S é m i n a i re s
&
C o n g r è s
C O L L E C T I O N S M F
Numéro 20
ÉCOLE DE THÉORIE ERGODIQUE
Y. Lacroix, P. Liardet, J.-P. Thouvenot, éds.
A NOTE ON THE GREEDY
β
-TRANSFORMATION
WITH ARBITRARY DIGITS
A NOTE ON THE GREEDY
β-TRANSFORMATION WITH
ARBITRARY DIGITS
by
Karma Dajani & Charlene Kalle
Abstract. —We consider a generalization of the greedy and lazyβ-expansions with digit setA={a0 < a1<· · ·< am}. We prove that the transformation generating
such expansions admits a unique absolutely continuous invariant ergodic measure. Furthermore, the support of this measure is an interval.
Résumé(Note sur laβ-transformation avec des chiffres arbitraires). —Nous consid´erons une g´en´eralisation des«greedy»et«lazy»β-d´eveloppements avec chiffres dans un al-phabetA={a0< a1<· · ·< am}. Nous montrons que la transformation qui donne
ces d´eveloppements poss`ede une unique mesure qui soit invariante, ergodique et ab-solument continue par rapport `a la mesure de Lebesgue. En outre, le support de cette mesure est un intervalle.
1. Introduction
Letβ >1 be a real number, and letTβ be the transformation of the unit interval [0,1) given by Tβx = βx(mod1). This transformation gives rise to the β-expansion introduced by R´enyi [19]: for any 0≤x <1,
(1) x=
∞ X
k=1
dkβ−k ,
where dk = dk(x, β) = bβTβk−1xc, k ≥1 (here bξc denotes the greatest integer not exceedingξ). R´enyi showed that for eachβ >1 theβ-transformationTβ is ergodic, and that there exists a unique probability measureνβ,equivalent to Lebesgue measure and invariant underTβ,such that for each Borel measurable setB∈ Bone has
νβ(B) = Z
B
hβ(x)dx ,
2000 Mathematics Subject Classification. — 37A05, 11K55.
wherehβ(x) is a measurable function satisfying 1− 1 β ≤ hβ(x) ≤ 1 1−1 β .
Shortly afterwards, Parry [15] gave an explicit formula for the density of the invariant measure, namely, hβ(x) = 1 F(β) X x<Tn(1) 1 βn x∈[0,1), whereF(β) =R01 P x<Tn(1)β1n dx is a normalizing constant.
The β-transformation given above can be defined geometrically in the following way. There exists a subdivision α0 = 0 < α1 < · · · < αm < αm+1 = 1 of [0,1),
such that Tβ is linear with slope β on each subinterval Ij = [αj, αj+1). Further,
T Ij = [0,1), T αj = 0 forj = 0,1,· · ·, m, and T1 = limx→1T x ≤ 1. This implies
that onIj,T is given byT x=βx−j. Iterations ofT give expansions of the form (1) with digitdk∈A={0,1,· · · , m} (notice thatm=bβc).
Expansion (1) is also sometimes known in the literature by thegreedy expansion. The reason is that, for each k the digit dk in (1) is the largest element of A = {0,1,· · ·,bβc} such that Pk
j=1djβ−j ≤ x. So at each step, the greedy expansion chooses the largest possible digit. Taking this point of view, Pedicini [17] generalized the above notion to greedy expansions with digits in some set A = {a0, . . . , am}, with a0 < · · · < am. More precisely, he studied the combinatorial and arithmetic
properties of expansions of the formx=P∞k=1dkβ−k such that for eachk,dk is the largest element ofA={a0, a1, . . . am}withPkj=1djβ−j ≤x(see [4] for more detail). He showed that every point in the interval a0
β−1,
am
β−1
has a greedy expansion with digits in the setA, if and only if
(2) max
0≤j≤m−1(aj+1−aj)≤
am−a0
β−1 .
He called such expansionsgreedy expansions with deleted digits. This terminology has already been used by several authors (see [11], [18], [12], [17], [4], [21]), but in most cases the digit sets under consideration contain only non-negative integers. Since our digit sets will contain arbitrary real numbers, we choose to adopt greedy expansions with arbitrary digits instead or we will refer explicitly to the digit set that we use. We will now give a geometrical description of the underlying map generating these expansions, which shows why condition (2) imposed by Pedicini is a natural one. Furthermore, it allows us to put these transformations in the general framework of piecewise linear maps, for which a rich theory has been developed, and which we use to analyze and understand the ergodic properties of theβ-transformations with arbitrary
digits. The description is similar to the one given for the classical greedy expansion defined above. We consider transformations T whose domain is some interval [0, α] with the following properties
(i) there exists a subdivision 0 = α0 < α1 < · · · < αm < αm+1 = αwith
corre-sponding interval partition Jj = [αj, αj+1), j = 0,1, . . . m−1, Jm = [αm, α] such thatT on eachJj is linear with slopeβ,
(ii) T αj = 0,j= 0,1, . . . , m, andT α=α, (iii) T Jj ⊂[0, α], j= 0,1, . . . m−1.
Note that from (ii) and the linearity of T, we have that T Jm = [0, α] so that T is surjective. Settingaj=βαjforj= 0,1, . . . m, one sees from (i) and (ii) that on each
Jj,T has the formT x =βx−aj, j = 0,1, . . . m. From the second equation in (ii), one has thatα= am
β−1. From (iii) one gets max0≤j≤m−1(aj+1−aj)≤α=
am
β−1, which
is condition (2) witha0= 0. See Figure1for the graph ofT. Iterations ofT generate
greedy expansions with digits inAas described by Pedicini.
The above geometrical description can be slightly modified in order to capture the case a0 6= 0. The interval [0, α] is replaced by [α0, α] (0 6= α0 < α), and in
condition (ii) we replace the first equality byT αj =α0 fori= 0,1, . . . , m. However,
a simple translation by α0 conjugates a transformation T in this class with domain
[α0, α] and subdivisionα0< α1<· · ·< αm< αm+1=αto a transformationSof the
previous class on [0, α−α0] with subdivision 0 =γ0< γ1<· · ·< γm< γm+1=α−α0,
γi=αi−α0,i= 0,1, . . . , m+1. Setting nowaj=βαj−α0, and using the conjugation
withS or the defining properties ofT, we see that
T x= βx−aj, ifx∈ ï a 0 β−1+ aj−a0 β , a0 β−1+ aj+1−a0 β ã , forj= 0, . . . , m−1, βx−am, ifx∈ ï a0 β−1+ am−a0 β , am β−1 ò .
We call T =Tβ,A thegreedy transformations with digits in the setA={a0, . . . , am} witha0<· · ·< am. Clearly,Asatisfies condition (2), and a set with this property is
called anallowable digit set forβ. From the above, and to keep the exposition simple, we will assume with no loss of generality thata0= 0 =α0.
As explained above, the greedy expansion chooses at each stage the largest possible digit. One can look at the other extreme case, namelyβ-expansionsP∞k=1dkβ−ksuch that for eachk,dk is the smallest member of Asatisfying
(3) x≤ k X i=1 di βi + ∞ X i=k+1 am βi.
These expansions are known aslazy expansions, and were studied in the classical case,
i.e.,A={0,1, . . .bβc}by many authors (see for example [5], [6], [8] and [7]), and for the general caseA={a0, . . . , am}by Pedicini [17] (see also [4]). Pedicini showed that under condition (2), everyx∈ a0
β−1,
am
β−1
has a lazy expansion. In [4] the underlying transformation generating lazy expansions with digit setAwas given, and was shown to be conjugate to the greedy expansion T = Tβ,A˜ with ˜A = {˜a0, . . . ,˜am}, where ˜
ai=a0+am−am−i,i∈ {0, . . . , m}. The isomorphismφis given by φ: ï a0 β−1, am β−1 ò −→ ï a0 β−1, am β−1 ò x7−→ a0+am β−1 −x.
So we haveL◦φ=φ◦T. The explicit definition of L=Lβ,Ais given by
Lx= βx−a0, ifx∈ ï a0 β−1, am β−1 − am−a0 β ò , βx−aj, ifx∈ Å am β−1− am−aj−1 β , am β−1− am−aj β ò , forj= 1, . . . , m.
As mentioned, throughout we will assume thata0= 0.
A number of articles have been published on invariant measures of piecewise mono-tonic transformations. Among others, the articles [1] by Buzzi and Sarig, [2] by Byers and Boyarsky, [3] by Byers, G´ora and Boyarsky, [9] and [10] by Hofbauer, [13] by Lasota and Yorke, [14] by Li and Yorke, [20] by Schweiger and [22] by Wilkinson state a variety of results regarding invariant measures of this kind of transformations and their ergodicity.
In the first section of this article we will prove the existence of a unique absolutely continuous invariant ergodic measureµfor the greedy transformation with arbitrary digits using the results found by Li and Yorke in [14]. We will show that the support of this measure is the smallest interval of the form [0, t) suchT([0, t))⊂[0, t), and we identifyt explicitly. This leads to the following theorem
Theorem 1.1. — The restriction ofT to the interval[0, t) admits a unique invariant ergodic measure that is equivalent to Lebesgue measure on this interval.
We give similar results for the lazy transformation with arbitrary digits. In the last section we consider in more detail two classes of greedy transformations with arbitrary digits, and we give an explicit formula for the density of their absolutely continuous invariant measures. For the first class an article by Wilkinson ([22]) has been an important source. For the second class we use an article by Byers and Boyarski ([2]), which is based on [16] by Parry.
2. Ergodic absolutely continuous measures
Let β > 1 andA ={a0, a1, . . . , am} be an allowable digit set with a0 = 0. Let
T =Tβ,A: 0, am β−1 → 0, am β−1
be the greedy transformation with digits in A. This is a piecewise linear, strictly increasing transformation, which has its discontinuities in the points ai
β fori= 1, . . . , m. LetJ denote the set containing these points. Then
J is finite and for each x ∈ 0, am
β−1
\J we haveT0(x) = β > 1. The points in J
give a partition ∆ ={∆i}mi=0 of the interval
0, am β−1 , where ∆m = am β , am β−1 and fori∈ {0, . . . , m−1}, ∆i =ai β, ai+1 β
. Define fori∈ {1, . . . , m} the values yi to be the values obtained from T by taking the limit from the left to the points ai
β, i.e., yi =ai−ai−1. (See Figure1) a1 Β a2 Β a3 Β a4 Β 0 y1 y2 y3 y4 a4 Β -1 D0 D1 D2 D3 D4
Figure 1. The greedyβ-transformation withβ= 2.5 andA={0,1.35,1.75,3.3,6}.
Let λ denote the Lebesgue measure. We begin by defining different notions of invariance under the transformation T and by stating the results found by Lasota and Yorke in [13] and by Li and Yorke in [14]. Letµbe a Borel measure on
0, am
β−1
. Aµ-integrable functionf is calledan invariant function underT if for all measurable setsE,R
Ef dµ= R
T−1Ef dµ. We call a Lebesgue measurable setEforward invariant
under T if T E = E modulo sets of λ-measure zero. It was shown in [13] that there exists an invariant measure, absolutely continuous with respect to the Lebesgue measure, for transformationsτthat are piecewise continuous with a finite set of points
of discontinuity and that have a derivative bigger than 1 for points outside this finite set. In [14], Li and Yorke studied these invariant measures in more detail. Their results translate in the following way to our particular greedy transformationT. For the transformation T, there exist sets B1, . . . , Bn and functions f1, . . . , fn, where
n≤m, such that all the following hold.
(c1) For each i ∈ {1, . . . n}, Bi is a finite union of closed subintervals of 0,βa−m1. EachBicontains at least one of the elements ofJin its interior. Moreover, each
Bi is forward invariant underT.
(c2) Bi∩Bjcontains at most a finite number of points, wheni=6 j,i, j∈ {1, . . . , n}. (c3) For almost allx∈
0, am
β−1
\J, there is ani∈ {1, . . . , n} such that the closure of the forward orbit ofxunderT equals the setBi, i.e.,
Λ(x) := ∞ \ N=1 {Tn(x)}∞ n=N =Bi.
(c4) For each i∈ {1, . . . n},Bi is the support of the function fi, i.e., fi >0 λa.e. onBi andfi = 0 onBic. Moreover,
R
Bifidλ= 1.
(c5) For eachi∈ {1, . . . n},fiis invariant underT and ifg is invariant underT and satisfies (c4) for somei, theng=fi λalmost everywhere.
(c6) Each functionf that is invariant underT can be written asf =Pn
i=1bifi with a suitably chosen set of constants{bi}ni=1.
(c7) Iff is an invariant function andEis a measurable set, such thatT Eis measur-able andT E⊆E λa.e., then f·1E is an invariant function, where 1E denotes the indicator function of the setE.
Remark 2.1. — The last result was proven in [14] for setsE such thatT E=E forλ
a.e. x∈E, however the proof of Li and Yorke still holds under the weaker assumption thatT E⊆E λa.e.
We will first make some observations about the setsBi. Lemma 2.2. — Let I⊆
0, am
β−1
be a closed interval.
(i) If I is forward invariant under T and contains at least one element ofJ in its interior, then 0∈I.
(ii) If I does not contain an element of J in its interior, then I is not forward invariant underT.
Proof. — The first part of the lemma follows immediately from the fact that for each
i∈ {1, . . . , m},T ai
β
= 0.
For the second part it is enough to notice that ifIdoes not contain an element of
Remark 2.3. — As an immediate consequence of this lemma, we have that there can-not exist two or more sets Bi satisfying (c1) and (c2). To see this, suppose that the sets Bi and Bj both satisfy (c1). Then they are both forward invariant under
T, so by the previous lemma there should exist numbers 0 < xi, xj ≤ βa−m1 such that [0, xi] ⊆ Bi and [0, xj] ⊆ Bj, but this contradicts (c2). So by the previously stated results from [14], there exists a number 0 < x ≤ am
β−1 and a finite number
of closed intervals I1, . . . , Ik ⊆
0, am
β−1
with T(Ij) containing a closed interval of positive Lebesgue measure, such that the set
(4) B:= [0, x]∪
k [
i=1
Ii
satisfies (c1) to (c7) for an invariant probability density function f. This implies that there exists a unique invariant measure forT that is equivalent to the Lebesgue measure on B. Notice that without loss of generality, we can assume that B is the finite union of disjoint closed intervals. The fact thatB is forward invariant implies thatT[0, x]⊆[0, x].
The next lemma states that ifB contains a closed interval whose image underT
is contained in itself, thenB is exactly this interval.
Lemma 2.4. — Let f be the density function of an invariant absolutely continuous probability measure as in Remark2.3and letB be its support. Suppose that[α1, α2]⊆
B is a closed interval. IfT[α1, α2]⊆[α1, α2]λa.e., then[α1, α2] =B. Consequently,
[α1, α2] is a forward invariant set.
Proof. — Consider the functiong=f ·1[α1,α2]. Sincef is an invariant function and
[α1, α2] satisfies T[α1, α2]⊆[α1, α2]λa.e., by (c7) we know that also the functiong
is invariant, with its support contained in the support of f. By (c6) there exists a constantc, such thatg=c·f. Now define the function
h=R g g dλ.
Thenhis an invariant probability density function andh=c0·f, withc0=c/R g dλ. This means that h=f λ a.e., so that 1[α1,α2](x) = 1 forλ almost allx∈B. Since B is a finite union of closed intervals, it follows thatB= [α1, α2]. By (c1), [α1, α2] is
forward invariant.
Remark 2.5. — By the same reasoning as in the proof of the previous lemma, it can be shown thatT is ergodic with respect to the invariant measure. To see this, let µ
be the measure given byµ(E) =R
Ef dλfor each measurable setE and suppose that
a.e., so by (c7) the functiong=f·1A is invariant. Following the idea of the proof of Lemma2.4gives that 1A= 1λa.e., soµ(A) = 1.
By Remark 2.3, there exists an element x ∈ 0, am
β−1
such that the support B
of f contains the interval [0, x] with T[0, x] ⊆ [0, x]. By Lemma 2.4, we see that
B = [0, x] = T[0, x] λ a.e. The next two lemmas specify the value of x. First we define the following value. Let yi0 = max
yi : aβi ≤ x and if there are two such values, then letyi0 be the one with the smallest index.
Lemma 2.6. — Let B = [0, x] be the support of the probability density function f as described above. ThenB= [0, yi0].
Proof. — SinceT[0, x] = [0, x]λa.e., we have thatyi≤xfor anyisuch that
ai
β ≤x.
Henceyi0 ≤x. Also T x≤x.
Supposex∈∆k for somek∈ {0, . . . , m}. Then by the definition ofyi0,
(5) T ï 0,ak β ã ⊆[0, yi0] λa.e. If yi0 ∈ 0,ak β
, then T[0, yi0] ⊆ [0, yi0] ⊆ [0, x] λ a.e. and thus by Lemma 2.4,
[0, yi0] = [0, x]λ a.e. If on the other handyi0 ∈
ak
β, x
, then sinceT x ≤x, we also have T yi0 ≤yi0 and this means that
T ï ak β , yi0 ò ⊆[0, yi0] λa.e.
Combining this with equation (5) gives thatT[0, yi0]⊆[0, yi0]⊆[0, x]λa.e., so again
by Lemma2.4we have that [0, yi0] = [0, x].
From the previous lemma we know thatxis one of the values yi, i∈ {1, . . . , m}. The next lemma states explicitly which of these values it is.
Lemma 2.7. — Let yi0 be defined as above. Then
(6) i0= min{i:T[0, yi]⊆[0, yi] λa.e.}.
Proof. — Since [0, x] = [0, yi0] is the support of the invariant probability density
functionf, we must have by Lemma 2.4thatT[0, yi]6⊆[0, yi]λa.e. for anyyi< yi0.
In particular, by the definition ofyi0 we have that ifi < i0, then ai
β < ai0
β ≤yi0.
This implies that yi < yi0 and thus that T[0, yi]6⊆[0, yi] λa.e. Hence i0 = min{i : T[0, yi]⊆[0, yi]λa.e.}.
In the previous lemmas and remarks we have established the existence of a unique absolutely continuous invariant measure for the greedy transformation with arbitrary digits, that is ergodic, and we have given its support. These results are summarized in the following theorem.
Theorem 2.8. — Let β > 1 and let A = {0, a1, . . . , am} be an allowable digit set
for β. If T : 0, am β−1 → 0, am β−1
is the greedy β-transformation with digits in A, then there exists a unique absolutely continuous invariant measure, that is ergodic. Furthermore, the support of the probability density function f is the interval [0, yi0],
wherei0= min{i:T[0, yi]⊆[0, yi]λ a.e.}.
As a corollary we get Theorem1.1, stated in the introduction.
Now consider the lazy transformation with arbitrary digits for β1 and allowable digit set ˜A={˜a0, . . . ,˜am}, where ˜a0 = 0. Indicate the points of discontinuity of L
in the following way. For i∈ {0, . . . , m−1}, let ˜`i= β˜a−m1−˜amβ−a˜i. In the same way as was done for the greedy transformation, we can make a partition{∆i˜ }m
i=0, using
these points of discontinuity. Let ˜∆0 = [0,`˜0] and ˜∆m = `˜m−1,β˜a−m1
and for all
i∈ {1, . . . , m−1}, define
˜
∆i= (˜`i−1,`˜i].
For eachi∈ {1, . . . , m}, let ˜yi denote the value of Lwhen taking the limit from the right to the point ˜`i−1,i.e., ˜yi=β˜a−m1−(˜ai−˜ai−1). (See Figure2)
{1 {2 {3 {0 0 y1 y2 y3 y4 a 4 Β -1 D0 D 1 D 2 D 3 D 4
The following corollary follows directly from Theorem2.8.
Corollary 2.9. — LetL be the lazy transformation forβ1and allowable digit set A˜= {˜a0, . . . ,a˜m}, for which ˜a0 = 0. Let T be the greedy transformation for the same
β >1and with allowable digit setA={a0, . . . , am}, such thatai= ˜am−˜am−i. Then
there exists a unique absolutely continuous invariant measureν forL, that is ergodic. Let i0 be defined for the greedy transformation as in equation (6). Then the support
of the measureν is given by the interval ˜ ym−i0+1, ˜ am β−1 .
Proof. — By Theorem 2.8 we have that the interval [0, yi0] is the support of the
density function of the invariant measure for the greedy transformation. Letµbe the absolutely continuous invariant measure for T. Since T and L are isomorphic with isomorphism φ, given by φ(x) = am
β−1−x,L also has a unique absolutely continuous
invariant measure, given byν =µ◦φ−1. This is an ergodic measure and its support
is given by φ([0, yi0]) = ï ˜ ym−i0+1, ˜ am β−1 ò .
3. Examples of explicitly calculable invariant measures
In the previous section it is shown that by the results of Li and Yorke in [14], on [0, yi0]T has a unique invariant measure that is equivalent to the normalized Lebesgue
measure. The same result holds for the lazy transformation. In general we can not give an explicit expression of this invariant measure, but in the next section we discuss two cases of which we do know what the invariant measure is.
3.1. Imposing a condition on the number of digits. — The first example is a particular case of the transformations studied by Wilkinson in [22]. In this paper, Wilkinson derives a formula for the density of an absolutely continuous invariant measure for certain piecewise linear transformations. Before we go into more detail, let us give some definitions. We consider our greedy transformation T with β > 1 and A = {0, a1, . . . , am} an allowable digit set forβ, but with the extra restriction that m <
β ≤m+ 1. In [4] it was shown that condition (2) implies that dβe ≤m+ 1, so this example is the case in which the smallest number of digits is considered.
By the previous section, we have that the support of the absolutely continuous invariant measure is given by [0, yi0]. Suppose that N is the largest index such that
aN
β < yi0. Then the points
ai
β, i= 1, . . . , N, give an interval partition of the interval [0, yi0] as before. Let ∆ ={∆0, . . . ,∆N} be the partition of [0, yi0], such that
∆0= ï 0,a1 β ã , ∆N = ï aN β , yi0 ò
and fori= 1, . . . , N−1, ∆i= ï ai β, ai+1 β ã .
An element ∆i ∈∆ is called afull interval of rank 1 ifλ(T∆i) = 1, where λis the normalized Lebesgue measure on the interval [0, yi0]. Otherwise we call it non-full.
Using ∆ andT, we can make the sequence of partitions{∆(n)}in the usual way. For
n ≥ 1, ∆(n) = Wn−1 k=0T
−k∆. The elements of ∆(n) are intervals and are called the
fundamental intervals of rank n. An elementE(n) ∈ ∆(n) is called full of rankn if
λ(TnE(n)) = 1 and non-full otherwise. Now let I(E(n)) be the number of non-full
fundamentals interval of rankn+ 1, that are contained inE(n)and let
In= sup E(n)∈∆(n)
I(E(n)).
So for each fundamental interval of rankn, we take the number of non-full fundamental intervals of rankn+ 1 and In indicates the supremum of these numbers over all the fundamental intervals of rankn. If we then take the supremum over all ranks, we get a numberI,i.e.,
I= sup n≥0
In,
where I0 is the number of non-full intervals of rank 1. In [22] Wilkinson derived
a formula for the absolutely continuous invariant measure under the condition that
β > I. We will adapt his result to our case and generalize it to our setting. For eachK ≥0, let ¯IK = supn≥KIn and let Bn denote the union of those fundamental intervals of ranknwhich are full, but which are not a subset of any full fundamental interval of lower rank. Notice thatI = ¯I0. We have the following lemma, which is a
generalization of Corollary 4.5 in [22].
Lemma 3.1. — LetI¯K andBn be as above and suppose thatβ >I¯K for someK≥0.
Then ∞
X
n=1
λ(Bn) = 1.
Proof. — Consider the support [0, yi0] and fill it as far as possible with full
funda-mental intervals of rank 1. Since every non-full fundafunda-mental interval of rank 1 has Lebesgue measure smaller than β1, the remaining part has Lebesgue measure smaller than I0
β. Now fill this part as far as possible with full fundamental intervals of rank 2. The remaining part has Lebesgue measure smaller than I0·I1
β2 . If we continue in this
manner, aftern+ 1 steps the remaining part will have Lebesgue measure smaller than
I0·I1· · · · ·In· 1
And by hypothesis we have lim n→∞I0·I1· · · · ·In· 1 βn ≤I0·I1· · · · ·IK−1nlim→∞ ů IK β ãn = 0,
which completes the proof.
The next theorem is an adaptation of the formula by Wilkinson and a generalization of Theorem 5.12 of [22]. It gives an explicit expression of the absolutely continuous invariant measure of the greedy transformation with arbitrary digits under the as-sumption that β >I¯K for some K ≥ 0. Before we state the theorem, we need the following notation. Let Dn be the collection of all non-full fundamental intervals of rank n, that are not subsets of any full fundamental interval of lower rank. Let
x∈[0, yi0]. Defineφ0(x) = 1 and forn≥1, let
(7) φn(x) = X E(n)∈D n 1 βn1TnE(n)(x).
Theorem 3.2. — If β > I¯K for some K ≥ 0, then the functions φn, n ≥ 0 and φ,
given by φ: [0, yi0]−→[0, yi0] x7−→ ∞ X n=0 φn(x)
are Lebesgue integrable and the functionhgiven by
h: [0, yi0]−→[0, yi0] x7−→ R φ(x)
φ(x)dλ(x)
is the density of the absolutely continuous invariant measure of T.
Proof. — The proof follows from Lemma 3.1 and a slight adaptation of the corre-sponding proof in [22].
Remark 3.3. — In the caseK= 0 Theorem3.2reduces to the theorem proved by K. Wilkinson.
The next theorem states that in the casem < β ≤m+ 1, we have that β > I, so we can immediately apply Theorem3.2.
Theorem 3.4. — Let β > 1 and A = {0, a1, . . . , am} be an allowable digit set, such
that m < β≤m+ 1. LetT be the greedy transformation for thisβ andA. Then the unique absolutely continuous invariant density is given by Theorem 3.2.
Proof. — It is enough to show that β > I. First notice that ∆i0−1 is a full
funda-mental interval, so that we haveI0 ≤N ≤m < β. By the definition ofyi0 we have
that ai0
β < yi0, which means that ∆i0−16= ∆N. Since for each fundamental interval
of rank n, E(n), we have that TnE(n) is an interval of the form [0, y] ⊆[0, y
i0], we
know thatE(n) can contain at mostN non-full fundamental intervals of rankn. So
In≤N for each n≥1, which means thatI < β, as we wanted.
We consider two examples. The first one satisfies the condition of Theorem 3.4. The second one does not satisfy the condition of this theorem, but in this case we can apply Theorem3.2. 0 1 2 2 2Β-3 1 Β 1 3 Β
Figure 3. The greedyβ-transformation withβ= 1+
√
2 andA={0,1,3}
on the interval [0,2].
Example 3.5. — First, letβ = 1 +√2 be the positive solution of the equation β2−
2β−1 = 0 and consider the allowable digit setA={0,1,3}. The interval [0,2] is the support of the invariant measure, see Figure 3. The orbits of the points 1 and 2 are as follows. T1 =β−1, T21 =T(β−1) = 1 β, T 31 =T(1 β) = 0, T2 = 2β−3, T22 =T(2β−3) =β−1, T32 =T(β−1) = 1β.
Notice that the condition of Theorem3.4is satisfied, so equation (7) gives forx∈[0,2],
φ(x) = 1 + 1 β1[0,1)(x) + 1 β1[0,2β−3)(x) + ∞ X k=0 ck+2 βk+21[0,1)(x) + ∞ X k=0 ck+1 βk+21[0,β−1)(x) + ∞ X k=0 ck βk+21[0,1 β)(x),
where ck is the k-th term of the tribonacci sequence,i.e., ck =ck−1+ck−2+ck−3,
the following identity, ∞ X k=0 ckxk= 2x 1−x−x2−x3.
Using this formula, we get that ∞ X k=0 ck+2 βk+2 = 2 β 1−1 β− 1 β2 − 1 β3 −2 β = 2− 1 β, ∞ X k=0 ck+1 βk+2 = 1 β " 2 β 1− 1 β − 1 β2 − 1 β3 # = 1, ∞ X k=0 ck βk+2 = 1 β2 " 2 β 1−1 β− 1 β2 − 1 β3 # = 1 β. So, now φ(x) = 1 + 1 β ·1[0,2β−3)(x) + 2·1[0,1)(x) + 1[0,β−1)(x) + 1 β ·1[0,1β)(x)
and rewriting this, leads to
φ(x) = 2β·1[0,1 β)(x) + (β+ 2)·1[ 1 β,1)(x) +β·1[1,β−1)(x) + (β−1)·1[β−1,2β−3)(x) + 1[2β−3,2](x). Since Z [0,2] φ(x)dλ(x) =8β−4 β ,
it is easily seen that the measureµgiven by
µ(E) = Z
E
β
8β−4φ(x)dλ(x),
for every measurable setE, is the unique invariant measure that is absolutely contin-uous with respect to Lebesgue measure.
Example 3.6. — For our second example we let β = 1+√5
2 be the golden mean,i.e.,
the positive solution of the equation β2−β−1 = 0, and we consider the allowable
digit set A={0,2β,5}. Notice that with this combination of β andAthe condition of Theorem3.4is not satisfied. The support of the invariant measure is given by the interval [0,2β], see Figure4. We now look at the orbits of the points 2β and 5−2β.
T(2β) = 2β−3, T2(2β) = 2−β, T3(2β) =β−1, T4(2β) = 1, T5(2β) =β+ 1, T6(2β) = 1,
T(5−2β) = 3β−2, T2(5−2β) = 3−β, T3(5−2β) = 2β−1,
T4(5−2β) = 2−β.
And we see that after the first iteration both orbits never hit ∆2again. This means
0 5-2Β 2Β 2Β 2Β-3 2 5 Β
Figure 4. The greedyβ-transformation withβ=1+
√
5
2 andA={0,2β,5}
on the interval [0,2β].
of any full interval of lower rank is at most 1, hence ¯I2= 1< β. By Theorem 3.2we
can apply formula (7) to get
φ(x) = 1 + 1 β ·1[0,5−2β)(x) + 1 β ·1[0,2β−3)(x) + 1 β2 ·1[0,3β−2)(x) + 2 β3 ·1[0,2−β)(x) + 1 β3 ·1[0,3−β)(x) + 2 β4 ·1[0,β−1)(x) + 1 β4 ·1[0,2β−1)(x) + " 1 β4 + 2 β7 ∞ X k=0 1 β3k # 1[0,1)(x) + " 1 β5 + 2 β8 ∞ X k=0 1 β3k # 1[0,β)(x) + " 1 β6 + 2 β9 ∞ X k=0 1 β3k # 1[0,β+1)(x).
Rewriting this will get you
φ(x) = (2β+ 1)·1[0,2β−3)(x) + (β+ 2)·1[2β−1,2−β)(x) + (8−3β)·1[2−β,β−1)(x) + (3β−2)·1[β−1,1)(x) + (β+ 1)·1[1,3−β)(x) + (4−β)·1[3−β,β)(x) + (2β−1)·1[β,5−2β)(x) +β·1[5−2β,2β−1)(x) + (4β−5)·1[2β−1,β+1)(x) + (3−β)·1[β+1,3β−2)(x) + 1[3β−2,2β](x). Furthermore, we have Z [0,2β] φ(x)dλ(x) =49−23β β ,
so the densityhof the unique absolutely continuous invariant measure is given by
h(x) = βφ(x) 49−23β.
Remark 3.7. — In [22] more is said about piecewise linear transformations with max-imal slopeβ for whichI < β. For example, Wilkinson proves that these transforma-tions are exact and weak Bernoulli. We can remark that greedy β-transformations with arbitrary digit for which the number of digits m+ 1 satisfies m < β ≤m+ 1 have these same properties, i.e., they are exact and weak Bernoulli.
3.2. Ultimately periodic endpoints. — The condition we impose on the system in this second example is that the endpoints of the transformation must have ultimately periodic orbits. What we mean by this is clarified in the following definition.
Let T be the greedy transformation for β > 1 and allowable digit set A = {0, a1, . . . , am}, restricted to [0, yi0] as given in Theorem 2.8. Let N ≥ 1 be the
largest index such that aN
β < yi0. We say that the endpoints of T have ultimately
periodic orbits if for each i ∈ {1, . . . , N} there exist numbers u(i), p(i), such that
Tu(i)+p(i)yi =Tu(i)yi. In this case we say that the pointsyi have ultimately periodic orbits of periodp(i).
In [2] Byers and Boyarsky proved some nice results about the absolutely continuous invariant measure of a certain class of piecewise linear functions, namely the piecewise linear Markov maps, which they defined as follows. Let 0 =α0< α1<· · ·< αn= 1 be a partition of the interval [0,1], denoted by P. A map τ : [0,1] → [0,1] is called a piecewise linear Markov map if τ|(αi,αi+1) is a linear homeomorphism onto
some interval (αj(i), αk(i)) for all i ∈ {0, . . . , n−1}. If τ is such a map, it induces
an n×n 0-1 matrix M = Mτ in the following way. The entry mij equals 1 if [αj, αj+1)⊆τ[αi, αi+1) and 0 otherwise. The fact thatτis a piecewise linear Markov
map guarantees that the nonzero entries in each row are contiguous. In [2], Byers and Boyarsky proved the following results.
Letτ be a piecewise linear Markov map, that is expanding and of constant slope and suppose that the 0-1 matrixM it induces is irreducible. Then
(d1) There exists a unique invariant probability measure µ, that is equivalent to Lebesgue measure and that maximizes entropy.
(d2) The entropy of τ with respect to this measure µ equals logβ, where β is the spectral radius ofM and is also equal to the slope ofτ.
A combination of these results, with those found by Parry in [16], gives that the invariant measureµofτcan be found in the following way. Letβbe as given by (d2). Letv= (v0, . . . , vn) be the right eigenvector ofM, belonging to the eigenvalueβ and
such that n X
i=0
belonging to eigenvalueβ and such that n X
i=0
uivi = 1. Then the function
φ: [0,1]−→[0,1] x7−→ n X i=0 ui·1[αi,αi+1)(x) (8)
is the density of the unique absolutely continuous invariant measure forτ that we are looking for.
Now, consider the greedy transformation with arbitrary digitsT that has ultimately periodic endpoints. Using the orbits of these endpoints, we make a partitionP of the interval [0, yi0]. Consider the set
I={Tkyi: 1≤i≤N, k≥0} ∪ ß ai β : 1≤i≤N} ∪ {0 ™ .
Since all the orbits of the pointsyiare periodic, this set only contains a finite number of elements, sayn+ 1 elements, so we can put them in increasing order to get a sequence 0 = α0 < α1 < · · · < αn = yi0. This gives us the partition P, i.e., P = {Pi}
n i=0
withPi= [αi, αi+1). The next lemma states that in this case T is a piecewise linear
Markov map for this partition.
Lemma 3.8. — LetT andP be as above. ThenT is linear on each of the elements of
P and for eachi∈ {0, . . . , n} we haveT Pi= [αj(i), αk(i)), for someαj(i), αk(i)∈I.
Proof. — Since all the points ai
β are in I, it is easy to see that T is linear on each element ofP. Now fix an element Pi ∈ P. If αi =
aj
β for somej ∈ {1, . . . N}, then
T Pi = [0, Tky`) for some`∈ {1, . . . , N}andk≥0. Supposeαi=Tky`. Ifαi+1=
aj
β, then T αi < yj , so T Pi = [Tk+1y`, yj). If, on the other hand αi+1 =Tmyj, then
T αi < T αi+1andT Pi= [Tk+1y`, Tm+1yj). In all casesT Piis of the desired form. The following lemma states that each element of P is eventually mapped in each other element ofP. This implies that the matrix thatT induces is irreducible. Lemma 3.9. — For eachi, j∈ {0, . . . , n} there exists ank≥0, such thatPj ⊆TkPi.
Proof. — Let Pi and Pj be two elements of P. Let µ be the measure given by Theorem2.8and letφbe its density. By (c4) we know thatφ >0 almost everywhere onPiandPj, soPiandPjboth have strictly positive measure. Moreover,µis ergodic, so there exists ak≥0, such that
(9) µ(Pi∩T−kPj)>0. By Lemma 3.8, we have that TkP
i = S`∈αP` for some index set α ⊆ {0, . . . , n}. Then (9) gives thatµ(Pj∩S`∈αP`)>0, so thatPj⊆TkPi.
It is easy to see that the transformation T, restricted to [0, yi0] is isomorphic to
a greedy transformation with arbitrary digits ¯T, defined on the interval [0,1]. The isomorphismψ is given byψ: [0,1]→[0, yi0] :x7→yi0xand we haveT◦ψ=ψ◦T¯.
The transformationT induces ann×n0-1 matrixM in the following way. The entry
mij = 1 ifPj⊆T Piandmij = 0 otherwise. By Lemma3.8the ones in each row ofM are consecutive and by Lemma3.9, the matrixM is irreducible. Letv= (v0, . . . , vn)
be the right eigenvector ofM, belonging to the eigenvalueβand such thatPn
i=0vi = 1 and suppose thatu= (u0, . . . , un) is the left eigenvector ofM belonging to eigenvalue
β and such thatPn
i=0uivi = 1. The transformation ¯T induces the same matrix and therefore, we have the following theorem.
Theorem 3.10. — Let β >1 andA={0, a1, . . . , am} be an allowable digit set.
Con-sider T on[0, yi0] and suppose that all the endpoints ofT have periodic orbits. LetT¯
be the greedy transformation with arbitrary digits, defined on [0,1], which is isomor-phic toT by the isomorphismψdefined above. Then the unique absolutely continuous invariant measureµT is the unique measure that maximizes entropy. This entropy is
given bylogβ. The measure µT is defined for all measurable setsB by
µT(B) =µT¯(ψ−1(B)),
whereµT¯ is the absolutely continuous invariant measure forT¯, of which the probability
density function is given by equation (8).
We consider the same two examples as in the previous paragraph.
Example 3.11. — In the first example, β = 1 +√2 was the positive solution of the equationβ2−2β−1 = 0 , and the allowable digit set considered wasA ={0,1,3}.
The support of the absolutely continuous measure was the interval [0,2]. We already saw that both endpoints have finite orbits and the partition P ={Pi}5i=0, given by
these orbits is as follows:
P0= ï 0, 1 β ã , P1= ï 1 β,1 ã , P2= ï 1,3 β ã , P3= ï 3 β, β−1 ã , P4= [β−1,2β−3), P5= [2β−3,2].
This gives us the following 0-1 matrix: M = 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 .
Let v= (v0, . . . , v5) be the right eigenvector of M with eigenvalueβ and such that
P5
i=0vi= 1 and let u= (u0, . . . , u5) be the left eigenvector of M for the eigenvalue
β and such thatP5
i=0uivi = 1. Then v= 1 2β2(β, β+ 1, β−1,1, β,1) and u= β 4β−2(2β, β+ 2, β, β, β−1,1).
This means that the invariant probability density for our transformation T is given by h(x) = 1 2· β 4β−2 [2β·1[0,1β)(x) + (β+ 2)·1[ 1 β,1)(x) +β·1[1,β−1)(x) + (β−1)·1[β−1,2β−3)(x) + 1[2β−3,2](x)],
just as we obtained before.
Example 3.12. — In the second example,β = 1+ √
5
2 was the golden mean andA =
{0,2β,5} was the allowable digit set. The support of the absolutely continuous mea-sure was [0,2β]. In this case, both the orbit of 2β and that of 5−2β were eventually periodic and the partitionP ={Pi}12i=0and 0-1 matrixM they give are the following.
For the partition we have
P0= [0,2β−3), P1= [2β−3,2−β), P2= ï 2−β, 1 β ã , P3= ï 1 β,1 ã , P4= [1,3−β), P5= [3−β, β), P6= [β,5−2β), P7= [5−2β,2), P8= [2,2β−1), P9= [2β−1, β+ 1), P10= [β+ 1,3β−2), P11= ï 3β−2,5 β ã , P12= ï 5 β,2β ò .
And for the matrix, M = 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 .
The right eigenvector, v, with eigenvalue β and such that the sum of its elements equals 1 is
v= 1
10β+ 6(β,1, β, β+ 1, β+ 1, β,1, β, β, β+ 1, β, β,1)
and the left eigenvector, u, belonging to the eigenvalue β and such that the dot product withvis 1 is
u=10β+ 6
29β+ 3(2β+ 1,2 +β,8−3β,3β−2, β+ 1,4−β,2β−1, β, β,4β−5,3−β,1,1). The invariant probability density then is
h(x) = 1 2β · 10β+ 6 29β+ 3 [(2β+ 1)·1[0,2β−3)(x) + (β+ 2)·1[2β−1,2−β)(x) + (8−3β)·1[2−β,β−1)(x) + (3β−2)·1[β−1,1)(x) + (β+ 1)·1[1,3−β)(x) + (4−β)·1[3−β,β)(x) + (2β−1)·1[β,5−2β)(x) +β·1[5−2β,2β−1)(x) + (4β−5)·1[2β−1,β+1)(x) + (3−β)·1[β+1,3β−2)(x) + 1[3β−2,2β](x)].
And again, this is equal to the result from the previous paragraph.
4. Conclusions
In the second section of this article we have established that the greedyβ -trans-formation with arbitrary digits has a unique invariant measure that is absolutely
continuous with respect to the Lebesgue measure. We saw that this measure is ergodic and gave the interval on which the density function is strictly positive.
In the last section we have studied two specific examples of the greedy transforma-tion in which this absolutely continuous invariant measure can be explicitly calculated. The first example put a restraint on the number of digits that can be chosen. If this numberm+ 1, satisfiesm < β≤m+ 1, then the density of the absolutely continuous invariant measure is given by Wilkinson’s formula. We remarked that in this case the absolutely continuous, invariant measure is exact and weak Bernoulli. The second example we studied is the case in which the endpoints of the transformation have ultimately periodic orbits. In that case the absolutely continuous invariant measure is also the measure of maximal entropy (with entropy equal to logβ) and its density is given by Parry’s formula.
Acknowledgement. — We would like to thank the referee for the detailed comments which greatly improved the presentation of the paper.
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K. Dajani, Department of Mathematics, Utrecht University, Postbus 80.000, 3508 TA Utrecht, the Netherlands • E-mail :[email protected]
C. Kalle, Department of Mathematics, Utrecht University, Postbus 80.000, 3508 TA Utrecht, the Netherlands • E-mail :[email protected]
