu

The Ordinal Efficiency of Betting Markets: an exploded logit approach

Keywords: market efficiency, betting, information

This paper offers a new perspective on efficiency in betting markets by examining the degree to which finishing order in horseraces corresponds to probabilities inherent in odds across different categories of horserace. The application of an exploded logit procedure reveals a significantly greater degree of ordinal efficiency in higher relative to lower class races. Explanations for the phenomenon include differences in prize-related incentives, cross-market distinctions in information markets and differential opportunity/incentive for market manipulation.

I Introduction

This paper offers a new perspective on the issue of the efficiency of horserace betting markets by the application of an exploded logit procedure to test the reliability of prices (odds) in predicting the order of finish of horses in races. The emphasis on ordinal efficiency which this technique permits offers a new and distinctive insight into the wider efficiency question and reveals important distinctions between markets associated with different classes of race.

From a betting perspective, enriching understanding of ordinal efficiency is important given that bet types which seek to predict the first and second or the first, second and third finishers in a race (e.g. `forecast' and `tricast' bets) are an established and popular feature of both bookmaker and pari-mutuel betting markets in the UK and elsewhere.

Important aspects of the regulatory framework within which British horseracing takes place underscore the significance of the concept of ordinal efficiency. The rules of racing, as explicitly require any horse participating in a race to be ridden so as to gain its best possible finishing position. Horses, trainers and jockeys investigated under the so-called `non-trier's rule' (prior to March 2000, Rule 151;now Rules 155-157, British Horseracing Board/Horseracing Regulatory Authority ( 2006)) may suffer penalties where they have been found to infringe its terms. In theory these regulatory arrangements should offer some reassurance to bettors. They imply that, ceteris paribus, if the regulatory regime operates effectively and if the odds on individual horses reflect accurately their relative chances of success, then the finishing order of horses might be expected to mirror the ascending profile of odds. This said, it is of course the case that the regulatory environment and the sanctions which form part of it, is just one of many potential factors influencing behaviour, performance and ultimately the outcome of a horserace. The particular advantage of the concept of ordinal efficiency introduced in this paper is therefore its explicit focus on the ability of the market to predict the order of finish, and not just the winner, in horseraces.

The paper proceeds as follows. Section II offers a brief discussion of the existing betting market efficiency literature in order to set the context for this contribution. This is followed in Section IV outlines data and methods employed. Section V presents results, which are interpreted in Section VI, and concluding remarks follow.

II Betting Market Efficiency

No topic has received more attention in the academic literature relating to betting markets than that of market efficiency. For half a century, the research agenda in this area has been dominated by investigations regarding the degree to which prices in betting markets reliably embody and reflect information relevant to the market outcome. This has involved consideration of various forms of efficiency (`weak', `semistrong' and `strong') which address the incorporation into prices of different forms or layers of more and less transparent information (for an extensive and recent survey of these areas, see Vaughan Williams ( 2005)). The literature has also embraced the study of betting markets in widely differing international contexts and the study of distinct forms of betting market (e.g. pari-mutuel, see for example, [START_REF] Hoerl | Reliability of Subjective Evaluations in a High Incentive Situation[END_REF], [START_REF] Asch | Racetrack betting and informed Behaviour[END_REF]; bookmaker, see, for example, 1976)). Within the study of particular market forms, a range of approaches to the exploration of pockets of market inefficiency has been employed, including segmentation of market activity by bet type, by timing of bet [START_REF] Kopelman | Towards a Psychology of Parimutuel Behaviour: test of Gluck's Laws[END_REF], [START_REF] Johnson | Gluck's Second Law: An Empirical Investigation of Horserace betting in Early and Late Races[END_REF]), or by type of bettor (e.g. motivation, utility function; see, for example, [START_REF] Snyder | Horse Racing: Testing the Efficient Markets Model[END_REF], [START_REF] Thaler | Parimutuel Betting Markets: Racetracks and Lotteries[END_REF], [START_REF] Bruce | Toward an explanation of betting as a leisure pursuit[END_REF] Busche ( 1994)).

Momentum in relation to the study of efficiency issues has been sustained in recent years by a number of factors. These include the development of traditional betting markets into new territories in terms of betting events, new forms of bet and betting market e.g. betting exchanges, spread betting and significant reconfiguration of regulatory arrangements in relation to betting. Each of these has prompted new and wider research into efficiency characteristics across a broad range of geographical and sporting contexts (see, for example, [START_REF] Gander | Searching for the favourite-longshot bias down under: an examination of the New Zealand pari-mutuel betting market[END_REF] (New Zealand horseracing); [START_REF] Sobel | An examination of the empirical derivatives of the favourite-longshot bias in racetrack betting[END_REF] (American greyhound racing); [START_REF] Dare | Efficiency in the NFL betting market: modifying and consolidating research methods[END_REF]Holland (2004), Boulier et al (2006) (American football), [START_REF] Winter | Do horses like vodka and sponging? -On market manipulation and the favouritelongshot bias[END_REF] (German horseracing) and for a collection of articles which reflects the diversity of themes relating to contemporary betting Whilst the volume and range of work in the area of betting market efficiency is clearly extensive, therefore, there is a reasonable consensus around two themes. First, there are established biases in betting markets in terms of patterns of misalignment between subjective and objective probabilities which appear generally (though not universally) robust to different market and national contexts. Second, despite the existence of these biases, the opportunities for developing operationally viable and profitable betting strategies appear highly limited.

As noted above, the dominant basis for evaluating efficiency in horserace betting markets has, in simple terms, relied on comparing the probabilities of success implied by particular sets of odds with the actual probability of success of horses with those odds, as revealed by race outcomes. This allows an insight into the reliability of odds in predicting winning outcomes, but it says nothing directly about the reliability of odds in predicting the order in which horses finish within a particular race or set of races. As such, it neglects a dimension of market efficiency, the ability of market prices to rank effectively the probability of outcomes within a market. This is where the application of an exploded logit procedure has the potential to enrich our understanding. The significance of this for rank-based bet types and in relation to the regulatory framework of horseracing was noted above. It is also worth noting here that the odds of a horse in a UK bookmaker market, whilst directly reflecting the market's view of its chance of winning and determining the return to a winning bet, also affect returns to bets invested on a horse to finish in second, third or fourth place. Depending on the race conditions, odds applied to a `placed' horse derive directly from the odds in the `win' market. For example, in a handicap race of 14 runners, the return to a placed horse (positions 2 and 3) is one quarter of its `to win' odds, whereas for a non-handicap race with eight runners, the returns to a placed horse (positions 2 and 3) are calculated as one fifth of its `to win' odds. These formulae are part of the set of rules which are overseen by the Horseracing Regulatory Authority, the governing body of UK racing, and which apply to all licensed bookmakers operating in the UK.

It is clear from the above that, in bookmaker markets, an implicit assumption is that the probability of a horse being placed is a function of the probability that the horse will win the race. This adds further significance to the testing for ordinal efficiency investigated here. In particular, the application of the exploded logit model challenges the simplistic assumption implicit in the [START_REF] Harville | Assigning Probabilities to the outcome of Multi-Entry Competitions[END_REF] approach, whereby the relationship between a horse's finishing position and its performance relative to other horses is seen as a given, irrespective of that finishing position.

III The Exploded Logit Technique

In general terms, the exploded logit technique offers a basis for analysing ranked data by decomposing a single observation of J ranked alternatives into J-1 `pseudo-observations' of rankings. This involves, initially, discarding the first-ranked alternative from the original observation and then repeating this process, discarding the subsequently first-ranked alternatives to leave successively smaller sets of ranked alternatives. Essentially, as [START_REF] Chapman | Exploiting Rank Ordered Choice Set Data within the Stochastic utility Model[END_REF] note, this explosion process, based on [START_REF] Luce | Preference Utility, and Subjective Probability[END_REF] Ranking Choice Theorem, exploits additional information inherent in the rank ordered choice sets. [START_REF] Bolton | Searching for positive returns at the track: a multinomial logit model for handicapping horse races[END_REF] point to the benefits in terms of more precise parameter estimates resulting from an increase in the number of independent choice sets available for analysis.

In the context of horserace betting data, taking the finishing order of J horses in a race as the initial observation, we remove the winner to produce a second `pseudo-observation' where the second placed horse is the `winner'; in a similar way, removal The exploded logit procedure enables testing of the reliability of the odds in correctly ranking the order of finishers in a horserace as successive elimination of winners reaches further down the ranking and the original observation `explodes' into multiple observations. The greater the `depth' of explosion possible, that is to say the degree to which successive subsets of data can be viably pooled as independent events, the more reliable is the ranking. The focus of interest in this study is on the efficiency of the market in predicting the order of finish of the first three horses in a race. This reflects the fact that, in the UK market, bet types which relate to the order of finishing are confined to prediction of first and second or first, second and third places, as noted above.

In framing the empirical enquiry, we draw on previous research relating to the efficiency characteristics of horserace betting markets which has investigated efficiency distinctions between different classes of race [START_REF] Vaughan Williams | Why is there a Favourite-Longshot Bias in British Racetrack Betting Markets[END_REF]).

The class of a horserace, as identified by its official designation under the auspices of the British Horseracing Board (now part of the Horseracing Regulatory Authority), reflects the fact that different races are designed to attract horses of differing levels (Bruce and Johnson (2005)). The explanation for these differing degrees of efficiency relates principally to the differences in the information environments between the high and low grade categories. Specifically, the higher class races (and associated betting markets) attract wide media interest and analysis where, prima facie, the opportunity to conceal, and benefit from, privileged information would appear to be limited. The latter category represents the opposite extreme, featuring low class events with little or no attendant media coverage. The contention, put simply, is that in betting markets associated with this latter category, the opportunity to conceal and profit Accordingly, drawing on this earlier class-based literature, the central proposition to be examined in this paper is whether, in terms of ordinal efficiency, the relative efficiency characteristics of, respectively, high grade and low grade races mimic the relative efficiencies in terms of the alignment of subjective (odds-based) and objective (revealed) probabilities. More specifically, are markets in higher grade races more ordinally efficient than their lower grade race equivalents? In pursuing this issue, the categories employed to probe cross-category distinctions in Bruce and Johnson (2005), is replicated here; the set of races is split between two separate categories; race classes A-C and race classes D-H (non-handicap only), this reflecting the system of race classification in operation when the races under scrutiny took place. The exclusion of handicap races in categories D-H requires explanation. The basis for focusing on non-handicap lower grade races is that the terms of handicap races, at any level, are determined by the decisions of the independent handicapper, whose role is to assign weights to horses to induce a competitive contest. In non-handicap races, however, this independent moderating influence is absent and the terms on which non-handicap races are run may be partly determined by horses' `connections' (owner, trainer). For horses' weights may be influenced by the value which connections place on the horse (in terms of the price for which the horse may be bought after the race). This element of discretion vested in connections, may create an incentive to influence race terms in order to profit from subsequent betting activity. It should be noted, in emphasising this point, that the financial rewards (in terms of prize money) in non-handicap low grade races are lower than in any other race type, so that the potential betting returns/prize money ratio in such events is significantly greater than for higher grade races.

IV Data and Method

The data employed are drawn from a dataset of races run at 893 race meetings, held between August 29 th 1997 and 31 st July 1998 inclusive at UK racecourses. Races with fewer than three runners were excluded from the sample, as were races where horses tied for first, second or third place and races where a horse or horses were withdrawn. This gave a total of 2184 races The basis of the exploded logit technique is the development of a conditional logit model to estimate the `winningness' of a horse. The `winningness index' is defined as follows: and where

ij odds j i ij p w + = ) l n( , (1) where
= = h i ij ij odds j i q q p 1 ,
1 1 + = ij ij o q ,
is the parameter to be estimated, ij is the error term. odds j i p , is the probability of winning (implied by the starting price) of horse i in race j; h is the total number of horses in race j and ij o is the starting price of horse i in race j.

Consequently, we can predict the probability of horse i winning race j ( j i p , ) as follows: The parameter is estimated by maximizing the joint probability of observing the results of all J races in the sample.

Formally, we define E as the depth of explosion, and J(E) refers to the number of independent choice set observations that could be generated from j races. There are two tests which address the issue of the appropriate depth of explosion.

The first, a chi-squared statistic based on [START_REF] Watson | Transferability of Disaggregate Mode Choice Models[END_REF] involves grouping observations by depth of explosion and sequentially testing the viability of pooling observations of successive explosion depths.

To test if a move from E=1 to E=2 is appropriate, we test the null hypothesis as:

) 2 ( ) 1 ( 0 : = H { } )] ( ) ( [ ) ( 2 ) 2 ( ) 1 ( ) 2 1 ( 2 = + = = = + L L L (2)
where

)
= = = + L L L
are the log-likelihood functions for corresponding estimations. This test statistic will be asymptotically distributed 2 with N degrees of freedom. The degree of freedom in our tests is 1, corresponding to the number of variables in the conditional logit model. [START_REF] Wald | Tests of Statistical Hypotheses Concerning Several Parameters, when the Number of Observations is Large[END_REF].

Similarly, to test if a move from E=2 to E=3 is appropriate, we test:

) 3 ( ) 2 ( 0 : = H { } )] ( ) ( [ ) ( 2 ) 3 ( ) 2 1 ( ) 3 2 1 ( 2 = + = = = + + + L L L
(3)

The second test involves calculating the value of the likelihood ratio index for successive depths of explosion such as:

) 0 ( ) ( 1 2 = = = L L R ~(0,1) (4)
where

) ( = L
is the value of the log-likelihood function at the estimated parameters and ) 0 ( = L is its value when all the parameters are set equal to zero.

Here, given that the index is not dependant on the number of observations, calculated R squared values should remain approximately constant as depth of explosion increases unless a

V Results

The results indicate that there is a significant distinction between high and low-grade races in terms of the depth of viability of explosion. Table 1 reports the log likelihood ratio and the chi-squared statistic.

[Table 1 about here]

For race classes A-C, the 2 statistic is 1.954 for an explosion depth of two (E=2), and 2.994 for an explosion depth of three.

(E=3). Neither value is rejected by the null hypothesis, given the critical value of 2 (1) is 6.635 at the 1% level of statistical significance. It suggests the reliability of exploding the data to a depth of 3.

For race class D-H (non-handicap only), the 2 statistics is 23.864 for E=2 and 42.9174 for E= 3. Both are rejected by the null-hypothesis. [Table 2 about here]

The results demonstrate that, for race classes A-C, with the increasing depth of explosion, R-squared is increasing slightly.

VI Interpretation

The effects revealed via the application of the exploded logit technique to different class-based race groups echo efficiency distinctions reported in earlier work (Bruce and Johnson (2005)). Specifically, there is evidence that, in terms of ordinal efficiency, the higher grade race category outperforms the lower class group over the critical range, in terms of rank-based betting options, of horses placed first, second and third.. where the results offer at least prima facie grounds for suggesting that the incentive to comply with the rules of racing may be less sharp than it is in higher grade events.

Explanations advanced in relation to the earlier results centred

  UK by the Horseracing Regulatory Authority,

  an introduction to the exploded logit technique.

  , including efficiency studies, see the Special Issue of this Journal, (2008, Volume 40:1)).

  placed horse produces a further observation, where the third placed horse is the `winner'.

  so that the aggregate population of racehorses is provided with regular opportunities to compete. As well as being highly correlated with the ability of participating horses, race classes are characterised by an extended hierarchy of prize money. When the races under scrutiny in this study took place, the class system designated races on a scale of A (highest) to H (lowest), with each category populated by both handicap and non-handicap races.A feature of class-based discrimination between sets of races has been an identifiably greater susceptibility of low class races to a well-established aspect of inefficiency, favourite-longshot bias (the propensity to overbet on long odds horses and underbet on short odds horses), compared with high class races

  is greater. A full exposition of this argument is provided inBruce and Johnson (2005).

  certain lower class events such as claiming races,

  and 26060 runners, 588 races (7030 runners) in Class A-C and 1596 races(19030 runners) in Class D-H (non-handicap). Race outcomes, including the order of finishers in each race, are range of sources. For the purposes of this study the statistical records published in the specialist horseracing media were employed.

  introduces `noisy' observations, in which case R squared values would decline significantly, signalling the inappropriateness of that round.

  (i.e., for classes A-C, R-Squared for E=1 is 0.40674 and it increases to 0.41115 for E=2 and 0.41984 for E=3); On the contrary, for race classes D-H (non-handicap only), the Rsquared is gradually decreasing.For E=1, E=2, E=3 respectively, R-squared is 0.45026, 0.42989 and 0.42063. The results from both tests indicate clearly that the viability of explosion for high class races exceeds that of low grade, nonhandicap races, across those finishing places relevant to rankbased betting activity.

  informational characteristics of the race groups and in particular the differential opportunities to profit from privately-held information across the two groups. In particular, the transparency of the betting environment for high class races was regarded as likely to inhibit strategies based on the retention and exploitation of privileged and valuable information, whereas profitable exploitation of such information advantages was regarded as viable in markets relating to lower grade events. The nature of the efficiency measure employed here is, of course, different. As such, the interpretation is required specifically to explain why the market odds constitute a more powerful basis for predicting ranking of horses in a race for high class vis-à-vis low class races.One element of explanation may relate to differential incentives to secure the highest possible finishing place for a horse as between high and low class races. Invariably, the class of a race and the prize money associated with winning or being placed in a race are highly correlated. The highest grade races in the UK feature levels of winning prize money which differ from the lowest grade prize equivalent by a factor of approximately 200.Prizes for placed horses (sometimes extending to awards for sixth place) in major events can significantly exceed winning prize money for low grade races. As such, the incentive to of a place prize in a high class event is significantly more potent than that in a low class race. This in itself would suggest that placed positions in high class races are more vigorously competed for than in low class races. Indeed, in explaining differential bias across race classes,Bruce and Johnson (2005) point to the relative financial benefits from betting vis-à-vis prize money in low class events.If the financial (prize money) incentives associated with achieving the best possible finishing place are more potent in high class races, then it could also be argued that the sanctions for not maximising finishing position are also less severe in low class races. This reflects the fact that media and public scrutiny of low class events is much less intense than for high class equivalents; the probability of `non-trying' behaviour being identified and acted upon by the horseracing authorities is consequently smaller.Rather more mundanely, it might be argued that the informationally impoverished betting environment in low class races might be expected to generate a more random pattern of betting activity compared with high class, information rich races, a factor which might help to explain the weaker ranking performance of prices in lower class events.

  which lower class races may offer holders of privileged information (insiders) greater opportunities to benefit from its exploitation than high class events is via market manipulation in lower grade races. In markets with relatively low betting volumes, in particular, which tend to be a characteristic of lower class races, there may exist an incentive to induce particular patterns of betting in the wider population which are ultimately to the advantage of the manipulator.[START_REF] Bruce | Trading Volume and Betting Market Efficiency: The Role of Insiders[END_REF] identify two types of strategy in this context, depending on the `insider' view of the demographic composition of the betting market. With each strategy, the point is that insiders send signals into the market, via their investment behaviour, which are unrelated to their real perceptions of likely outcome, with a view to triggering responses from the general population of bettors. These responses contribute to more advantageous rates of return to insiders, but because they are essentially `false signals' they introduce informational inefficiencies into the set of price data, or odds. These are evidenced in terms of a distribution of odds in lower grade races which more closely approximates to a random pattern of betting activity. exploited the exploded logit technique to develop a richer understanding of the efficiency characteristics of betting markets. This technique permits analysis of the degree to which the order of finishers in a horserace reflects the order inherent in the odds of horses in the associated betting market (`ordinal efficiency'). This perspective adds to more established views of market efficiency which rely more on comparing the objective (i.e. realised) probabilities of success for horses across different odds/odds ranges with the subjective probabilities embedded in the odds.The results point to significant differences in across established groups of races defined in terms of the standard industry classification, with higher class races displaying marked superiority in ordinal efficiency. Possible explanations for this effect are considered, including differences in prize-related incentives, informational environments and opportunities for market manipulation.In terms of the implications of this study for those directly involved in the racing and betting industries, the superior ordinal efficiency in the higher class race category would who engage in rank-based betting in such races may benefit from the signals embodied in price information, whereas such signals appear unreliable in lower class events. From the regulatory perspective, there may be grounds for considering increased scrutiny of lower grade races,

Table 2

 2

presents the R-Squared results for race classes A-C and race classes D-H (non-handicap only).

Table 2 . Conditional Logit Model Results--Estimated with Explosion Depths of 1,2 and 3 for each Race Category This

 2 table reports the estimated results using conditional logit models with explosion depths of 1, 2 and 3 for the two race categories (A-C and D-H).

	Class ABC
	The Depth of Explosion		Standard Error	T-ratio	R-Squared
	E=1 E=2	1.167277 1.098443	7.06E-02 4.97E-02	16.536 22.093	0.40674 0.41115
	E=3	1.049485	4.05E-02	25.926	0.41984
	Class D-H (non-handicap only)
	The Depth
	of Explosion E=1	1.28337	Standard Error 3.31E-02	T-ratio 38.821	R-Squared 0.45026
	E=2 E=3	1.17194 1.08882	2.26E-02 1.82E-02	51.641 59.658	0.42989 0.42063

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript