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CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE
UNIVERSITE de ROUEN
PUBLICATION de l’UMR 6085
LABORATOIRE DE MATHEMATIQUES RAPHAEL SALEM
#
SELF-SIMILAR CORRECTIONS TO THE ERGODIC THEOREM
FOR THE PASCAL-ADIC TRANSFORMATION.
E. Janvresse, T. de la Rue, Y. Velenik
" !
Document 2004-03
Universite de Rouen UFR des sciences
Mathematiques, Site Colbert, UMR 6085
F 76821 MONT SAINT AIGNAN Cedex
Tel: (33)(0) 235 14 71 00 Fax: (33)(0) 232 10 37 94Self-Similar Corrections to the Ergodic Theorem for
the Pascal-Adic Transformation
E. Janvresse, T. de la Rue, Y. Velenik
June 9, 2004
Abstract
Let T be the Pascal-adic transformation. For any measurable function
g, we consider the corrections to the ergodic theorem
j 1 ‘ 1X Xjk kg(T x) g(T x).

k=0 k=0
When seen as graphs of functions de ned on {0,...,‘ 1}, we show for a
suitable class of functions g that these quantities, once properly renormal-
ized, converge to (part of) the graph of a self-a ne function depending
only on the ergodic component of x.
Key words: Pascal-adic transformation, ergodic theorem, self-a ne function.
AMS subject classi cation: 37A30, 28A80.
1 Introduction
1.1 The Pascal-adic transformation and its basic blocks
The notion of adic transformation has been introduced by Vershik (see e.g.
[7], [6]), as a model in which the transformation acts on in nite paths in some
graphs, called Bratteli diagrams. As shown by Vershik, every ergodic automor-
phism of the Lebesgue space is isomorphic to some adic transformation, with
a Bratteli diagram which may be quite complicated. Vershik also proposed to
study the ergodic properties of an adic transformation in a given simple graph,
such as the Pascal graph which gives rise to the so-called Pascal adic transfor-
mation. We recall the construction of the latter by the cutting-and-stacking
method in the appendix.
When studying the Pascal-adic transformation, one is naturally led to con-
sider the family of words B (n 1, 0 k n) on the alphabet {a,b},n,k
inductively dened by (see Figure 1)
B := a, B := b, (n 1)n,0 n,n
and for 0<k <n
B := B B .n,k n 1,k 1 n 1,k
2a b
a ab b
a aab abb b
a aaab aababb abbb b
Figure 1: The beginning of the words triangle.
Figure 2: The graph F associated to the word B =6,3 6,3
aaabaababbaababbabbb.
It follows easily from this denition that the length of the block B is givenn,k
nby the binomial coe cient .
k
In order to describe the large-scale structure of the basic blocks B , wen,k
associatetoeachofthemthegraphofareal-valuedfunctionF . Letusdenoten,k
by B (‘) the ‘th letter of B . For each n 2 and 0 k n, we considern,k n,k
nthe functionF : [0, ]→R de ned from the basic block B as follows (seen,k n,kk
Figure 2):
F (0) = 0;n,k
(
F (‘ 1)+1 if B (‘) =a,n,k n,kn if 1‘ is an integer, F (‘) =n,kk F (‘ 1) 1 if B (‘) =b;n,k n,k
F is linearly interpolated between two consecutive integers.n,k
As we will see in Section 2.1, the ergodic theorem implies that the graph of
the function
1 nt 7 → F tn,kn k
k
converges to a straight line as n → ∞ and k/n → p. In order to extract the
nontrivialstructureofthisgraph,wehavetoremovethisdominantcontribution
andlookatthecorrection(seeSection2). Oncethisisdone, itappearsthatthe
resulting graph converges to the graph of a self-a ne function depending only
on p = limk/n, described in the following section. Examples of such limiting
graphs are shown in Figure 3.
1.2 A one-parameter family of self-a ne maps
L RFor any 0<p< 1, we consider the two a nities and de ned byp p
L (x,y) := (px,py+x),p
3Figure 3: Limiting observed shape for B with p = 0.5 (left) andn,pn
p = 0.8 (right).
and
R (x,y) := (1 p)x+p,(1 p)y x+1 .p
These maps are strict contractions of [0,1]R, thus there exists a unique
compact set E such thatp
L RE = (E )∪ (E ).p p pp p
Asshownin[1],E isthegraphofacontinuousself-a nemap : [0,1]→R,p p
whose construction is illustrated in Figure 4 (see also [2, Chapter 11]).
Acknowledgments
WewishtothankXavierMelaforhavingshownusthebeautifulshapeofB ,2n,n
which prompted our interest in this topic.
2 Results
As mentioned before, we have to renormalize F into a new function ϕn,k n,k
de ned on [0 ,1], vanishing at 0 and 1, and vertically scaled so that the point
corresponding to the end of B is mapped to 1:n 1,k 1

n nF t tFn,k n,kk k
ϕ (t) := . (1) n,k n 1 ( )n 1 k 1 nF Fn,k n n,kk 1 k( )k
Theorem 2.1. Let ϕ be the renormalized curve associated to the basic blockn,k
B (see (1)). For any p∈]0,1[, for any sequence (k(n)) such that k(n)/n→p,n,k
we have

L
ϕ → . (2)n,k(n) p
n→∞

n1Moreover, the denominator in (1) is of order .
n k(n)
4MME
C
D
A B
Figure 4: The rst four steps in the construction of , for p = 0.4.p
In the rst step, we transform the original interval AB into the polygonal
L Rline ACB, where C = (B) = (A). In the second step, we similarlyp p
map the segment AC onto the segments AD and DC, and the segment
CB onto CE and EB, by applying the same a ne transformations. The
procedure is then iterated yielding an increasing sequence of piecewise-
linear functions, converging to the self-a ne function .p
2.1 Ergodic interpretation
The functions F and ϕ introduced before can be interpreted as partic-n,k n,k
ular cases of the following general situation: Consider a real-valued function
g de ned on a probability space ( X,) on which acts a measure-preserving
transformation T. Given a point x ∈ X and an integer ‘ 1, we construct
g gthe continuous function F : [0,‘] → R by F (0) := 0; for each integer j,
x,‘ x,‘
1j ‘,
j 1 X
g kF (j) := g T x ; (3)
x,‘
k=0
gand F is linearly interpolated between the integers.
x,‘
Ifg isintegrable(whichwehenceforthassume), theergodictheorem implies
that, for 0<t< 1, for almost every x,
X X 1 1j jlim g T x = t lim g T x .
‘→∞ ‘ ‘→∞ ‘
0j<t‘ 0j<‘
Therefore, when dividing by ‘ both the abscissa and the ordinate, the graph
gof F for large ‘ looks very much like a straight line with slope the empiricalx,‘ P
jmean 1/‘ g T x . If we want to study small uctuations in the ergodic0j<‘
theorem, it is natural to remove the dominant contribution of this straight
line, and rescale the ordinate to make the uctuations appear. This leads to
gintroduce a renormalized function ϕ : [0,1]→R by settingx,‘
g gF (t‘) tF (‘)
g x,‘ x,‘ϕ (t) := ,gx,‘ R
x,‘
5MMgwhere R is the renormalization in the y-direction, which we can canonically
x,‘
de ne by
(
g gmax |F (t‘) tF (‘)| provided this quantity does not vanish,g 0t1 x,‘ x,‘R :=x,‘
1 otherwise.
(4)
gIt will be useful in the sequel to note that ϕ is not changed when we add a
x,‘
constant to g.
kIfweconsideraBernoullishiftinwhichthefunctionsgT arei.i.d. random
variables, Donsker invariance principle shows that these corrections to the law
of large numbers are given by a suitably scaled Brownian bridge.
Wearegoingtoinvestigatethecorrespondingquestionsinthecontextofthe
Pascal-adic transformation. Let us recall, see Appendix A, that the sequence
n
1( )kof letters a and b in the word B encodes the trajectory x,Tx,...,T xn,k
of a point x lying in the basis of the tower with respect to the partitionn,k
[0,1/2[ (labelled by “a”) and [1/2,1[ (labelled by “b”). Thus, the function Fn,k
gis nothing else but F for x in the basis of , with the function g de nedn n,k
x,( )k
by
g := .[0,1/2[ [1/2,1[
Now, the vertical renormalization chosen to de ne ϕ was not exactly the onen,k
de ned by (4), but it is not di cult to restate Theorem 2.1 in the following
way, where we de ne
g gϕ := ϕ (5)nn,k x,( )k
for any point x in the basis of .n,k
Theorem 2.2. Let g = . If k(n)/n→p, then[0,1/2[ [1/2,1[
∞Lg pϕ → . (6)
n,k(n)
n→∞ k kp ∞
This result shows that the corrections to the ergodic theorem are given by a
deterministicfunction, whenweconsidersumsalongtheRokhlintowers . Itn,k
is possible to derive an analogous pointwise statement at the cost of extracting
a subsequence.
Theorem 2.3. Let g = . For almost every x∈ X, therep[0,1/2[ [1/2,1[
exists a sequence ‘ such thatn
∞g L p
ϕ → . (7)x,‘n n→∞ k kp ∞
2.2 Limit for general dyadic functions
N0LetN 1. We consider the dyadic partitionP ofthe interval [0,1[ into 20 N0
sub-intervals. We want to extend Theorem 2.2 to a generalP -measurableN0
real-valued function g.
6M1M1MM1111

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