Limiting curves for polynomial adic systems

1.3 MB·

by  Aleksei Minabutdinov

Limiting curves for polynomial adic systems.∗
7 A. R. Minabutdinov†
1
0
January 27, 2017
2

n
a
J
6 Abstract
2
We prove the existence and describe limiting curves resulting from

deviations in partial sums in the ergodic theorem for cylindrical func-
]
S tions and polynomial adic systems. For a general ergodic measure-
D preserving transformation and a summable function we give a neces-
. sary condition for a limiting curve to exist. Our work generalizes re-
h
sultsbyÉ.Janvresse,T.delaRueandY.Velenikandanswersseveral
t
a questions from their work.
m
[ Key words: Polynomial adic systems, ergodic theorem, deviations in er-

godic theorem.
1
v MSC: 37A30, 28A80
7
1
6 1 Introduction
7
0
. Inthis paperwedevelop thenotionofa limiting curveintroducedbyÉ.Jan-
1
vresse, T. de la Rue and Y. Velenik in [16]. Limiting curves were studied for
0
7 the Pascal adic in [16] and [11]. In this paper we study it for a wider class
1 of adic transformations.
:
v Let T be a measure preserving transformation defined on a Lebesgue
i
X probability space (X,B,µ) with an invariant ergodic probability measure µ.
r Let g denote a function in L1(X,µ). Following [16] for a point x ∈ X and a
a
j−1
positiveintegerj wedenotethepartialsum (cid:80) g(cid:0)Tkx(cid:1)bySg(j). Weextend
x
k=0
the function Sg(j) to a real valued argument by a linear interpolation and
x
denote extended function by Fg(j) or simply F(j),j ≥ 0.
x
∗Supported by the RFBR (grant 14-01-00373)
†National Research University Higher School of Economics, Department of
Applied Mathematics and Business Informatics, St.Petersburg, Russia, e-mail:
[email protected].
1
Let(l )∞ beasequenceofpositiveintegers. Weconsidercontinuouson
n n=1
[0,1] functions ϕ (t) = F(t·ln(x))−t·F(ln)(cid:0) ≡ ϕg (t)(cid:1), where the normalizing
n Rn x,ln
coefficient R is canonically defined to be equal to the maximum in t ∈ [0,1]
n
of |F(t·l (x))−t·F(l )|.
n n
Definition 1. If there is a sequence lg(x) ∈ N such that functions ϕg
n x,lng(x)
convergetoa(continuous)functionϕg insup-metricon[0,1],thenthegraph
x
ofthelimitingfunctionϕ = ϕg iscalledalimiting curve, sequencel = lg(x)
x n n
is called a stabilizing sequence and the sequence R = Rg is called a
(cid:16) n x,lng(x(cid:17))
(cid:0) (cid:1)∞ (cid:0) (cid:1)∞
normalizing sequence. The quadruple x, l , R ,ϕ is called a
n n=1 n n=1
limiting bridge.
Heuristically, the limiting curve describes small fluctuations (of certainly
renormalized) ergodic sums 1F(l),l ∈ (l ), along the forward trajectory
l n
x,T(x),T2(x).... More specifically, for l ∈ (l ) it holds F(t·l) = tF(l)+
n
R ϕ(t)+o(R ), where t ∈ [0,1].
l l
In this paper we will always assume that T is an adic transformation.
Adic transformations were introduced into ergodic theory by A. M. Vershik
in [1]and were extensivelystudiedsince thattime. Thefollowing important
theorem shows that adicity assumption is not restrictive at all:
Theorem. (A.M.Vershik,[2]). Anyergodicmeasurepreservingtransforma-
tion on a Lebesgue space is isomorphic to some adic transformation. More-
over, one can find such an isomorphism that any given countable dense in-
variant subalgebra of measurable sets goes over into the algebra of cylinder
sets.
In [2,3, 5]authors encouraged studyingdifferentapproaches to combina-
torics of Markov’s compacts (sets of paths in Bratteli diagrams). In particu-
lar, it is interesting to find a natural class of adic transformations such that
the limiting bridges exist for cylindric functions. Moreover, it is interesting
to study joint growth rates of stabilizing and normalizing sequences.
In this paper we give necessary condition for a limiting curve to exist.
Next we find necessary and sufficient conditions for almost sure (in x) ex-
istence of limiting curves for a class of self-similar adic transformations and
cylindricfunctions. Thesetransformations(inaslightlylessgenerality)were
considered by X. Mela and S. Bailey in [19] and [12]. Our work extends [16]
and answers several questions from this research.
2
2 Limiting curves and cohomologous to a constant
functions
In this section we show that a necessary condition for limiting curves to
exist is unbounded growth of the normalizing coefficient R . Contrariwise
n
we show that normalizing coefficients are bounded if and only if function g
is cohomologous to a constant. In particular this implies that there are no
limiting curves for cylindric functions for an ordinary odometer.
2.1 Notions and definitions
Let B = B(V,E) denote a Bratteli diagram defined by the set of vertices
V and the set of edges E. Vertices at the level n are numbered k = 0
through L(n). We associate to a Bratteli diagram B the space X = X(B) of
infiniteedgepathsbeginningatthevertexv = (0,0).Followingfundamental
0
paper [1] we assume that there is a linear order ≤ defined on the set of
n,k
edgeswithaterminatevertex(n,k),0 ≤ k ≤ L(n). Theselinearordersdefine
alexicographicalorderonthesetofedgespathsinX thatbelongtothesame
class of the tail partition. We denote by (cid:22) corresponding partial order on
X. Thesetofmaximal(minimal)pathsisdefinedbyX (correspondingly,
max
X ).
min
(cid:0) (cid:1)
Definition 2. Adic transformation T is defined on X \ X ∪X by
max min
setting Tx, x ∈ X, equal to the successor of x, that is, the smallest y that
satisfies y (cid:31) x.
Let ω be a path in X. We denote by (n,k (ω)) a vertex through which ω
n
passesatleveln. Forafinitepathc = (c ,...,c )wedenotek (c)simplyby
1 n n
k(c). A cylinder set C = [c c ...c ] = {ω ∈ X|ω = c ,ω = c ,...,ω =
1 2 n 1 1 2 2 n
c }ofaranknistotallydefinedbyafinitepathfromthevertex(0,0)tothe
n
vertex (n,k) = (n,k(c)). Sets π of lexicographically ordered finite paths
n,k
c = (c ,c ,...,c ), k(c) = n, areinonetoonecorrespondencewithtowers
0 1 n−1
τ made up of corresponding cylinder sets C = τ (j),1 (cid:54) j (cid:54) dim(n,k).
n,k j n,k
The dimension dim(n,k) of the vertex (n,k) is the total number of such
finite paths (rungs of the tower).
We denote by Num(c) the number of finite paths in lexicographically
ordered set π . Evidently, 1 ≤ Num(c) ≤ dim(n,k). For a given level n
n,k
the set of towers {τ } defines approximation of transformation T,
n,k 0≤k≤L(n)
see [1], [3].
We can consider a vertex (n,k) of Bratteli diagram B as an origin in
a new diagram B(cid:48) = (V(cid:48),E(cid:48)). The set of vertices V(cid:48), edges E(cid:48) and edges
n,k
3
paths X(B(cid:48) ) are naturally defined. As above partial order (cid:22)(cid:48) on X(B(cid:48)) is
n,k
induced by linear orders ≤ ,n(cid:48) > n.
n(cid:48),k(cid:48)
Definition 3. Ordered Bratteli diagram (B,(cid:22)) is self-similar if ordered
diagrams (B,(cid:22)) and (B(cid:48) ,(cid:22)(cid:48) ) are isomorphic n ∈ N, 0 (cid:54) k (cid:54) L(n).
n,k n,k
Let F denote the set of all functions f : X → R. We denote by F the
N
space of cylindric functions of rank N (i.e. functions that are constant on
cylinders of rank N).
Letg ∈ F , N < n.WedenotebyFg linearlyinterpolatedpartialsums
N n,k
Sg . Assume that self-similar Bratteli diagram B has L+1 vertices at
x∈τk(1)
n
level N and let ω ∈ π ,0 (cid:54) k (cid:54) L(n), be a finite path such that its
n,k
initial segment ω(cid:48) = (ω ,ω ,...,ω ) is a maximal path, i.e. Num(ω(cid:48)) =
1 2 N
dim(N,k(ω(cid:48))). Let EN,l denote the number of paths from (0,0) to (n,k)
n,k
passing through the vertex (N,l),0 ≤ l ≤ L, and not exceeding path ω. We
denote by ∂N,l(ω) the ratio of EN,l to dim(N,l). It is not hard to see that a
n,k n,k
partial sum Fg evaluated at j = Num(ω) has the following expression:
n,k
L
Fg (j) = (cid:88)hg ∂N,l(ω), (1)
n,k N,l n,k
l=0
where coefficients hg are equal to Fg (H ), 0 (cid:54) l (cid:54) L.
N,l N,l N,l
Expression(1)isageneralizationofVandermonde’sconvolutionformula.
2.2 A necessary condition for existence of limiting curves
Let (X,T) be an ergodic measure-preserving transformation with invariant
measure µ. Let g be a summable function and a point x ∈ X. We consider
a sequence of functions ϕg and normalizing coefficients Rg given by the
x,ln x,ln
identity
Sg([t·l (x)])−t·Sg(l (x))
ϕg (t) = x n x n ,
x,ln(x) Rg
x,ln(x)
whereRg equalsmaximumofabsolutevalueofthenumerator. Without
x,ln(x)
loss of generality, we assume that the limit g∗(x) = lim 1Sg exists at the
n→∞ n x
point x. The following theorem generalizes Lemma 2.1 from [16] for an
arbitrary summable function.
Theorem 1. If a continuous limiting curve ϕg = lim ϕg exists for µ-a.e.
x n x,ln
x, then the normalizing coefficients Rg are unbounded in n.
x,ln
4
Proof. Assume the contrary that |Rg | (cid:54) K. For simplicity we introduce
x,ln
the following notation: S = Sg, ϕ = ϕg , R = Rg and ϕ = ϕ .
x n x,ln n x,ln x
Since ϕ (cid:54)= 0, there is j ∈ N such that 1S(j) (cid:54)= g∗. This in turn implies
j
lim(cid:12) infn(cid:12)(cid:12)ϕn(ljn)(cid:12)(cid:12) = liminfn R1n(cid:12)(cid:12)S(j)− jSl(nln)(cid:12)(cid:12) (cid:62) K1|S(j)−jg∗| = Kj (cid:12)(cid:12)1jS(i)−
g∗(cid:12) > 0, contradicting continuity of the limiting curve ϕ at the origin.
Definition 4. Afunctiong ∈ L∞(X,µ)(µ-a.e.) oftheformg = c+h◦T−h
for some c ∈ R and h ∈ L∞(X,µ) is called cohomologous to a constant in
L∞.
Theorem 2. Normalizing sequence Rg is bounded if and only if function
x,ln
g is cohomologous to a constant.
n−1
Proof. Sums (cid:80)(g−g∗)◦Tj ofacohomologousfunctionareµ-a.e. bounded,
j=0
therefore normalizing coefficients Rg are µ-a.e. bounded too.
x,ln
TheproofoftheconversestatementexploitstheresultbyA.G.Kachurovskiy
from [7]. Assume that the normalizing coefficients Rg are bounded. Then
for µ-a.e. point x ∈ X and for any j ∈ N the follxo,wlning inequality holds
j
|Sg(j)− j Sg(l )| (cid:54) C. Going to the limit in n, we see that |(cid:80)f◦Ti(x)| (cid:54)
x ln x n
i=1
C, where f = g − g∗. Theorem 19 from [7] (see also G. Halasz, [14]), in-
equality |Sf| (cid:54) C, is equivalent to existence of a function h ∈ L∞, such that
x
f = h◦T −h. Therefore g equals to h◦T −h+g∗.
Definition 5. Let B be a Bratteli diagram such that there is only one ver-
tex at each level, and let the edge ordering be such that the edges increase
from left to right. This transformation is called an odometer. A stationary
odometer is an odometer for which the number of edges connecting consec-
utive levels is constant.
Theorem 3. Let (X,T) be an odometer. Any cylindric function g ∈ F
N
is cohomologous to a constant. Therefore there is no limiting curve for a
cylindric function.
Proof. There is only one vertex (n,0) at each level n. Expression (1) for
the partial sum Fg (i) is evidently valid for any odometer (even without
n,0
assumption of self-similarity). Moreover, expression (1) is defined by the
only coefficient hg and therefore is proportional to H = dim(N,0). We
N,0 N
5
b
b
b
b
Figure 1: A Bratteli diagram of an odometer.
cansubtractsuchconstantC tothefunctiong thatequalityhg−C = 0holds.
N,0
But this is equivalent to the following: The function function g−C belongs
to the linear space spanned by the functions f −f ◦T,1 (cid:54) j (cid:54) H , where
j j N
f is the indicator-function of the j-th rung in the tower τ . Therefore
j N,0
function g−C is cohomologous to zero.
3 Existence of limiting curves for polynomial adic
systems
In this part we will show that any not cohomologous to a constant cylindric
function in a polynomial adic system has a limiting curve. These generalizes
Theorem 2.4. from [16].
3.1 Polynomial adic systems
Let p(x) = a + a x··· + a xd be an integer polynomial of degree d ∈ N
0 1 d
with positive integer coefficients a ,0 ≤ i ≤ d. Bratteli diagram B = (V,E)
i p p
associated to polynomial p(x) is defined as follows:
1. Number of vertices grows linearly: |V | = 1 è |V | = |V | + d =
0 n n−1
nd+1,n ∈ N.
2. If 0 (cid:54) j (cid:54) d vertices (n,k) and (n+1,k+j) are connected by a edges.
j
Polynomial p(x) is called a generating polynomial of the diagram B , see
p
paper [12] by S. Bailey.
6
b
b b b
b bb bbb bb b
b bb bbb bbb bbb bb b
b bb bbb bbb bbb bbb bbb bb b
Figure 2: Bratteli diagram associated to polynomial 1+x+3x2.
Since the number of edges into vertex (n,k) is exactly p(1) = a +a +
0 1
···+a it is natural to use the alphabet A = {0,1,...,a +a +···+a −1}
d 0 1 d
for edges labeling. We call a lexicographical order defined in [12] a canonical
order. It is defined as follows: Edges connecting (0,0) with (1,d) are labeled
through 0 to a −1 (from left to right); edges connecting (0,0) and (1,d−1),
d
are indexed by a to a +a −1, etc. Edges connecting (0,0) and (1,0),
d d d−1
are indexed through a +a +···+a to a +a +···+a .
0 1 d−1 0 1 d
Infinite paths are totally defined by this labeling and may be considered
N
as one sided infinite sequences in A . We denote the path space by X .
p
b 2
4 3 1
0
b b b
Figure 3: Labeling of the polynomial system associated to 1+x+3x2.
We denote by T the adic transformation associated with the canonical
p
ordering.
Remark. Any self-similar Bratteli diagram is either a diagram of a
stationary odometer or is associated to some polynomial p(x). Any non-
canonical ordering is obtained from canonical by some substitution σ.
Everywhere below we stick to the canonical order. Case of general order
needs several straightforward changes that are left to the reader.
Dimension of the vertex (n,k) from diagram B equals to the coefficient
p
ofxk inthepolynomial(p(x))n andiscalled generalized binomial coefficient.
We denote it by C (n,k). For n > 1 coefficients C (n,k) can be evaluated
p p
7
by a recursive expression C (n,k) = (cid:80)d a C (n−1,k−j).
p j=0 j d
In [19] and [12] X. Méla and S. Bailey showed that the fully supported
invariant ergodic measures of the system (X ,T ) are the one-parameter
p p
family of Bernoulli measures:
Theorem 4. (S. Bailey, [12], X. Méla, [19]) 1. Let q ∈ (0, 1 ) and t is the
a0 q
unique solution in (0,1) to the equation
a qd+a qd−1t+···+a td−qd−1 = 0,
0 1 d
then the invariant, fully supported, ergodic probability measures for the adic
transformationT aretheone-parameterfamilyofBernoullimeasuresµ ,q ∈ (0, 1 ),
p q a0
(cid:89)∞ (cid:18) t2q t2q tdq tdq (cid:19)
µ = q,...,q,t ,...,t , ,..., ,..., ,..., .
q q q q q qd−1 qd−1
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
0 a0 a1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
a2 ad
2.Invariant measures that are not fully supported are
∞ (cid:18) (cid:19) ∞ (cid:18) (cid:19)
(cid:89) 1 1 (cid:89) 1 1
,..., ,0,...,0 and 0,...,0, ,..., , .
a a a a
0 0 d d
0 (cid:124) (cid:123)(cid:122) (cid:125) 0 (cid:124) (cid:123)(cid:122) (cid:125)
a0 ad
Definition 6. Polynomial adic system associated with polynomial p(x), is a
triple (X ,T ,µ ), q ∈ (0, 1 ).
p p q a0
In particular, if p(x) = 1+x system (X ,T ,µ ), q ∈ (0,1), is the well-
p p q
known Pascal adic transformation. Transformation was defined in [2] by
A. M. Vershik 1 and was studied in many works [20, 15, 4, 6], see more
complete list in the last two papers. For the Pascal adic space X is an
p
infinite dimensional unit cube I = {0,1}∞, while measures µ are dyadic
q
Bernoulli measures (cid:81)∞(q,1−q). Transformation T = P is defined by the
1 p
following formula (see [2])2:
x (cid:55)→ Px; P(0m−l1l10...) = 1l0m−l01... (2)
(thatisonlyfirstm+2coordinatesofxarebeingchanged). De-Finetti’sthe-
oremandHewitt-Savage0–1lawimplythatallP-invariantergodicmeasures
are the Bernoulli measures µ = (cid:81)∞(p,1−p), where 0 < p < 1.
p 1
Below we enlist several known properties of the polynomial systems:
1However earlier isomorphic transformation was used by [13] and [17].
2Pk(x), k ∈Z, is defined for all x except eventually diagonal, i.e., except those x for
which there exists n∈N such that either x =0 for all k≥n or x =1 for all k≥n
k k
8
1. Polynomial systems are weakly bernoulli (the proof essentially follows
[15] and is performed in [19] and [12]).
2. Complexity function has polynomial growth rate (for the Pascal adic
firsttermofasymptoticexpansionisknowntobeequalto n3,see[20]).
6
3. Polynomialsystem(X ,T ,µ )definedbyapolynomialp(x) = a +a x
p p q 0 1
with a a > 1 has a non-empty set of non-constant eigenfunctions.
0 1
Authorsof[16]studiedlimitingcurvesforthePascaladictransformation
(I,P,µ ),q ∈ (0,1).
q
Theorem 5. ([16], Theorem 2.4.) Let P be the Pascal adic transformation
definedonLebesgueprobabilityspace(I,B,µ ),q ∈ (0,1),andg beacylindric
q
function from F . Then for µ -a.e. x limiting curve ϕg ∈ C[0,1] exists if
N q x
and only if g is not cohomologous to a constant.
For the Pascal adic limiting curves can be described by nowhere differ-
entiable functions, that generalizes Takagi curve.
Theorem 6. ([11], Theorem 1.) Let P be the Pascal adic transformation
defined on the Lebesgue space (I,B,µ ), N ∈ N and g ∈ F be a not co-
q N
homologous to a constant cylindric function. Then for µ -a.e. x there is
q
a stabilizing sequence l (x) such that the limiting function is α T1, where
n g,x q
α ∈ {−1,1}, and T1 is given by the identity
g,x q
∂F
T1(x) = µq ◦F−1(x), x ∈ [0,1],
q ∂q µq
where F is the distribution function3 of µ .
µq q
The graph of 1T1 is the famous Takagi curve, see [22].
2 1/2
Forafunctiong correlatedwiththeindicatorfunctionsofi-thcoordinate
1 , x = (x )∞ ∈ X Theorem 6 was proved in [16].
{xi=0} j j=1
3.2 Combinatorics of finite paths in the polynomial adic sys-
tems
In this section we’ll specify representation (1) for the polynomial adic sys-
tems.
3More precisely F is distribution function of measure µ˜ , that is image of µ under
µq q q
∞
canonical mapping φ:I →[0,1], φ(x)= (cid:80) xi.
2i
i=1
9
For a finite path ω = (ω ,ω ,...,ω ) we set k1(ω) equal to nd−k(ω).
1 2 n
Usingself-similarityofthediagramB wecaninductivelyprovethefollowing
p
explicit expression for Num(ω):
Proposition 1. Index Num(ω) of a finite path ω = (ω )n in lexicographi-
j j=1
cally ordered set π is defined by equality:
n,k(ω)
r ωaj−1
(cid:88) (cid:88)
Num(ω) = C (a −1;k1(ω)−k1(i)−m )+Num(ω ), (3)
P j j 1
j=2 i=0
r−1
where m = (cid:80) k1(ω ),2 (cid:54) j (cid:54) r−1, m = 0, and polynomial P(x) is given
j at r
t=j
by the identity P(x) = xdp(x−1).
Remark. If initial segment (ω ,ω ,...,ω ) of ω ∈ π is a maximal
1 2 N n,k
path to some vertex (N,l), then (3) can be rewritten as follows:
r ωaj−1
(cid:88) (cid:88)
Num(ω) = C (a −1;k1(ω)−k1(i)−m )+C (a ;k1(ω)−m ).
P j j P N l
j=N+1 i=0
(4)
Let N and l, 0 (cid:54) l (cid:54) Nd, be positive integers and ω ∈ π be a finite
n,k
path. Function ∂N,l : Z → Z ,k1 = nd−k, is defined by the identity
k1 + +
r ωaj−1
∂N,lω = (cid:88) (cid:88) C (a −1−N;k1−k1(i)−m −l), (5)
k1 P j j
j=N+1 i=0
where positive integers a ,k1(i),m , are defined as in (4). Parameters N
j j
and l correspond to shifting the origin vertex (0,0) to the vertex (N,l).
Therefore value of the function ∂N,lω,k1 = k1(ω), equals to the number of
k1
paths from the vertex (0,0) going through the vertex (N,l) to the vertex
(n,k),k = nd−k1, and non-exceding path ω divided by dim(N,l).
Let K , 1 (cid:54) M (cid:54) n, denote indexes (in lexicographical order)
M,n,k
of those paths ω = (ω ,...,ω ) ∈ π , such that their initial segment
1 n n,k
(ω ,ω ,...,ω ) is maximal (as a path from (0,0) to some vertex (M,l)).
1 2 M
Let g ∈ F . Function F˜g,M : K → R (where M,N (cid:54) M (cid:54) n is a
N n,k M,n,k
positive integer) is defined by the identity
Nd
F˜g,M(j) = (cid:88)hg ∂M,l ω, (6)
n,k M,l nd−k
l=0
10