ON THE METRICAL THEORY OF A PECULIAR CONTINUED FRACTION EXPANSION

on his 70th birthday

ON THE METRICAL THEORY OF A PECULIAR CONTINUED FRACTION EXPANSION

MARIUS IOSIFESCU and GABRIELA ILEANA SEBE

Consider the transformationT of the unit intervalI= [0,1]dened as T(x) =1[0,1/2)(x) x

1−x+1[1/2,1](x)1−x

x , x∈I,

that was studied by Ito [7], especially from the point of view of ergodic theory.

Our work should be seen as complementary to his. First, we take up the Gauss problem forT in the manner of [6]. This amounts to the asymptotic behaviour ofm T⁻ⁿ(A)

asn→ ∞for probability measuresmλ(Lebesgue measure on theσ-algebraBI of Borel subsets ofI) and setsA∈ BI. Second, we are concerned with the sequence(εn)n∈N₊, N+={1,2, . . .},of{0,1}-valued random variables on(I,BI), whereεn=ε1◦Tⁿ⁻¹, n∈N+, withε1=1_[1/2,1]andT⁰(x) =x, x∈I. Theεn, n∈N+, occur in a peculiar continued fraction expansion of irrationals in I, allowing to recover the so-called Lehner continued fraction expansion see [2]

and [5]. We also consider the random variableszn, n∈N+, dened recursively byzn= 1 + (1−2εn)/(εn+ 1/zn−1)with an arbitrarily prescribedz0. We prove that for anyn∈N+ the conditional probabilityλ(εn+1= 0|ε1, . . . , εn)is equal to (zn+ 1)/(zn+ 2) when z0 = 0. Also, when z0 is arbitrary, (zn)n∈N₊ is a [0,∞)-valued Markov process with transition probability function

P(z, A) = 1 z+ 21A

1

z+ 1

+z+ 1

z+ 21A(z+ 1), z∈[0,∞], A∈ B[0,∞), that has as stationary distribution the σ-nite, innite measure with density 1/(z+ 1)on[0,∞).

AMS 2000 Subject Classication: Primary 11K55;

Secondary 47B38, 60J05, 60J10.

Key words: Continued fraction expansion, Gauss' problem, Perron-Frobenius ope- rator, Markov chain, Markov process.

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56, 465477

1. PREREQUISITES

Consider, cf. Ito [7], the transformationT of the unit interval I = [0,1]

dened as

T(x) =1_[0,1/2)(x) x

1−x +1_[1/2,1](x)1−x

x , x∈I,

with 1A standing for the indicator function of the set A, and the inverse branches V₀ and V₁ ofT dened by

V₀(x) = x

x+ 1, 0≤x <1, and

V1(x) = 1

x+ 1, x∈I,

that are modular transformations. This suggests matrix representations for them, namely,

A₀ =

1 1 0 1

0 + 1·x

1 + 1·x =V₀(x)

and

A1 =

1 1 1 0

1 + 0·x

1 + 1·x =V1(x)

.

Write1_[1/2,1] =ε1 and for any x∈ I deneεn(x) =ε1 Tⁿ⁻¹(x)

,x ∈I, n∈N₊={1,2, . . .}, withT⁰ the identity transformation.

It is easy to check that

(1) x= ε1(x) + (1−ε1(x))T(x)

1 +T(x) = 1−ε1(x) +2ε1(x)−1

1 +T(x) , x∈I.

Hence

(2) x= 1−ε₁(x) + 2ε₁(x)−1

1 + 1−ε1(T(x)) +2ε1(T(x))−1 1 +T²(x)

= 1−ε1(x) + 2ε1(x)−1 2−ε₂(x) + 2ε₂(x)−1

1 +T²(x)

=· · ·

= 1−ε1(x) + 2ε1(x)−1 2−ε₂(x) + 2ε₂(x)−1

2−ε3(x) +

... ⁺2ε_n(x)−1 1 +Tⁿ(x) for any x ∈ I and n ∈ N₊ such that Tⁿ(x) 6= 0. This expansion can be continued indenitely if and only if x ∈ Ω = the set of irrationals in I while

for any x ∈ I \Ω there exists m = m(x) ∈ N+ such that Tⁿ(x) = 0 for all n≥m,and the expansion terminates. Clearly, theε_n, n∈N₊, can be viewed as {0,1}-valued random variables on (I,B_I), where B_I is the Borel σ-algebra inI.

Since (1) is associated with a modular transformation with matrix rep- resentation A_ε1(x), equation (2) can be also written as

(3) x= t_n(x) +Tⁿ(x)u_n(x)

rn(x) +Tⁿ(x)sn(x), x∈I, n∈N, where

r_n(x)s_n(x) t_n(x) u_n(x)

=

n

Y

i=1

A_εi(x), x∈I.

In particular, r₀(x) = 1, r₁(x) = 1, s₀(x) = 0,s₁(x) = 1, t₀(x) = 0, t₁(x) = ε₁(x),u₀(x) = 1,u₁(x) = 1−ε₁(x),x∈I. As

Aεn =

1 1 ε_n 1−ε_n

, n∈N+, we have

rn sn

t_n u_n

=

rn−1 sn−1

tn−1 un−1

1 1

ε_n 1−ε_n

, hence

(4) rn=rn−1+εnsn−1, sn=rn−1+ (1−εn)sn−1

for n ∈N₊. Thus, the sequence (r_n)n∈N+ of positive integers is clearly non- decreasing. It converges to ∞ for x ∈Ωsince in the sequence(ε_n(x))n∈N₊ = (ε₁ Tⁿ⁻¹(x)

n∈N+ the value 1 occurs innitely often while r_n+1 > r_n if ε_n(x) = 1.Consequently, by (3) and taking into account that

(5) |r_nu_n−s_nt_n|=

det

n

Y

i=1

A_εi

= 1, n∈N₊, we have

x− t_n(x) +u_n(x) rn(x) +sn(x)

= 1−Tⁿ(x)

(rn(x) +sn(x)) (rn(x) +Tⁿ(x)sn(x)), hence

x= lim

n→∞

tn(x) +un(x) r_n(x) +s_n(x) for any x∈Ω.

Remark. Let a : I → R be any real-valued function dened on I. We actually have

x= lim

n→∞

t_n(x) +a(x)u_n(x)

rn(x) +a(x)sn(x), x∈Ω, since

x− t_n(x) +a(x)u_n(x) rn(x) +a(x)sn(x)

= |a(x)−Tⁿ(x)|

(rn(x) +a(x)sn(x)) (rn(x) +Tⁿ(x)sn(x)). The importance of the special case a(x) ≡ 1 lies in the fact that the set of convergents

tn(x) +un(x)

rn(x) +sn(x), n∈N+,

is the union of the sets of the regular continued fraction and mediant conver- gents of x. See Ito [7, p. 562].

To conclude this section, let us note that by (2) we have (6) y= 2−ε₁(y−1)+ 2ε₁(y−1)−1

2−ε₂(y−1)+ 2ε2(y−1)−1 2−ε₃(y−1) +

... +2εn(y−1)−1 1 +Tⁿ(y−1) for anyy∈[1,2]andn∈N₊such thatTⁿ(y−1)6= 0. This allows to recover the so-called Lehner continued fraction expansion. Recall that any number y ∈ [1,3/2)∪(3/2,2) has a unique innite continued fraction expansion of the form

b₀+ e₁ b1+ e₂

b2+· · ·

:= [b₀;e₁/b₁, e₂/b₂, . . .],

where (bn, en+1) is equal to either (1,1) or (2,−1), n ∈ N. Cf. Dajani and Kraaikamp [2]. See also Iosifescu and Kraaikamp [5, Subsection 4.4.2]. Clearly, (6) is of this kind, so that using results in the references above we can state the results below.

Proposition 1. Almost everywhere in Ωwe have

n→∞lim 1 n

n

X

k=1

1 2−ε_k=1

2, lim

n→∞

pn

(2−ε1)· · ·(2−εn) = 2, lim

n→∞

1 n

n

X

k=1

εk= 0.

The last equation above implies that the asymptotic theoretical frequency of value 1, resp. 0, in the sequence (εn)n∈N+ is 0, resp. 1.

An explanation of this asymmetry, will be given later in Section 2.

2. THE PERRON-FROBENIUS OPERATOR AND GAUSS' PROBLEM

Daniels [3] showed that the transformation T has a σ-nite, innite in- variant measure µ with density 1/xon (0,1], 0 at x = 0, and the dynamical system(I,B_I, T, µ) is ergodic.

Let L¹(µ) and L^∞(µ) denote the usual L¹- and L^∞-Banach spaces on (I,B_I, µ). In what follows we parallel the discussion in [6], where a similar situation has been considered.

According to the general theory (cf. [1, p. 33]), the Perron-Frobenius operator U_T,µ of T underµtakes L¹(µ) into itself and satises the equation (7)

Z

I

(UT,µh)gdµ= Z

I

(g◦T)hdµ for any h∈L¹(µ) andg∈L^∞(µ). Hence

(8)

Z

I

U_T,µⁿ h gdµ=

Z

I

(g◦Tⁿ)hdµ for any n∈N₊. Let ν_h(A) =R

Ahdµ for h∈L¹(µ) and A∈ B_I. Then U_T,µh can be expressed as a Radon-Nikodym derivative, namely, d(ν_h◦T⁻¹)/dµ. In our case we have

Z

T⁻¹([a,b])

hdµ= Z _b+1^b

a a+1

hdx x +

Z _a+1¹

1 b+1

hdx x

for any 0 ≤ a < b ≤1. After the changes of variable x = _y+1^y and x = _y+1¹ , respectively, the right-hand member of the last equation becomes

Z b a

h y

y+ 1

y+ 1 y

dy (y+ 1)² +

Z b a

h 1

y+ 1

(y+ 1) dy (y+ 1)²

= Z b

a

h y

y+ 1 1

y+ 1µ(dy) + Z b

a

h 1

y+ 1 y

y+ 1µ(dy). It follows that

U_T,µh(y) = 1 y+ 1h

y y+ 1

+ y

y+ 1h 1

y+ 1

forµ-almost ally∈I and anyh∈L¹(µ). Clearly,UT,µ1 = 1even if1∈/ L¹(µ). Actually,U_T,µhmakes sense for any functionh:I →Cbut, of course, without satisfying (7) for all of them.

The Gauss problem for the transformationT amounts to the asymptotic behaviour ofm(T⁻ⁿ(A))asn→ ∞for probability measuresmλ(Lebesgue

measure onB_I) and sets A∈ B_I. It follows from (8) by takingg= 1A that Z

A

U_T,µⁿ hdµ= Z

T⁻ⁿ(A)

hdµ.

Hence

(9) m T⁻ⁿ(A)

= Z

A

U_T,µⁿ fdµ for any n∈Nand A∈ B_I, where

f = dm dµ = dm

dλ / dµ dλ, that is, f(x) =x^dm_dλ(x) for almost allx∈I.

Therefore, the asymptotic behaviour of m(T⁻ⁿ(A)) depends on the asymptotic behaviour ofU_T,µⁿ asn→ ∞.By Proposition 1.1.3 in [1] the trans- formationT is conservative. SinceT is conservative and ergodic, by [1, Exercise 2.2.1, p. 61], we have

(10) lim

n→∞

1 n

n−1

X

k=0

h◦T^k = 0 µ-a.e.,

hence λ-a.e., for any h ∈ L¹(µ). In particular, for h = 1_A with A ∈ B_I and µ(A)<∞ we get

(11) lim

n→∞

1 n

n−1

X

k=0

1_A◦T^k= 0 bothµ-a.e. and λ-a.e.

By dominated convergence, this clearly implies

(12) lim

n→∞

1 n

n−1

X

k=0

m T^−k(A)

= Z

I n→∞lim

n−1

P

k=0

1A◦T^k

n pdλ= 0 for any A∈ B_I such thatµ(A)<∞.This shows that if the limit

(13) lim

n→∞m T⁻ⁿ(A) exists, then it should be 0; and if so then, from (9),

0 = lim inf

n→∞

Z

A

U_T,µⁿ fdµ≥ Z

A

lim inf

n→∞ U_T,µⁿ fdµ, hence

lim inf

n→∞ U_T,µⁿ f = 0 µ-a.e.

on any A∈ B_I such thatµ(A)<∞. This actually means that lim inf

n→∞ U_T,µⁿ f = 0 λ-a.e. in [0,1].

In any case, one can assert that we always have lim inf

n→∞

1 n

n−1

X

k=0

U_T,µ^k f = 0 λ-a.e. inI.

Recall that f above is dened byf(x)≡x^dm_dλ(x)for almost allx∈I.

Coming back to equation (11), let A∈ B_I be such that µ(A) <∞. Let A^c =I\A. Since1_A+ 1_A^c = 1, we have

1 n

n−1

X

k=0

1_A◦T^k+ 1 n

n−1

X

k=0

1_A^c◦T^k= 1.

It follows from (11) that

(14) lim

n→∞

1 n

n−1

X

k=0

1A^c◦T^k= 1 bothµ-a.e. and λ-a.e.

This means that both µ-almost and λ-almost all orbits (x, T(x), T²(x), . . .), x∈Ω, hitA^cwith relative frequency asymptotical to1. Also, similarly to (11), we have

(15) lim

n→∞

1 n

n−1

X

k=0

m T^−k(A^c)

= 1.

Hence, if the limit

(16) lim

n→∞m T⁻ⁿ(A^c) exists, then it should be 1.

Note that (14) makes more precise the assertion following Proposition 1 concerning the asymptotic relative frequency of value 0 in the sequence (εn)n∈N+.

We conjecture that both limits (13) and (16) do not exist, so that (12) and (15) appear to be the solution of Gauss' problem for T.

3. AN INFINITE ORDER CHAIN AND A MARKOV CHAIN Let us dene

z_n= s_n rn

, n∈N.

It follows from (4) that

(17) z_n= (1−ε_n)zn−1+ 1

ε_nzn−1+ 1 = 1 + 1−2ε_n ε_n+ 1/zn−1

,

hence

zn= 1 + 1−2εn

εn+ 1

1 + 1−2εn−1

εn−1 +

... + 1

1 + 1−2ε₂

ε₂+ 1

1 +(1−2ε1)z0

ε1z0+ 1

for any n ∈N₊ with z₀ = s₀/r₀ = 0.(Further on, values z₀ 6= 0 will be also considered.)

Alternatively, for anyn∈N₊ we have (18) z_n+ 1 = (zn−1+ 1) + 1

1−ε_n+ε_n(zn−1+ 1) = 1 εn+ 1−2εn

1 + (zn−1+ 1) , hence

z_n+ 1 = 1

ε_n+ 1−2εn

1 + 1

εn−1 +

...+ 1

ε1+ 1−2ε1

1 + (z₀+ 1) .

Let us note that by (17) and (18) we can assert that z_n= a_nz₀+b_n

cnz0+dn

, with

an bn

c_n d_n

:=

n

Y

i=1

1−εi 1 ε_i 1

, n∈N+, and

zn+ 1 = a⁰_n(z0+ 1) +b⁰_n c⁰_n(z₀+ 1) +d⁰_n,

with

a⁰_n b⁰_n c⁰_n d⁰_n

:=Aε1· · ·Aεn, n∈N+.

We shall now show that the dependence structure of (εn)n∈N+ can be described in terms of the zn, n ∈ N+. First, we dene the cylinder sets or fundamental intervals. For any n ∈ N₊ and any i_k ∈ {0,1}, 1 ≤ k ≤ n, consider the n-tuple i⁽ⁿ⁾ := (i1, . . . , in). Let I(i(ⁿ)) denote the set of

x ∈ Ω for which εk(x) = ik, 1 ≤ k ≤ n. Remark that all n-tuples are ad- missible, to mean that all I(i(ⁿ)), n ∈ N+, are non-empty sets. These sets are called cylinder sets or fundamental intervals. For example, I(0) = 0,¹₂

∩ Ω, I(1) = ₁

2,1

∩ Ω, I(0,0) = 0,¹₃

∩ Ω, I(0,1) = ₁

3,¹₂

∩ Ω, I(1,1) =₁

2,²₃

∩Ω, I(1,0) =₂

3,1

∩Ω, I(0,0,0) = 0,¹₄

∩Ω, I(0,0,1) = ₁

4,¹₃

∩ Ω, I(0,1,1) = ₁

3,²₅

∩ Ω, I(0,1,0) = ₂

5,¹₂

∩ Ω, I(1,1,0) = ₁

2,³₅

∩Ω, I(1,1,1) =₃

5,²₃

∩Ω, I(1,0,1) =₂

3,³₄

∩Ω, I(1,0,0) =₃

4,1

∩Ω, etc. We shall prove that any I i⁽ⁿ⁾

is the set of irrationals from a certain interval with rational end points.

Proposition 2. For any n∈N+ a cylinder set I i⁽ⁿ⁾

is either the set of irrationals in the interval with end points

tn−1

rn−1 and tn−1+un−1/2 rn−1+sn−1/2

when i_n= 0 or the set of irrationals in the interval with end points tn−1+un−1/2

rn−1+sn−1/2 and tn−1+un−1

rn−1+sn−1

when i_n = 1. Here the r_k, s_k, t_k, u_k, k ∈ N₊, are computed using the rules given in Section 1.

Proof. If x∈I i⁽ⁿ⁾

and i_n = 0, thenTⁿ⁻¹(x)∈I₀ while if x∈I i⁽ⁿ⁾ and i_n= 1,thenTⁿ⁻¹(x)∈I₁.By (3) we have

x= tn−1(x) +Tⁿ⁻¹(x)un−1(x) rn−1(x) +Tⁿ⁻¹(x)sn−1(x)

with 0 ≤Tⁿ⁻¹ <1/2 in the rst case and1/2≤ Tⁿ⁻¹(x) ≤1 in the second one, which leads to the assertion in the statement.

Corollary. The Lebesgue measure of a cylinder I i⁽ⁿ⁾ is λ I(i⁽ⁿ⁾)

= 1

2 (rn−1+sn−1/2) (rn−1+δ(in,1)sn−1). Proof. This follows by direct computation using (5).

Proposition 3. For any n∈N₊ we have λ(ε_n+1 = 0|ε₁, . . . , ε_n) = 1 +z_n

2 +zn

and

λ(εn+1= 1 |ε1, . . . , εn) = 1 2 +z_n.

Proof. The equations are simple consequences of the above corollary and equations (4). Clearly,

λ(I(ε1, . . . , εn)) = 1

2 (rn−1+sn−1/2) (rn−1+δ(εn,1)sn−1) and

λ(I(ε₁, . . . , ε_n,0)) = 1

2 (r_n+s_n/2)r_n

= 1

(rn−1+εnsn−1+ (rn−1+ (1−εn)sn−1/2) 2 (rn−1+εnsn−1),

where thernandsn, n∈N+, are computed according to equations (4). Hence λ(ε_n+1 = 0 |ε₁, . . . , ε_n) = λ(I(ε₁, . . . , ε_n,0))

λ(I(ε₁, . . . , ε_n))

= (rn−1+sn−1/2) (rn−1+δ(εn,1)sn−1)

(rn−1+ε_nsn−1+ (rn−1+ (1−ε_n)sn−1)/2) (rn−1+ε_nsn−1)

= (2 +zn−1) (1 +δ(εn,1)zn−1)

(2 (1 +ε_nzn−1) + 1 + (1−ε_n)zn−1) (1 +ε_nzn−1)

= 2 +zn−1

3 + (1 +ε_n)zn−1

. Finally, remark that for all n≥2 we have

(z_n>1)≡(ε_n= 0)≡(zn−1=z_n−1), (zn<1)≡(εn= 1)≡

zn−1 = 1 z_n −1

. Hence

2 +zn−1

3 + (1 +εn)zn−1

=



















2 +z_n−1

3 +z_n−1 = 1 +z_n

2 +z_n if ε_n= 0 2 + 1

z_n −1 3 + 2

zn

−2

= 1 +zn

2 +z_n if εn= 1

for all n≥2 while for n= 1the above equation holds trivially as z0 = 0 and z1 = 1.The proof is complete.

Proposition 3 shows that the sequence(ε_n)n∈N+ is a{0,1}-valued innite order chain, cf. Iosifescu and Grigorescu [4, Section 5.5]. On the other hand, the sequence (z_n)n∈N₊ on (I,B_I, λ) is a Q₊-valued Markov chain starting at z1 = 1 with the following transition mechanism: from state z ∈ Q+ the

only possible transitions are to states 1/(z+ 1) and z+ 1 with probabilities 1/(z+ 2)and (z+ 1)/(z+ 2),respectively.

More generally, we can also consider anR₊-valued Markov process with the same transition mechanism. Clearly, the transition operator U of such a process can be expressed as

U h(z) = z+ 1

z+ 2h(z+ 1) + 1 z+ 2h

1 z+ 1

, z∈R+= [0,∞), for any bounded measurable function on R₊.

Proposition 4. The innite σ-nite Gauss' measure γ on R₊ whose density is 1/(x+ 1), x∈R+, is invariant for the R+-valued Markov process (zn)n∈N+.

Proof. The assertion amounts to the validity of the equationR

R+U hdγ= R

R+hdγ for any boundedγ-integrable function onR₊. This is easy to check as Z

R+

U hdγ= Z ∞

0

f(z+ 1) z+ 2 dz+

Z ∞ 0

f(1/(z+ 1))

(z+ 1) (z+ 2)dz:=I1+I2. Making the changes of variable z+ 1 =w and1/(z+ 1) =w, respectively, in the two integrals above we obtain

I1= Z ∞

1

f(w) w+ 1dw=

Z ∞ 1

fdγ and

I₂= Z 1

0

wf(w) (1 + 1/w)

dw w² =

Z 1 0

fdγ, and the proof is complete.

Let us note that if we consider the transformationτ of (0,∞) dened as τ(x) =

( (1−x)/x if x∈(0,1), x−1 if x≥1,

thenz_i=τ(z_i+1) for anyi∈Nwhile, as easily checked, the Perron-Frobenius operatorU_τ,γ ofτ underγ has the same analytical expression asU, but acting now on functions h∈L¹(γ). Clearlyτ(x) =τ(1/x), x∈(0,∞), and

T(x) =





 1

τ(x) if 0< x≤1/2, τ(x) if 1/2≤x≤1.

Note also that τ was considered more than 45 years ago in another context by Parry [8, p. 153].

Coming back to Proposition 4, we note that the existence of a stationary measure dierent from γ of the R+-valued Markov process (zn)n∈N+ is an open question. Such a measure would be a τ-invariant one, leaving just the possibility of a λ-singular measure. Cf. [1, p. 45] on unicity of a λ-absolutely continuous invariant measure.

Concerning both Perron-Frobenius operators UT,µ and U =Uτ,γ, we re- mark that they are operators dealt with in dependence-with-complete-con- nection theory. Cf. [4], especially Chapters 1 and 5. No known result from that theory can be used in the present case to derive convergence of U_T,µⁿ h or Uⁿhasn→ ∞for certain functionsh. Note that by Proposition 3 the proba- bility λ(ε_n+1 = 0) that ε_n+1 takes on value 0 is equal to Uⁿh(0), n ∈ N, where

h(z) = z+ 1

z+ 2, z∈[0,∞).

This follows from the equations

λ(ε_n+1= 0|ε₁, . . . , ε_n) =λ(z_n+1>1 |z_n) =U1_(1,∞)(z_n) =z_n+ 1

zn+ 2, n∈N₊, λ(ε₁ = 0) = 1

2 =h(0) =

= z₀+ 1 z0+ 2

with z0 = 0.

It follows from Proposition 1 that

n→∞lim 1 n

n

X

k=1

λ(ε_k= 1) = lim

n→∞E_λ 1 n

n

X

k=1

ε_k

!

= 0, hence

n→∞lim 1 n

n

X

k=1

λ(ε_k= 0) = 1.

We conjecture that, moreover, we even have

n→∞lim λ(ε_k= 0) = 1.

Acknowledgement. Marius Iosifescu gratefully acknowledges support from Contract 2-CEx-06-11-97 of the Romanian Authority for Research.

REFERENCES

[1] J. Aaronson, An Introduction to Innite Ergodic Theory. Amer. Math. Soc., Providence, RI, 1997.

[2] K. Dajani and C. Kraaikamp, The mother of all continued fractions. Colloq. Math. 84/85 (2000), 109123.

[3] H.E. Daniels, Processes generating permutation expansions. Biometrika 49 (1962), 139149.

[4] M. Iosifescu and “. Grigorescu, Dependence with Complete Connections and its Applica- tions. Cambridge Univ. Press, Cambridge, 1990.

[5] M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions. Kluwer, Dor- drecht, 2002.

[6] M. Iosifescu and C. Kraaikamp, Metric properties of Denjoy's canonical continued frac- tion expansion. Tokyo J. Math. 31 (2008), 495510.

[7] S. Ito, Algorithms with mediant convergents and their metrical theory. Osaka J. Math.

26 (1989), 557578.

[8] W. Parry, Ergodic properties of some permutation processes. Biometrika 49 (1962), 151156.

Received 3 April 2008 Romanian Academy

Gheorghe MihocCaius Iacob Institute of Mathematical Statistics and Applied Mathematics

Calea 13 Septembrie nr. 13 050711 Bucharest 5, Romania

miosifes@acad.ro and

Politehnica University of Bucharest Department of Mathematics I

Splaiul Independenµei 313 060042 Bucharest, Romania

gisebe@mathem.pub.ro