Ergodic properties of generalized L¨ uroth series

LXXIV.4 (1996)

Ergodic properties of generalized L¨ uroth series

by

Jose Barrionuevo (Mobile, Ala.), Robert M. Burton (Corvallis, Ore.),

Karma Dajani (Utrecht) and Cor Kraaikamp (Delft)

1. Introduction

1.1. L¨uroth series. In 1883 J. L¨ uroth [Lu] introduced and studied the following series expansion, which can be viewed as a generalization of the decimal expansion. Let x ∈ (0, 1]. Then

(1) x = 1

a

+ 1

a

(a

− 1)a

+ . . . + 1

a

(a

− 1) . . . a

(a

− 1)a

+ . . . , where a

≥ 2, n ≥ 1. L¨ uroth showed, among other things, that each irra- tional number has a unique infinite expansion (1), and that each rational number has either a finite or a periodic expansion.

Underlying the series expansion (1) is the operator T

: [0, 1] → [0, 1], defined by

(2) T

x :=

 1 x

 1 x

 + 1

 x −

 1 x



, x 6= 0; T 0 := 0

(see also Figure 1), where bξc denotes the greatest integer not exceeding ξ.

For x ∈ [0, 1] we define a(x) := b1/xc + 1, x 6= 0; a(0) := ∞ and a

(x) = a(T

x) for n ≥ 1. From (2) it follows that T

x = a

(a

− 1)x − (a

− 1), and therefore

x = 1 a

+ 1

a

(a

− 1) T

x

= 1

a

+ 1

a

(a

− 1)a

+ . . . + T

x

a

(a

− 1) . . . a

(a

− 1) .

Since a

≥ 2 and 0 ≤ T

x ≤ 1 one easily sees that the infinite series from (1) converges to x.

1991 Mathematics Subject Classification: 28D05, 11K55.

[311]

Recently S. Kalpazidou, A. and J. Knopfmacher introduced and studied in [K

1,2] the so-called alternating L¨uroth series. For each x ∈ (0, 1] one has

x = 1

a

− 1 − 1

a

(a

− 1)(a

− 1) + . . .

+ (−1)

a

(a

− 1) . . . a

(a

− 1)(a

− 1) + . . . , where a

≥ 2, n ≥ 1. Dynamically the alternating series expansion is gen- erated by the operator S

: [0, 1] → [0, 1] defined by

(3) S

x := 1 +

 1 x



−

 1 x

 1 x

 + 1



x, x 6= 0, and S

0 := 0, i.e. S

x = 1 − T

x (see also Figure 1).

Fig. 1

L¨ uroth series have been extensively studied; for further reference we mention here papers by H. Jager and C. de Vroedt [JdV], T. ˇ Sal´at [Sa], and books by J. Galambos [G], O. Perron [Pe] and W. Vervaat [V].

In [JdV] it was shown that the stochastic variables a

(x), a

(x), . . . are

independent with λ(a

= k) = 1/(k(k − 1)) for k ≥ 2, and that T

is

measure preserving with respect to Lebesgue measure (

) and ergodic (

).

Using similar techniques analogous results were obtained in [K

1,2] for the operator S

from (3). In fact, much stronger results can be obtained easily, not only in the case of the (alternating) L¨ uroth series, but also in a more general setting.

1.2. Generalized L¨uroth series. Let I

= (l

, r

], n ∈ D ⊂ N = {0, 1, 2, . . .}, be a finite or infinite collection of disjoint intervals of length L

:= r

− l

, such that

(4) X

n∈D

L

= 1 and

(5) 0 < L

≤ L

< 1 for all i, j ∈ D, i > j.

The set D is called the digit set. Usually such a digit set is either a finite or infinite set of consecutive positive integers, see also the examples at the end of this section.

Furthermore, let I

:= [0, 1] \ S

n∈D

I

, L

:= 0 and define the maps T, S : [0, 1] → [0, 1] by

T x :=

( x − l

r

− l

, x ∈ I

, n ∈ D, 0, x ∈ I

, Sx :=

( r

− x r

− l

, x ∈ I

, n ∈ D, 0, x ∈ I

. Define for x ∈ Ω := [0, 1] \ I

= S

n∈D

I

, s(x) := 1

r

− l

and h(x) := l

r

− l

, in case x ∈ I

, n ∈ D, s

= s

(x) :=

 s(T

x), T

x 6∈ I

,

∞, T

x ∈ I

, and

h

= h

(x) :=

 h(T

x), T

x 6∈ I

, 1, T

x ∈ I

. For x ∈ (0, 1) such that T

x 6∈ I

, one has

T x = s(x)x − h(x) = s

x − h

.

(

) That is, λ(T

(A)) = λ(A) for every Lebesgue measurable set A.

(

) That is, λ(A M T

A) = 0 ⇒ λ(A) ∈ {0, 1}.

Inductively we find x = h

s

+ 1

s

T x = h

s

+ 1

s

 h

s

+ 1

s

T

x



= . . . (6)

= h

s

+ h

s

s

+ . . . + h

s

. . . s

+ 1

s

. . . s

T

x.

Since Sx = 1 − T x = −s

x + 1 + h

, for x ∈ Ω, one finds

(7) x = h

+ 1

s

− Sx s

.

Continuing in this way we obtain an alternating series expansion (see also [K

1,2]). Figure 1 illustrates the case D = N \ {0, 1}, I

:= (1/n, 1/(n − 1)].

Now let ε = (ε(n))

be an arbitrary, fixed sequence of zeroes and ones.

We define the map T

: [0, 1] → [0, 1] by

(8) T

x := ε(x)Sx + (1 − ε(x))T x, x ∈ [0, 1], where

ε(x) :=

 ε(n), x ∈ I

, n ∈ D, 0, x ∈ I

. Let ε

:= ε(T

x),

s

= s

(x) :=

 s(T

x), T

x 6∈ I

,

∞, T

x ∈ I

, and h

defined similarly. By (6) and (7) one finds that

x = h

+ ε

s

+ (−1)

s

T

x

= h

+ ε

s

+ (−1)

s

 h

+ ε

s

+ (−1)

s

T

x



= . . .

= h

+ ε

s

+ (−1)

h

+ ε

s

s

+ (−1)

h

+ ε

s

s

s

+ . . . + (−1)

h

+ ε

s

. . . s

+ (−1)

s

. . . s

T

x.

For each k ≥ 1 and 1 ≤ i ≤ k one has s

≥ 1/L > 1, where L = max

L

, and |T

x| ≤ 1. Thus,

(9)

   x − p

q

  = T

x s

. . . s

≤ L

→ 0 as k → ∞, where

p

q

= h

+ ε

s

+ (−1)

h

+ ε

s

s

+ (−1)

h

+ ε

s

s

s

+ . . . (10)

+ (−1)

h

+ ε

s

. . . s

and q

= s

. . . s

. In general p

and q

need not be relatively prime (see also Section 3.1). Let ε

:= 0, then for each x ∈ [0, 1] one has

(∗) x =

X

(−1)

h

+ ε

s

. . . s

.

For each x ∈ [0, 1] we define its sequence of digits a

= a

(x), n ≥ 1, as follows:

a

= k ⇔ T

x ∈ I

,

for k ∈ D ∪ {∞}. The expansion (∗) is called the (I, ε)-generalized L¨uroth series (GLS ) of x. Notice that for each x ∈ [0, 1] \ I

one finds a unique ex- pansion (∗), and therefore a unique sequence of digits a

, n ≥ 1. Conversely, each sequence of digits a

, n ≥ 1, with a

∈ D ∪ {∞} and a

6= ∞ defines a unique series expansion (∗). We denote (∗) by

(11) x =

 ε

, ε

, ε

, . . . , ε

, . . . a

, a

, a

, . . . , a

, . . .

 . Since ε

= ε(a

), n ≥ 1, we might as well replace (11) by (12) x = [ a

, a

, a

, . . . , a

, . . . ]

No new information is obtained using (11) instead of (12). However, we will see in Section 3 that it is sometimes adventageous to use (11) instead of (12).

Finite truncations of the series in (∗) yield the sequence p

/q

of (I, ε)- GLS convergents of x. We denote p

/q

by

p

q

=

 ε

, ε

, ε

, . . . , ε

a

, a

, a

, . . . , a

 .

Examples. 1. Let I

:= (1/n, 1/(n − 1)], n ≥ 2. In case ε

= 0 for n ≥ 2, one gets the classical L¨ uroth series, while ε

= 1 for n ≥ 2 yields the alternating L¨ uroth series.

2. For n ∈ N, n ≥ 2, put I

= (i/n, (i + 1)/n], i = 0, 1, . . . , n − 1. In case ε(i) = 0 for all i, T

yields the n-adic expansion. In case n = 2 and ε(0) = 0, ε(1) = 1, T

is the tent map.

See also Sections 3.2 and 3.3 for more intricate examples.

2. Ergodic properties of generalized L¨ uroth series. Let Ω be as in Section 1.2, B be the collection of Borel subsets of Ω, and λ be the Lebesgue measure on (Ω, B). Let (I, ε) be as in the previous section, viz. I = (I

)

satisfies (4) and (5), while ε = (ε(n))

is a sequence of zeroes and ones.

We have the following lemma.

Lemma 1. The stochastic variables a

(x), a

(x), . . . , corresponding to

the (I, ε)-GLS operator T

from (8) are i.i.d. with respect to the Lebesgue

measure λ, and

λ(a

= k) = L

for k ∈ D ∪ {∞}.

Furthermore, (I

)

is a generating partition.

P r o o f. Define for (k

, . . . , k

) ∈ D

, n ≥ 1, the so-called fundamental intervals of order n by

(13) ∆

:= {x ∈ Ω : a

(x) = k

, . . . , a

(x) = k

}.

Let p

/q

∈ Q be defined as in (10) (and recall that the s

’s, h

’s and ε

’s are uniquely determined by k

, . . . , k

), then obviously one has

x ∈ ∆

⇔ ∃y ∈ [0, 1] such that x = p

q

+ (−1)

s

. . . s

y.

Thus ∆

1...k_n

is an interval with p

/q

as one endpoint, and having length 1/(s

. . . s

). Therefore,

λ(∆

) = λ(a

= k

, . . . , a

= k

) = 1 s

. . . s

. Since

s

= 1

r

− l

= 1

L

, i = 1, . . . , n, one finds

λ(∆

) = λ(a

= k

, . . . , a

= k

) = Y

L

.

The independence of the a

(x)’s and the equality λ(a

= k) = L

for k ∈ D ∪ {∞} now easily follow from

X

k_i∈D

L

= 1 for all n ≥ 1 and all 1 ≤ i ≤ n.

That (I

)

is a generating partition is immediate from (9).

Theorem 1. The (I, ε)-GLS operator T

from (8) is measure preserving with respect to Lebesgue measure and Bernoulli.

P r o o f. For any k

, . . . , k

∈ D, n ≥ 1, T

∆

= [

k∈D

∆

is a disjoint union of fundamental intervals of order n + 1, so that λ(T

∆

) = X

k∈D

λ(∆

) = L

. . . L

X

k∈D

L

= L

. . . L

= λ(∆

).

Since the collection {∆

: n ≥ 1, k

∈ D} generates B, it follows

from [W, Theorem 1.1, p. 20] that λ is T

-invariant. From Lemma 1, viz.

λ(∆

) = Y

λ(∆

), we conclude that ([0, 1], B, λ, T

) is a Bernoulli system.

R e m a r k s. 1. The Bernoullicity of the L¨ uroth operator T

was already noticed by P. Liardet in [Li].

2. From the fact that T

is Bernoulli, and therefore ergodic, one can draw a great number of easy consequences, using Birkhoff’s Ergodic Theorem. See also [JdV] and [K

2]. We mention here some results:

For almost every x the sequence (T

x)

is uniformly distributed over [0, 1]. Furthermore,

n→∞

lim 1 n

n−1

X

k=0

T

x = 1

2 , lim

n→∞



Y

k=0

T

x



= 1 e a.e.

and

n→∞

lim (a

. . . a

)

= e

a.e., where c = P

k∈D

L

log k (

).

Define the map T

: [0, 1] × [0, 1] → [0, 1] × [0, 1] by (14) T

(x, y) :=



T

(x), h(x) + ε(x)

s(x) + (−1)

s(x) y

 . Notice that for

x =

 ε

, ε

, ε

, . . . , ε

, . . . a

, a

, a

, . . . , a

, . . .



one has

T

(x, 0) =

 ε

, ε

, . . . , ε

, . . . a

, a

, . . . , a

, . . .

 ,

 ε

a



,

where 

ε

a



= (−1)

h

+ s

s

= h

+ s

s

. Now

T

(x, 0) =

 ε

, ε

, . . . , ε

, . . . a

, a

, . . . , a

, . . .

 ,

 ε

, ε

a

, a



, where

 ε

, ε

a

, a



= (−1)

h

+ ε

s

+ (−1)

h

+ ε

s

s

= h

+ ε

s

+ (−1)

h

+ ε

s

s

.

(

) In case 0 ∈ D we put e

:= 0.

Putting T

(x, 0) =: (T

, V

), n ≥ 0, where T

= T

x =

 ε

, ε

, . . . a

, a

, . . .



, n ≥ 0, and

V

=

 ε

, ε

, . . . , ε

a

, a

, . . . , a



, n ≥ 1, V

:= 0, we see inductively that

V

= h

+ ε

s

+ (−1)

s

V

.

As in the case of the continued fraction we will call T

= T

x the future of x at time n, while V

= V

(x) is the past of x at time n (see also [K]).

We have the following theorem.

Theorem 2. The system ([0, 1] × [0, 1], B × B, λ × λ, T

) is the natural extension of ([0, 1], B, λ, T

). Furthermore, ([0, 1] × [0, 1], B × B, λ × λ, T

) is Bernoulli.

P r o o f. For any two vectors (k

, . . . , k

) ∈ D

, (l

, . . . , l

) ∈ D

one has

∆

× ∆

= T

(∆

× [0, 1]).

Since {∆

× ∆

: k

, l

∈ D, 1 ≤ i ≤ n, 1 ≤ j ≤ m, n, m ≥ 1}

generates B × B, it follows that _

m≥0

T

(B × [0, 1]) = B × B.

Now, for any ∆

× ∆

one has

T

(∆

× ∆

) = ∆

× ∆

. Thus,

λ × λ(T

(∆

× ∆

)) = λ(∆

)λ(∆

)

= λ(∆

)λ(∆

)

= λ × λ(∆

× ∆

).

Since cylinders of the form ∆

1...k_n

× ∆

1...l_m

generate the σ-algebra B × B, it follows that T

is measure preserving with respect to Lebesgue measure. Thus, T

is the natural extension of T

(see [R] for details). Since T

is Bernoulli it is an exercise to show that T

is Bernoulli (see also [B]).

Corollary 1. For almost all x the two-dimensional sequence (15) T

(x, 0) = (T

, V

), n ≥ 0,

is uniformly distributed over [0, 1] × [0, 1].

P r o o f. Denote by A that subset of [0, 1] for which the sequence (T

, V

)

is not uniformly distributed over [0, 1] × [0, 1]. It follows from Lemma 1 and the definition of T

that for all x, y, y

∈ [0, 1] one has

|T

(x, y) − T

(x, y

)| < L

, n ≥ 0,

and we see that (T

(x, y) − T

(x, y

))

is a null-sequence. Hence, if A :=

A × [0, 1], then for every pair (x, y) ∈ A the sequence T

(x, y), n ≥ 0, is not uniformly distributed over [0, 1] × [0, 1]. Now, if A had, as a subset of [0, 1], positive Lebesgue measure, so would A as a subset of [0, 1] × [0, 1]. However, this is impossible in view of Theorem 2.

The partition ξ = {I

× [0, 1]}

is a generator for T

, which implies that the entropy h(T

) of T

equals h(T

, ξ) (see also [W], p. 96). There- fore,

h(T

) = − X

k∈D

L

log L

.

Now let (I

)

and (I

)

be two partitions of [0, 1] satisfying (4) and (5), and suppose that L

= L

for k ∈ D. Furthermore, let ε = (ε

)

and ε

= (ε

)

be two arbitrary sequences of zeroes and ones. It follows at once from Ornstein’s Isomorphism Theorem (see [W], p. 105) and Theorem 2 that T

and T

are metrically isomorphic. We conclude this section with the following theorem, which gives a concrete isomorphism.

Theorem 3. Let (I

)

and (I

)

be two partitions of [0, 1], sat- isfying (4) and (5). Suppose that L

= L

for k ∈ D. Furthermore, let ε = (ε

)

and ε

= (ε

)

be two sequences of zeroes and ones. Let T

and T

be defined as in (8). Finally, define Ψ : [0, 1] × [0, 1] → [0, 1] × [0, 1]

by Ψ

 ε

, ε

, . . . a

, a

, . . .

 ,

 ε

, ε

, . . . a

, a

, . . .



:=

 ε

(a

), ε

(a

), . . . a

, a

, . . .

 ,

 ε

(a

), ε

(a

), . . . a

, a

, . . .



. Then Ψ is a measure preserving isomorphism.

P r o o f. Since almost every x ∈ [0, 1] has unique (I, ε)-, (I

, ε

)-GLS expansions, it follows that Ψ is injective. Now, for any cylinder ∆

×

∆

1...l_m

,

∆

× ∆

= Ψ

(∆

× ∆

)

and

(λ × λ)(∆

× ∆

) = L

. . . L

L

. . . L

= L

. . . L

L

. . . L

= (λ × λ)(∆

× ∆

)

= (λ × λ) Ψ

(∆

× ∆

).

This shows that Ψ is measurable and measure preserving.

Finally, let (x, y) ∈ [0, 1] × [0, 1] with x =

 ε

, ε

, . . . a

, a

, . . .



, y =

 ε

, ε

, . . . a

, a

, . . .

 . Then

Ψ T

(x, y) =

 ε

(a

), ε

(a

), . . . a

, a

, . . .

 ,

 ε

(a

), ε

(a

), ε

(a

), . . . a

, a

, a

, . . .



= T

Ψ (x, y),

therefore Ψ is a measure preserving isomorphism.

3. Applications and examples

3.1. Approximation coefficients and their distribution. As before let I = (I

)

be a partition of [0,1] which satisfies (4) and (5), and let ε = (ε(n))

be a sequence of zeroes and ones. Putting q

= s

. . . s

, it follows from (9) and Corollary 1 that for a.e. x the approximation coefficients θ

, n ≥ 0, defined by

θ

= θ

(x) := q

   x − p

q

 , n ≥ 0,

have the same distribution as T

x, n ≥ 0. Viz., for a.e. x the sequence (θ

)

is uniformly distributed on [0,1].

In fact, for many partitions (I

)

more information on the distribution of the θ

’s can be obtained by a more careful definition of q

. As an example we will treat here the case of the classical L¨ uroth series, and all other GLS expansions with the same partition (I

)

(see also the examples at the end of Section 1.2).

In this case

s

= s(T

x) = 1

1 an−1

−

n

= a

(a

− 1), h

= a

− 1, and

h

+ ε

s

. . . s

= a

− 1 + ε

a

(a

− 1)a

(a

− 1) . . . a

(a

− 1) = 1

a

(a

− 1) . . . (a

− ε

) . Therefore it is more appropriate to put

q

= a

−ε

, q

= a

(a

−1)a

(a

−1) . . . a

(a

−1)(a

−ε

), n ≥ 2,

and we see

(16) θ

(x) = a

− ε

a

(a

− 1) T

x, n ≥ 1.

We have the following theorem.

Theorem 4. Let (I

)

be the L¨uroth partition, that is, I

:= (1/n, 1/(n − 1)] for n ≥ 2, and let ε(n) ∈ {0, 1} for n ≥ 2. Then for a.e. x and for every z ∈ (0, 1] the limit

N →∞

lim 1

N #{1 ≤ j ≤ N : θ

(x) < z}

exists and equals F

(z), where F

(z) :=

b1/zc+1−ε(b1/zc+1)

X

k=2

z

k − ε(k) + 1

b1/zc + 1 − ε(b1/zc + 1) , 0 < z ≤ 1.

P r o o f. Let z ∈ (0, 1]. From (15) and (16) it follows that (17) θ

< z ⇔ (T

, V

) ∈ Ξ(z) =

[

Ξ

(z), where

Ξ

(z) :=



0, k(k − 1) k − ε(k) z



∩ [0, 1]



× ∆

, k ≥ 2.

For k ≥ 2 we have the following two cases (of which the first one might be void).

(A) 2 ≤ k ≤ b1/zc + 1 − ε(b1/zc + 1). In this case Ξ

(z) =



0, k(k − 1) k − ε(k) z



× ∆

.

(B) k > b1/zc + 1 − ε(b1/zc + 1). In this case Ξ

(z) = [0, 1] × ∆

. From (A) and (B) one finds, that

(λ × λ)(Ξ(z))

=

b1/zc+1−ε(b1/zc+1)

X

k=2

(λ × λ)



0, k(k − 1) k − ε(k) z



× ∆



+ 1

b1/zc + 1 − ε(b1/zc + 1)

=

b1/zc+1−ε(b1/zc+1)

X

k=2

z

k − ε(k) + 1

b1/zc + 1 − ε(b1/zc + 1) .

The theorem at once follows from Corollary 1.

R e m a r k s . 1. Although the map x → (1/x) mod 1 generating the con- tinued fraction is not piecewise linear, which leads to complications in esti- mations, a similar result as in Theorem 4 was obtained for continued frac- tions (see also [BJW]).

2. If ε(n) = 0, n ≥ 2 (the classical L¨ uroth case) (

), the distribution function F

reduces to

F

(z) =

b1/zc+1

X

k=2

z

k + 1

b1/zc + 1 , 0 < z ≤ 1.

Furthermore

F

(z) =

b1/zc

X

k=2

z

k − 1 + 1

b1/zc , 0 < z ≤ 1;

see also Figure 2. Notice that F

≤ F

≤ F

.

Fig. 2

We have the following corollary.

(

) From now on the classical (resp. the alternating) L¨ uroth case will be indicated by

a subscript L (resp. A).

Corollary 2. Let (I

)

be the L¨uroth partition and let ε(n) ∈ {0, 1}

for n ≥ 2. Then there exists a constant M

such that for a.e. x,

N →∞

lim 1 N

X

θ

= M

.

Moreover , M

can be calculated explicitly, and M

≤ M

≤ M

, where M

= 1 −

ζ(2) = 0.177533 . . . and M

=

(ζ(2) − 1) = 0.322467 . . .

P r o o f. By definition M

is the first moment of F

and thus M

= R

0

(1 − F

(x)) dx.

R e m a r k s . 1. From Corollary 2 it follows that among all (I, ε)-GLS expansions with I the L¨ uroth partition the alternating L¨ uroth series has the best approximation properties.

2. The presentation of Corollary 2 suggests that by choosing ε = (ε(n))

appropriately, each value in the interval [M

, M

] = [0.177533 . . . , 0.322467 . . .] might be attained. This is certainly incorrect, as the following example shows. Let ε = (ε(n))

be given by ε(2) = 1 and ε(n) = 0 for n ≥ 3, and let ε

= (ε

(n))

be given by ε

(2) = 0 and ε

(n) = 1 for n ≥ 3. A simple calculation yields that M

= M

− 1/8 = 0.197467 . . . and M

= M

+ 1/8 = 0.302533 . . . ; we thus see that M

< M

and from this one can easily deduce that there does not exist a sequence ε

= (ε

(n))

of zeroes and ones for which M

∈ [M

, M

]. Some further investigations even suggest that the set

Υ := {M

: ε(n) ∈ {0, 1} for n ≥ 2}

is a Cantor set.

3.2. Jump transformations. Let T

be a (I, ε)-GLS operator with digit set D, and let a ∈ D. For x ∈ Ω, put

n

= n

(x) := min

n≥1

{a

(x) : a

(x) = a}

(and n

= ∞ in case a

(x) 6= a for all n ≥ 1). Define the jump transforma- tion J

: Ω → Ω by

(18) J

x :=

 T

x, n

∈ N, 0, n

= ∞.

Jump transformations were first studied by H. Jager [J] for the particular case that T

x = 10x (mod 1). Jager showed that such jump transformations are stronly mixing. Here, in the more general setting, we have a stronger result.

Theorem 5. Let T

be an (I, ε)-GLS operator with digit set D. For each

a ∈ D the corresponding jump transformation J

, as defined in (18), is an

(I

, ε

)-GLS operator , with

I

= {∆

: n ≥ 1, a

= a and a

6= a for 1 ≤ i ≤ n − 1}

and for each ∆

∈ I

the corresponding value of ε is given by ε

(∆

) = ε(a

) + . . . + ε(a

) (mod 2).

3.3. β-expansions and pseudo golden mean numbers. For an irrational number β > 1 the β-transformation T

: [0, 1] → [0, 1] is defined by

T

x = βx (mod 1)

(see also [FS] for further references). Clearly, β-transformations do not be- long to the class of GLS-transformations. However, in some cases there exists an intimate relation between both types of transformations, as the following example shows.

Let β > 1 be the positive root of the polynomial X

−X

−. . .−X −1, where (

) m ≥ 2. Due to C. Frougny and B. Solomyak [FS] we know that such β’s are Pisot numbers and that the β-expansion d(1, β) is finite, and equals

1 = 1 β + 1

β

+ . . . + 1 β

.

Notice that T

1 = β

+ . . . + β

, 0 ≤ i ≤ m − 1, and T

1 = 0 for i ≥ m. Furthermore, let

X :=

m−1

[

k=0

(T

1, T

1] × [0, T

1]

(see also Figure 3 for m = 4), and define T

: X → X by T

(x, y) :=

 T

x, 1

β (bβxc + y)

 .

Let Y := [0, 1] × [0, 1/β] and W : Y → Y the corresponding induced trans- formation under T

, that is

W(x, y) = T

(x, y),

where n(x, y) = min{s > 0 : T

(x, y) ∈ Y }. Clearly one has W(x, y) = T

(x, y),

where k = k(x) ∈ {0, 1, . . . , m − 1} is such that x ∈ (T

1, T

1].

Notice that W maps rectangles to rectangles; see also Figure 3.

Finally, let T

be the (I, ε)-GLS operator with partition I given by (T

1, T

1], 0 ≤ i ≤ m − 1 (see also Figure 3), and ε(n) = 0 for

(

) For m = 2 one has β = ( √

5 + 1)/2, which is the golden mean. For m ≥ 3 we call

these β’s pseudo golden mean numbers.

Fig. 3

each digit n. Notice that for x ∈ (T

1, T

1], 0 ≤ i ≤ m − 1, one has

(19) T

x = T

x.

We have the following lemma.

Lemma 2. Let Ψ : [0, 1] × [0, 1] → Y be defined by Ψ (x, y) := (x, y/β).

Then Ψ is a measurable bijection which satisfies Ψ ◦ T

= W ◦ Ψ.

P r o o f. For (x, y) ∈ [0, 1] × [0, 1] let i = i(x) ∈ {0, 1, . . . , m − 1} be such that x ∈ (T

1, T

1]. From (14) it then follows that

T

(x, y) =



T

x, T

1 + y β



and therefore

(20) Ψ (T

(x, y)) =

 T

x, 1

β T

1 + y β

 .

From (19), (20) and the definitions of W and Ψ it now follows that

W(Ψ (x, y)) = Ψ (T

(x, y)) for i = 0 and in case i 6= 0 one has

W(Ψ (x, y)) = W

 x, y

β



= T

 x, y

β



=



T

x, 1 β

 0 +

 1 β



1 + . . . 1 β

 1 + 1

β

 1

| {z }

i times

+ y β



. . .



=

 T

x, 1

β

+ . . . + 1

β

+ y β



= Ψ (T

(x, y)).

Let % be the measure on Y defined by

%(A) := (λ × λ)(Ψ

(A)) for each Borel set A ⊂ Y.

It follows from Lemma 2 and the fact that λ × λ is an invariant measure for T

that % is invariant with respect to W, and % = β(λ × λ). Lemma 2 now at once yields the following proposition.

Proposition. The dynamical systems ([0, 1] × [0, 1], λ × λ, T

) and (Y, %, W) are isomorphic.

Using standard techniques (see [CFS], p. 21) one obtains the measure µ on X which is invariant with respect to T

, viz.

µ(A) = β

1 β + 2

β

+ . . . + m β

(λ × λ)(A)

for each Borel set A ⊂ X. One also easily shows that (X, µ, T

) forms the natural extension of ([0, 1], ν, T

), where ν is the invariant measure with respect to T

[So]. Projecting µ on the first coordinate of X yields ν; one finds that ν has density h(x), where

h(x) = 1

1 β + 2

β + . . . + m β

X

x<T_βⁱ1

1 β

, as given by W. Parry [Pa].

Acknowledgements. We want to thank the referee for several helpful suggestions concerning the presentation of this paper.

References

[B] J. R. B r o w n, Ergodic Theory and Topological Dynamics, Academic Press, New York, 1976.

[BJW] W. B o s m a, H. J a g e r and F. W i e d i j k, Some metrical observations on the approximation by continued fractions, Indag. Math. 45 (1983), 281–299.

[CFS] I. P. C o r n f e l d, S. V. F o m i n and Ya. G. S i n a i, Ergodic Theory, Grundlehren

Math. Wiss. 245, Springer, New York, 1982.

[FS] C. F r o u g n y and B. S o l o m y a k, Finite beta-expansions, Ergodic Theory Dy- namical Systems 12 (1992), 713–723.

[G] J. G a l a m b o s, Representations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer, Berlin, 1982.

[J] H. J a g e r, On decimal expansions, Zahlentheorie, Berichte aus dem Mathema- tische Forschungsinstitut Oberwolfach 5 (1971), 67–75.

[JdV] H. J a g e r and C. d e V r o e d t, L¨ uroth series and their ergodic properties, Indag.

Math. 31 (1968), 31–42.

[K

1] S. K a l p a z i d o u, A. K n o p f m a c h e r and J. K n o p f m a c h e r, L¨ uroth-type alter- nating series representations for real numbers, Acta Arith. 55 (1990), 311–322.

[K

2] —, —, —, Metric properties of alternating L¨ uroth series, Portugal. Math. 48 (1991), 319–325.

[K] C. K r a a i k a m p, A new class of continued fraction expansions, Acta Arith. 57 (1991), 1–39.

[Li] P. L i a r d e t, MR: 93m:11077.

[Lu] J. L ¨ u r o t h, Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Ann. 21 (1883), 411–423.

[Pa] W. P a r r y, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar.

11 (1960), 401–416.

[Pe] O. P e r r o n, Irrationalzahlen, de Gruyter, Berlin, 1960.

[R] V. A. R o h l i n, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960) (in Russian); English translation: Amer. Math. Soc. Transl.

Ser. 2, 39 (1969), 1–36.

[Sa] T. ˇ S a l ´a t, Zur metrischen Theorie der L¨ urothschen Entwicklungen der reellen Zahlen, Czech. Math. J. 18 (1968), 489–522.

[So] B. S o l o m y a k, Personal communication with C. Kraaikamp, Seattle, July 9, 1991.

[V] W. V e r v a a t, Success Epochs in Bernoulli Trails with Applications in Number Theory, Math. Centre Tracts 42, Amsterdam, 1972.

[W] P. W a l t e r s, An Introduction to Ergodic Theory, Grad. Texts in Math. 79, Springer, New York, 1982.

Mathematics Department Department of Mathematics

University of South Alabama University of Oregon

Mobile, Alabama 36688 Corvallis, Oregon 97331

U.S.A. U.S.A.

E-mail: jose@mathstat.usouthal.edu E-mail: burton@math.orst.edu

Universiteit Utrecht Technische Universiteit Delft

Department of Mathematics TWI (SSOR)

Budapestlaan 6 Mekelweg 4

P.O. Box 80.000 2628 CD Delft, the Netherlands

3508TA Utrecht, the Netherlands E-mail: C.Kraaikamp@twi.tudelft.nl E-mail: Dajani@math.ruu.nl

Received on 29.12.1994

and in revised form on 1.8.1995 (2721)