Limiting curves for polynomial adic systems
Aleksei Minabutdinov




Limiting curves for   polynomial adic systems.[Supported by the RFBR (grant 14-01-00373)]
    A. R. MinabutdinovNational Research University Higher School of Economics, Department of Applied Mathematics and Business Informatics, St.Petersburg, Russia, e-mail: 
    December 30, 2023
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We prove the existence and describe limiting curves resulting from deviations in partial sums in the ergodic theorem for  cylindrical functions and   polynomial  adic systems. For a general ergodic measure-preserving transformation and a summable function we give a necessary condition for  a limiting curve to exist. Our work generalizes results by É. Janvresse, T. de la Rue and Y. Velenik and answers several questions from their work.



Key words: Polynomial adic systems, ergodic theorem, deviations in ergodic theorem.

MSC:   37A30, 28A80



§ INTRODUCTION


In this paper we develop the notion of a limiting curve introduced  by    É. Janvresse, T. de la Rue and Y. Velenik in <cit.>. Limiting curves were studied for   the Pascal adic in <cit.> and <cit.>. In this paper we study it  for  a wider class of adic transformations.


Let  T be a measure preserving transformation defined on a Lebesgue probability space (X, ℬ,μ) with an invariant ergodic probability measure μ. Let    g denote a function in L^1(X,μ). Following <cit.> for a point x∈ X and a positive integer j we denote the partial sum ∑_k=0^j-1g(T^kx) by S_x^g(j). We extend the function S_x^g(j) to a real valued argument by a linear interpolation and denote extended function by  F_x^g(j) or simply F(j),j≥ 0.

Let  (l_n)_n=1^∞ be a sequence of positive integers. We consider continuous on [0,1] functions φ_n(t) = F(t· l_n(x)) - t · F(l_n)/R_n( ≡φ_x,l_n^g(t)  ), where the normalizing coefficient R_n is canonically defined to be equal to the maximum in t∈[0,1] of |F(t· l_n(x)) - t · F(l_n)|.



If there is a sequence l^g_n(x)∈ℕ  such that functions φ_x,l^g_n(x)^g converge to a (continuous) function φ_x^g in sup-metric on [0,1], then
the graph of the limiting function φ=φ^g_x is called a limiting curve, sequence l_n=l_n^g(x) is called a stabilizing sequence and the sequence R_n = R_x,l^g_n(x)^g is called a normalizing sequence.  The quadruple (x, (l_n)_n=1^∞, (R_n)_n=1^∞, φ) is called a limiting bridge.




Heuristically, the limiting curve describes   small fluctuations (of certainly renormalized) ergodic sums 1/lF(l), l∈(l_n), along the forward trajectory x, T(x), T^2(x)…. More specifically, for l∈(l_n) it holds F(t· l) = t F(l)+R_l φ(t)+o(R_l), where t∈[0,1].





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  <cit.> and were extensively  studied since that time.  The following important theorem shows that adicity assumption is not restrictive at all:

(A. M. Vershik, <cit.>).

Any ergodic measure preserving transformation  on a Lebesgue space  is  isomorphic to some adic transformation. Moreover, one can find such an isomorphism that any given countable dense invariant subalgebra of measurable sets goes over into the algebra of cylinder sets.



In <cit.> authors encouraged studying different approaches to combinatorics of Markov's compacts (sets of paths in Bratteli diagrams).
In particular, 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) existence of limiting curves  for a class
of self-similar adic transformations and cylindric functions.  These transformations (in a slightly less generality) were considered by   X. Mela and S. Bailey in <cit.> and <cit.>. Our work 	extends  <cit.> and answers several questions from this research.




§ 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_n. Contrariwise 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.



 §.§ 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 infinite edge paths  beginning at the vertex v_0 = (0, 0). Following fundamental paper <cit.> we assume that there is a linear order  ≤_n,k defined on the set of edges with a terminate vertex (n,k),  0≤ k≤ L(n). These linear orders define a lexicographical order on the set of edges paths in X that belong to the same class of the tail partition.  We denote by   ≼ corresponding partial order on X. The set of maximal (minimal) paths is defined by X_max (correspondingly, X_min).


 Adic transformation  T is defined on  X∖(X_max∪ X_min) by setting Tx,   x∈ X, equal to the successor  of x, that is, the smallest y that satisfies y ≻ x.



Let  ω be a path in X. We denote by (n,k_n(ω)) a vertex through which ω passes at level n.
For a finite path c=(c_1,…,c_n) we denote   k_n(c) simply by k(c). A cylinder set C=[c_1c_2… c_n]={ω∈ X|ω_1=c_1,ω_2=c_2,…, ω_n=c_n} of a rank n is totally defined by a finite path from the vertex (0,0)  to the vertex (n,k)=(n,k(c)). Sets  π_n,k of lexicographically ordered finite paths c=(c_0,c_1,…,c_n-1), k(c)=n, are  in one to one correspondence with  towers τ_n,k made up of corresponding cylinder sets C_j = τ_n,k(j),1⩽ j⩽(n,k).
The  dimension (n,k) of the vertex (n,k) is the total number of such finite paths (rungs of the tower).

We denote by  (c) the number  of  finite paths in lexicographically ordered set π_n,k. Evidently, 1≤(c)≤(n,k).  For a given level n the set of towers {τ_n,k}_0≤ k≤ L(n) defines approximation of transformation T, see <cit.>, <cit.>.



We can consider a vertex (n,k) of Bratteli diagram B as an origin  in a new diagram  B'_n,k=(𝒱',ℰ'). The set of vertices 𝒱', edges ℰ'  and edges paths X(B'_n,k) are naturally defined. As above  partial order  ≼'    on X(B') is induced by linear orders ≤_n',k', n'>n.


 Ordered Bratteli diagram  (B,≼) is  self-similar if ordered diagrams (B,≼) and (B'_n,k,≼'_n,k) are isomorphic n∈ℕ, 0⩽ k⩽ L(n).



Let ℱ denote the set of all functions f:X→ℝ.
We denote by  ℱ_N the space of cylindric functions of rank N (i.e. functions that are constant on cylinders of rank  N).


Let g∈ℱ_N,   N< n. We denote by F_n,k^g linearly interpolated partial sums S^g_x∈τ_n^k(1). Assume that self-similar Bratteli  diagram B has  L+1 vertices at level N and let ω∈π_n,k, 0⩽ k ⩽ L(n), be a finite path such that its initial segment ω'=(ω_1,ω_2,…, ω_N) is a maximal path, i.e. (ω')=(N,k(ω')). Let E^N,l_n,k denote the number of paths from (0,0) to (n,k) passing through the vertex (N,l),0≤ l≤ L, and not exceeding path ω. We denote by ∂^N,l_n,k(ω) the ratio of  E^N,l_n,k to (N,l). It is not hard to see
that a partial sum F_n,k^g evaluated at   j=(ω) has the following expression:


    F^g_n,k(j) = ∑_l=0^L h^g_N,l∂^N,l_n,k(ω),

where coefficients h^g_N,l are equal to  F_N,l^g(H_N,l),  0⩽ l⩽ L.

Expression (<ref>) is a generalization of Vandermonde's convolution formula.




 §.§ 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  φ^g_x,l_n and normalizing coefficients R_x,l_n^g given  by the identity 
    φ_x,l_n(x)^g(t) = S_x^g([t· l_n(x)]) - t · S_x^g(l_n(x))/R_x,l_n(x)^g,
 where R_x,l_n(x)^g equals maximum of absolute value of the numerator. Without loss of generality, we assume that the limit g^*(x)=lim_n→∞1/nS^g_x exists at the point x. The following theorem generalizes Lemma 2.1 from <cit.> for an arbitrary summable function.


If a continuous limiting curve φ_x^g=lim_nφ^g_x,l_n exists for μ-a.e. x, then the normalizing coefficients  R_x,l_n^g are unbounded in n.




Assume the contrary that  |R_x,l_n^g|⩽ K. For simplicity we introduce the following notation:  S=S_x^g, φ_n = φ^g_x,l_n, R_n = R_x,l_n^g and  φ = φ_x. Since φ≠0, there is j∈ℕ such that  1/jS(j)≠ g^*. This in turn implies lim inf_n|φ_n(j/l_n)|= lim inf_n 1/R_n|S(j)-j S(l_n)/l_n|⩾1/K|S(j)-jg^*|=j/K|1/jS(i)-g^*|>0,
contradicting continuity of the limiting curve φ at the origin.




A function g∈ L^∞(X,μ)  (μ-a.e.) of the form g = c+h∘ T - h for some c∈ℝ and h∈ L^∞(X,μ) is called cohomologous to a constant in L^∞.




Normalizing sequence R_x,l_n^g is bounded if and only if function g is cohomologous to a constant.


Sums ∑_j=0^n-1 (g-g^*)∘ T^j of a cohomologous function are μ-a.e. bounded, therefore normalizing  coefficients R_x,l_n^g are μ-a.e. bounded too.

The proof of the converse statement exploits the result by A. G. Kachurovskiy from <cit.>. Assume that the normalizing coefficients R_x,l_n^g are bounded. Then for μ-a.e. point x∈ X and for any j∈ℕ the following inequality holds |S^g_x(j)-j/l_nS^g_x(l_n)|⩽ C. Going to the limit in n, we see that
|∑_i=1^jf∘ T^i(x)|⩽ C,
where f = g-g^*. Theorem 19  from <cit.> (see also  G. Halasz, <cit.>), inequality |S^f_x|⩽ C, is equivalent to existence of a function h∈ L^∞, such that f = h∘ T-h. Therefore g equals to h∘ T-h+g^*.



Let B be a Bratteli diagram such that there is only one vertex 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 consecutive levels is constant.




 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.


 There is only one vertex  (n,0) at each level n. Expression  (<ref>) for the partial sum F_n,0^g(i) is  evidently  valid for any odometer (even without assumption of self-similarity).  Moreover, expression  (<ref>)  is defined by the only coefficient  h^g_N,0 and therefore is proportional to H_N = (N,0). We can subtract such  constant  C to the function g
that equality  h^g-C_N,0=0 holds. But this is  equivalent to the following: The function function g-C belongs to the linear space spanned by the functions f_j - f_j∘ T, 1⩽ j⩽ H_N, where f_j  is the indicator-function of the j-th rung in the tower τ_N,0. Therefore function g-C is cohomologous to zero.




§ 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 <cit.>.






 §.§ Polynomial adic systems


Let p(x) = a_0 + a_1 x… + a_d x^d  be an integer polynomial of degree   d∈ℕ with positive integer coefficients a_i, 0≤ i≤ d. Bratteli diagram B_p =(𝒱,ℰ)_p associated to  polynomial  p(x) is defined as follows:


  * Number of vertices grows linearly: |𝒱_0|=1  |𝒱_n| = |𝒱_n-1|+d=nd+1, n∈ℕ.

  * If 0⩽ j⩽ d vertices (n,k) and (n+1,k+j) are connected by   a_j edges. 

Polynomial p(x) is called a generating polynomial of the diagram B_p, see paper <cit.> by S. Bailey.



Since the number  of edges into vertex (n, k)
is exactly p(1)=a_0+a_1+…+a_d it is natural to use the alphabet 𝒜 = {0,1,…,a_0+a_1+…+a_d-1} for edges labeling. We call  a lexicographical order  defined in <cit.> a canonical order. It is defined  as follows: Edges connecting (0, 0) with
(1, d) are labeled through  0 to  a_d-1 (from left to right); edges connecting (0,0) and (1,d-1), are indexed by a_d to a_d+a_d-1-1,  etc. Edges connecting  (0,0) and (1,0), are indexed through a_0+a_1+…+a_d-1 to a_0+a_1+…+a_d.

Infinite paths are totally defined by this labeling and may be considered as one sided infinite sequences in  𝒜^ℕ. We denote the path space by  X_p.








We denote by T_p the adic transformation associated with the canonical 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_p equals to  the coefficient of x^k in the polynomial (p(x))^n and is called  generalized binomial coefficient. We denote it by C_p(n,k). For n>1 coefficients C_p(n,k) can be evaluated by a  recursive expression C_p(n,k)=∑_j=0^da_jC_d(n-1,k-j).



In  <cit.> and <cit.> X.  Méla and S. Bailey showed that the fully supported invariant ergodic measures of the system (X_p,T_p) are the one-parameter family of Bernoulli measures:


(S. Bailey, <cit.>, X.  Méla, <cit.>) 1. Let q∈(0,1/a_0) and t_q is the unique solution in (0,1)  to the equation

    a_0q^d+a_1q^d-1t+…+a_dt^d-q^d-1=0,

then the invariant, fully supported, ergodic probability measures for the adic transformation T_p are the one-parameter family of Bernoulli measures μ_q, q∈(0,1/a_0),

    μ_q=∏_0^∞(q,…,q_a_0,t_q,…,t_q_a_1,t_q^2/q,…,t_q^2/q_a_2,…,t_q^d/q^d-1,…,t_q^d/q^d-1_a_d).
  2.Invariant measures that
are not fully supported are
    ∏_0^∞(1/a_0,…,1/a_0_a_0,0,…,0)   and  ∏_0^∞(0,…,0,1/a_d,…,1/a_d_a_d,).



 Polynomial adic system associated with polynomial p(x), is a triple (X_p,T_p,μ_q), q∈(0,1/a_0).



In particular, if p(x)=1+x system (X_p,T_p,μ_q), q∈(0,1), is the well-known Pascal adic transformation. Transformation was defined  in <cit.> by A. M. Vershik  [However earlier isomorphic transformation was used by <cit.> and <cit.>.] and was studied in many works <cit.>, see more complete list in the last two papers. For the Pascal adic space X_p is an infinite dimensional unit cube I={0,1}^∞, while measures   μ_q  are dyadic Bernoulli measures ∏_1^∞(q,1-q). Transformation T_p=P is defined by the following formula (see <cit.>)[P^k(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_k = 0 for all k ≥ n or x_k = 1 for all k ≥ n]:


    x↦ Px;   P(0^m-l1^l10…)=1^l0^m-l01…

 (that is only first m+2 coordinates of x are being changed). De-Finetti's theorem and Hewitt-Savage 0–1 law imply that all P-invariant ergodic measures are the Bernoulli measures  μ_p = ∏_1^∞(p,1-p), where 0<p<1.



Below we enlist several known properties of the polynomial systems:


  * Polynomial systems are weakly bernoulli  (the proof  essentially  follows <cit.> and is performed in <cit.> and <cit.>).

  * Complexity function has polynomial growth rate (for the Pascal adic first term of asymptotic expansion is known to be equal to n^3/6, see  <cit.>).

  * Polynomial system (X_p,T_p,μ_q) defined by a polynomial p(x) = a_0+a_1x with  a_0 a_1>1 has a non-empty set of non-constant eigenfunctions.


Authors of  <cit.> studied limiting curves  for the Pascal adic transformation (I,P,μ_q), q∈(0,1).



 (<cit.>, Theorem 2.4.) Let  P be the Pascal adic transformation defined on Lebesgue  probability space (I, ℬ,μ_q),q∈(0,1),  and g  be a cylindric function from ℱ_N. Then for μ_q-a.e. x
limiting curve φ^g_x ∈ C[0,1] exists if and only if  g is not cohomologous to a constant.


For the Pascal adic limiting curves can be  described by nowhere differentiable functions, that generalizes Takagi curve.

 (<cit.>, Theorem 1.) Let  P be the Pascal adic transformation defined on the Lebesgue space  (I, ℬ,μ_q), N∈ℕ and g∈ℱ_N be a not cohomologous to a constant cylindric function. Then for  μ_q-a.e. x there is a stabilizing sequence l_n(x) such that  the limiting function is α_g,x𝒯^1_q, where α_g,x∈{-1,1}, and  𝒯^1_q is given by the identity 
    𝒯^1_q (x) = ∂ F_μ_q/∂ q∘ F_μ_q^-1(x),   x∈[0,1],

where F_μ_q is the distribution function[More precisely F_μ_q is distribution function of measure μ̃_q, that is image of μ_q under canonical mapping ϕ:I→[0,1], ϕ(x)= ∑_i=1^∞x_i/2^i. ] of μ_q.



The graph of   1/2𝒯^1_1/2 is the famous Takagi curve, see <cit.>.


For a function g correlated with the  indicator functions of i-th  coordinate 1_{x_i=0},  x=(x_j)_j=1^∞∈ X Theorem <ref> was proved in <cit.>.



 §.§ Combinatorics of finite paths in the polynomial adic systems


In this section we'll specify representation (<ref>) for the  polynomial adic systems.


For a finite path ω=(ω_1,ω_2,…, ω_n) we set  k^1(ω) equal to nd-k(ω). Using self-similarity of the diagram B_p we can inductively prove the following explicit expression for  (ω):


Index (ω) of a finite path ω=(ω_j)_j=1^n in lexicographically ordered set  π_n,k(ω) is defined by equality:

    (ω)  = ∑_j=2^r ∑_i=0^ω_a_j-1C_P(a_j-1;k^1(ω)-k^1(i)-m_j) +(ω_1),

where m_j =∑_t=j^r-1k^1(ω_a_t),2⩽ j⩽ r-1, m_r=0, and polynomial P(x) is given by the identity P(x) = x^dp(x^-1).




Remark.
If initial segment (ω_1,ω_2,…,ω_N) of ω∈π_n,k is a maximal path to some vertex (N,l), then (<ref>) can be rewritten as follows:

    Num(ω)  = ∑_j=N+1^r ∑_i=0^ω_a_j-1C_P(a_j-1;k^1(ω)-k^1(i)-m_j) +C_P(a_N;k^1(ω)-m_l) .



Let N and l,  0⩽ l⩽ Nd, be positive integers and ω∈π_n,k be a finite path. Function ∂_k^1^N,l: ℤ_+→ℤ_+ , k^1=nd-k, is defined by the identity


    ∂_k^1^N,lω =  ∑_j=N+1^r ∑_i=0^ω_a_j-1C_P(a_j-1-N;k^1-k^1(i)-m_j-l),

where positive integers  a_j,k^1(i),m_j, are defined as in (<ref>).
Parameters N  and l correspond to shifting the origin vertex (0,0) to the vertex (N,l). Therefore value of the function  ∂_k^1^N,lω,k^1=k^1(ω),  equals to the number of paths from the vertex (0,0) going through the vertex (N,l) to the vertex  (n,k),k=nd-k^1, and non-exceding path ω divided by (N,l).

Let K_M,n,k, 1⩽ M⩽ n,   denote indexes (in lexicographical order) of those paths ω=(ω_1,…, ω_n)∈π_n,k, such that their initial segment (ω_1,ω_2,…,ω_M) is maximal (as a path from (0,0) to some vertex (M,l)).

Let  g∈ℱ_N. Function F̃^g,M_n,k: K_M,n,k→ℝ (where M, N⩽ M⩽ n is a positive integer) is defined by the identity

    F̃_n,k^g,M(j) =∑_l=0^Ndh_M,l^g  ∂_nd-k^M,l ω,

where (ω)=j, j∈ K_M,n,k, ω∈π_n,k.
We  extend domain of the function F̃_n,k^g,M  to the whole interval [1,H_n,k] using linear interpolation. Expression (<ref>) implies that for  j∈ K_M,n,k the identity F̃^g,M_n,k(j)=F^g_n,k(j) holds. Non strictly speaking,
higher values of parameter M ,M>N, makes functions F̃_n,k^g,M to be more and more rough approximation of function F^g_n,k and points from K_M,n,k correspond to nodes of this approximation.



Let  1⩽ j⩽ H_n,k and g∈ℱ_N. There exists a constant C=C(g), such that the following inequality holds for all n,k: 
    |F̃^g,N_n,k(j)-F_n,k^g(j)|⩽ C.







Remark. If the function g equals 1 and (ω)=(n,k)≡ C_P(n,k^1(ω)), then expression (<ref>) (as well as (<ref>)) reduces to:

    C_P(n,k) = ∑_l=0^NdC_P(N,l)C_P(n-N,k-l),

that is Vandermonde's convolution formula for generalized binomial coefficients.




 §.§ A generalized  r-adic number system on [0,1]


Let parameter q∈(0,1/a_0) and number t_q∈(0,1/a_1) be defined as in Theorem <ref>. We denote by r=p(1) number of letters in the alphabet 𝒜.

Let  ω=(ω_i)_i=1^∞∈ X_p,ω_i∈𝒜, be an infinite path. It is also natural to consider ω as a path in  an infinite  perfectly balanced tree ℳ_r.

By s̅_n = (s_n^0,…s_n^r-1)^T we denote  r-dimensional vector with j-th, 0⩽ j⩽ r-1, component equal to number of occurrences of letter  j among (ω_1,ω_2,…,ω_n). Let a̅_i,0⩽ i⩽ r-1, denote r-dimensional vector 
    (0,0,…,0_∑_j=0^i-1a_j,1,1,…,1_a_i,0,0,…,0_∑_j=i+1^da_j),




Let u· v denote scalar product of r-dimensional vectors u and v. We define mapping θ_q: X → [0,1] by the following identity:

    x = ∑_j=1^∞ I_q(ω_j)q^j(t_q/q)^a̅_1·s̅_j+2a̅_2·s̅_j+…+d a̅_d·s̅_j,

where I_q(w) = a_0q^h+1/t_q^h+1+a_1t_qq^h/t_q^h+1+…+a_hq/t_q+s with w=∑_i=0^ha_i+s, 0⩽ s<a_h+1,0⩽ h<d.

Let  X_0 denote the set of stationary paths.
Function θ_1/r  is a canonical bijection ϕ=θ_1/r: X∖ X_0→[0,1]∖ G,  G = ϕ(X_0).  Function ϕ maps measure μ_q,q∈(0,1), defined on X to measure μ̃_q on  [0,1], the family of towers {τ_n,k}_k=0^nd to the family {τ̃_n,k}_k=0^nd of disjunctive intervals. That defines isomorphic realization  T̃_p on  [0,1]∖ G of polynomial adic transformation T_p.


As shown by A. M. Vershik any adic transformation has a cutting and stacking realization on the subset of a full measure of [0,1] interval. However, nice explicit expression   (<ref>) needs some regularity from the Bratteli diagram.


Conversely, any point  x∈[0,1] could be represented by series (<ref>). We call this representation q-r-adic representation associated to the polynomial p(x). (If  r=2  representation (<ref>) for  q=1/2 is a usual dyadic representation of x∈(0,1).) Let G_q^m denote the set (vector) of all stationary numbers of rang m, m∈ℕ, i.e. numbers with a finite representation

    x = ∑_j=1^m I(ω_j)∏_i=0^r-1p_i^s_j^i,

and let  G_q=∪_m G_q^m be the set of all q-r-stationary numbers.



Let l∈ℕ and x be a path in X_p. We consider  r^l-dimensional vectors K̃_n=K_n-l,n,k_n(x)   and renormalization mappings D_n,k:[1, C_p(n,k)]→[0,1] defined by D_n,k(j)=j/C_p(n,k). Using ergodic theorem it is straightforward to show that for μ_q-a.e. x  it holds lim_n→∞R_n,k_n(x)(K̃_n)=G^l_q (where convergence is the componentwise convergence of vectors).




 §.§ Existence of limiting curves for polynomial adic systems


In this part we generalize Theorem 2.4 from <cit.> for polynomial adic systems (X_p,T_p) associated with  positive integer polynomial p.

First we prove a combinatorial variant of the theorem. Let, as above, x∈ X_p be an infinite path going through vertices  (n,k_n(x))∈ B_p. Below we write vertex (n,k_n(x)) as (n,k_n) or simply as (n,k). To simplify notation, the dimension  (n,k)=C_p(n,k) is  denoted  by H_n,k.


We define function  φ_n,k^g=φ_x∈τ_n,k(1),H_n,k^g:[0,1]→ℝ by identity

    φ_n,k^g(t) = F_n,k^g(tH_n,k)-tF_n,k^g(H_n,k)/R_n,k^g.

Let F be a function defined on [1,H_n,k]. Define function ψ_F,n,k on [0,1] by

    ψ_F(t) = F(tH_n,k)-tF(H_n,k)/R_n,k,

where  R_n,k is a canonically defined normalization coefficient. Then the following identity holds ψ_F_n,k^g=φ_n,k^g.

Let  g∈ℱ_N be not cohomologous to a constant cylindric function.
Theorem <ref> implies that normalization sequence (R_n,k_n^g)_n⩾1 monotonically increases. Lemma <ref> shows that

    || ψ_F_n,k^g - ψ_F_n,k^g,N||_∞0.




We want to show that there is a sequence (n_j)_j⩾ 1 and a continuous function  φ(t),t∈[0,1],  such that

    lim_j→∞|| ψ_F_n_j,k_n_j^g,N - φ||_∞=0.


Following <cit.>, we consider an auxiliary object: a family of polygonal functions  ψ^M_n = ψ_F_n,k^g,n-M+N, N+1⩽ M⩽ n. Graph of each function ψ^M_n is defined by (2r)^M-dimensional array (x^M_i(n),y^M_i(n))_i=1^r^M, such that  ψ^M_n(x^M_i(n))=y^M_i(n). Results from Section 3.3 show that vector (x^M_i(n))_i=1^r^M converges pointwise to q-r-stationary numbers G^M_q of rank M, given by polynomial p(x).

Let l and M be positive integers, such that N+1⩽ l<M<n. Functions F_n,k^g,n-M and F_n,k^g,n-l coincide at each point from  K_n-l,n,k_n, therefore functions  ψ^M_n and ψ^l_n also coincide at (x^l_i(n))_i=1^r^l. Moreover, Proposition <ref> (it generalizes Proposition 3.1 from <cit.>) provides the following estimate:

    || ψ^M_n_j -ψ^l_n_j||_∞⩽ C_1e^-C_2(M-l),

with C_1,C_2>0. For a fixed  M we can extract a subsequence (n_j)  such that polygonal functions ψ^M_n_j converge to a polygonal function φ^M in sup-metric. Then, as in <cit.>,  using a standard diagonalization  procedure we can find subsequence (that again will be denoted by (n_j)_j) such that convergence to some continuous on [0,1] function holds for any M: 
    lim_M→∞lim sup_j→∞|| ψ_n_j^M - φ||_∞=0.

Auxiliary functions  φ^M are polygonal approximations to the function φ.



Therefore we have proved the following claim, generalizing Theorem <ref>:

 Let   (X,T,μ_q ),q∈(0,1/a_0),  be a polynomial adic transformation defined on Lebesgue  probability space (I, ℬ,μ_q),q∈(0,1),  and g  be a not cohomologous to a constant cylindric function from ℱ_N. Then for μ_q-a.e. x passing through vertices (n,k_n(x)) we can extract a subsequence (n_j) such that  φ_n_j,k_n_j(x)^g  converges in sup-metric to a continuous function on [0,1].



Each limiting curve φ is a limit in  j of polygonal curves ψ_n_j^m,m⩾ 1, with nodes at stationary points  G_q⊂[0,1]. Therefore, its values
φ(t) can be obtained as limits lim_j→∞ψ_n_j^m((ω)/H_n_j,k_n_j), where t∈ G^m_q and lim_j→∞(ω)/H_n_j,k_n_j = t, with (ω)∈ K_n_j-m,n_j,k_n_j.

Self-similar structure of towers simplifies this task. We write simply n for n_j(x), F for F_n,k^g and R_n= R_n,k^g.  The  following lemma in fact generalizes results from Section 3.1. of <cit.>:

Limiting curve is totally defined by the following limits n→∞:

    lim_n→∞1/R_n(F(L_m,i,n,k) - L_m,i,n,k/H_n,kF(H_n,k)),

where  L_m,i,n,k=∑_j=0^ma_jH_n-i,k(ω)+j-di,0⩽ m⩽ d.


 We may assume that  δ<k_n/n(x)<δ d for some δ>0.
First, we suppose that some typical   n=n(x)>>m and k_n are taken.
We consider a set of ingoing finite paths of length m to the vertex (n,k), d≤ k≤ d(n-1), n>>m. Self-similarity of B_p implies that these paths can be considered as paths going from the origin to some vertex (m,j), 0≤ j≤ md, of B_p, see Fig. <ref>. As shown in Section 3.3 above, each such path correspond to a point from G_q^m, which in its turn correspond to q-r-adic interval of rank m. Let  x_m,j, 0⩽ j⩽ md denote length of such interval and y_m,j,  denote increment of the function ψ_n^M on the interval  (m,j). Values (x_m,j, y_m,j) may be defined inductively: For  m=0 by x_0,0=1, y_0,0=0, and for m>0  and indices  j such that (m-1)d<j⩽ md by x_m,j=∑_i=0^j a_iC_p(n-m,nd-k+j-di)/H_n,k,  y_m,j =ψ_n^M(x_m,j); for other values of j by recursive expression x_m-1,i=∑_j=0^d a_jx_m,i-j, y_m-1,i=∑_j=0^d a_jy_m,i-j. Therefore function ψ_n_j^M is totally defined by its values at x_m,j, 1⩽ m⩽ M, (m-1)d<j⩽ md. Going to the limit we obtain the claim.
 






Stochastic version of Theorem <ref> is obtained from the following claim: for any ε>0 for μ_q-a.e. x there exists subsequence  n_j(x) such that (w^j), w^j=(x_1,… x_n_j), satisfies the following condition (w^j)/H_n_j,k_n_j<ε. In fact, even more strong result holds. It follows from the recurrence property of one-dimensional random walk and was first proved by É. Janvresse and T. de la Rue in <cit.> to show that the Pascal adic transformation is loosely Bernoulli. Later it was  generalized in
 <cit.> for the polynomial adic systems.


For any ε>0 and μ_q×μ_q-a.e. pair of paths (x,y)∈ X× X there is a subsequence n_j such that k_n_j(x)=k_n_j(y) and indices (ω_x), (ω_y) of paths ω_x=(x_1,x_2,… x_n_j) and ω_y=(y_1,y_2,… y_n_j) satisfy the following inequlity (ω_z)/H_n_j,k_n_j(x)< ε, z∈{x,y}, for each j∈ℕ.





(Stochastic variant of Theorem <ref>.)
Let  (X,T,μ_q ), q∈(0,1/a_0),   and g  be a cylindric function from ℱ_N. Then for μ_q-a.e. x
limiting curve φ^g_x ∈ C[0,1] exists if and only if function g is not cohomologous to a constant.


 Follows from Lemma <ref>, Theorem <ref> and Theorem <ref>.


Remark
Lemma <ref> implies that appropriate choice of stabilizing sequence l_n(x) can provide the same limiting curve φ_x^g, lim_j→∞||φ_x,l_j(x)^g-φ_x^g||=0,  for μ_q-a.e. x.


Finally we prove Proposition <ref> used above. It generalizes Proposition 3.1 from <cit.>. However, its proof needs an additional statement due to the non unimodality of generalized binomial coefficients C_p(n,k):

 Let  p(x)=a_0+a_1x+…+a_dx^d be a positive integer polynomial. Then the following holds:

 1. There exist n_1∈ℕ and  C_1>0, depending only on {a_0,…,a_d}, such that  max_k{C_p(n,k+1)/C_p(n,k),C_p(n,k)/C_p(n,k+1)}⩽  C_1n for n>n_1.


 2. C_p(n-1, k-i)⩽1/a_imax{k/n, 1-k/n}C_p(n,k),  0⩽ i ⩽ d.
 


1.
Let  X be a discrete random variable on {0,1,…, d} with distribution associated to the polynomial p(x), that is Prob(X=k) =a_k/p(1), 0⩽ k⩽ d. Distribution of a sum Y_n=X_1+X_2+… X_n of i.i.d. random  variables X_k,0⩽ k⩽ n, with distributions associated to the polynomial p(x), is associated to the polynomial p^n(x), i.e. Prob(Y_n=k) =C_p(n,k)/p^n(1), 0⩽ k⩽ nd. A. Oldyzko and L. Richmond showed in <cit.> that the function  f_n(k)≡Prob(Y_n=k) is asymptotically unimodal, i.e. for n≥ n_1,   coefficients C_p(n,k), 0⩽ k ⩽ nd,n⩾ n_1, first increase  (in k) and decrease then.



We denote by C and c the maximum and the minimum values of the coefficients  {C_p(n_1,k)}_k=0^n_1d of the polynomial p^n_1(x).  Let also  denote the maximum of the coefficients  {a_0,…,a_d} of the polynomial p(x).
We will use induction in n to prove that  C_p(n,k+1)/C_p(n,k)⩽C/cdn, 0⩽ k⩽ nd-1, n⩾ n_1. (The second estimate C_p(n,k)/C_p(n,k+1)⩽C/cdn can be  proved in the same way).
We start now with the base case: For n=n_1 it obviously holds that C_p(n_1, k+1)/C_p(n_1, k)⩽C/c⩽Cd/c, 0⩽ k⩽ n, hence
we have shown the base case.

Now assume that we have already shown  C_p(n-1, k)/C_p(n-1, k-1)⩽Cd/c(n-1), where  1⩽ k⩽ d (n-1)  and n≥ n_1. We need to show that C_p(n, k)/C_p(n, k-1)⩽Cd/cn,  1⩽ k⩽ dn.

    C_p(n, k+1)/C_p(n, k) = ∑_i=0^d a_iC_p(n-1,k+1-i)/∑_i=0^d a_iC_p(n-1,k-i)⩽
    ⩽C_p(n-1,k)(a_0+a_1+…+a_d-1+d  a_dC/c(n-1))/a_dC_p(n-1, k)⩽
    ⩽a_0+a_1+…+a_d-1-d/a_d+ C d n/c⩽ Cd/c  n.

2. The  statement follows directly from the following identity for the generalized binomial coefficients: 
    ∑_i=1^dC_p(n-1,k-i)a_ii=k/nC_p(n,k).

To show it we differentiate identity p^n(x)=∑_k≥0 C_p(n,k) x^k resulting
np^n-1(x)p'(x) = ∑_k≥0 kC_p(n,k) x^k-1, p'(x)=a_1+2a_2x+…+da_dx^d-1.
It remains to equate exponents from the two sides.



The following proposition generalizes Propositon 3.1 from <cit.>. We  preserved the original notation where it was possible.

 Let N⩾ 1 be a positive integer and δ∈(0,1/4) be a small parameter. Let   A=A(n̅,k̅)∈ B_p be a vertex with coordinates (n̅,k̅) satisfying
2δn̅⩽ k⩽(d-2δ)n̅ and
2δn̅⩽ nd-k⩽(d-2δ)n̅.

Let α_l,0⩽ l⩽ Nd, be real numbers, such that ∑_l=0^Ndα_l^2>0. Let n, N⩽ n⩽n̅, and  B(n,k)=(n,k) be a vertex with coordinates satisfying 0⩽ k⩽k̅, 0⩽ n-k⩽n̅-k̅.  Define

    γ_n,k = 1/R∑_l=0 ^Ndα_l  C_d(n-N,k-l),

where R = R(A,B,δ) is a renormalization constant such that |γ_n,k| are uniformly in n and k  from 0⩽ k⩽k̅,0⩽ n-k⩽n̅- k̅, N⩽ n⩽n̅ bounded by 2. Then there exist a constant  C = C(δ,N), such that, provided n̅ is large enough, the following inequality holds for all n,k:

    |γ_n,k|⩽ 3e^-C(n̅-n).




Conditions on the vertex A δ-separate it from "boundary"  vertices (n̅,0) and (n̅,dn̅). Conditions on the vertex B=B(n,k) provides it can be considered as a vertex in a "flipped"  graph and that it can be connected with the vertex   A, see Fig. <ref>.




 We can assume that n̅>2n_1, where n_1=n_1(a_0,a_1,…,a_d), is defined in the proof of Lemma <ref>.
 Let l_0, 0⩽ l_0⩽ Nd, be such that coefficient α_l_0 is nonzero. We can rewrite the right hand side of  (<ref>) as follows:

    Rγ_n,k= C_d(n-N,k-l_0)P(n,k,l_0), N⩽ n⩽n̅, 0⩽ k⩽ nd,

where P(n,k,l_0) is defined by ∑_l=0 ^Nα_l C_d(n-N,k-l)/C_d(n-N,k-l_0).
Let α denote the maximum of |α_l|, 0⩽ l⩽ Nd.
We want to show that there is a polynomial Q(x) of degree deg(Q)≤ Nd such that

    |P(n,k,l_0) - P(n̅,k̅,l_0)|⩽ Q(n̅).

It is enough to show that there is c_1>0 such that |C_d(n-N,k-l)/C_d(n-N,k-l_0)|⩽ c_1 n^Nd, 0⩽ l⩽ Nd, N⩽ n⩽n̅.
The latter inequality follows from Nd fold  application of  part 1 of Lemma <ref>.
Define function Q̃  by  Q̃ =P(n,k,l_0) - P(n̅,k̅,l_0). We can write


    γ_n,k =1/RC_d(n-N,k-l_0)P(n,k,l_0) =
     =C_d(n-N,k-l_0)/C_d(n̅-N,k̅-l_0)C_d(n̅-N,k̅-l_0)P(n,k,l_0)/R  =
    =C_d(n-N,k-l_0)/C_d(n̅-N,k̅-l_0)C_d(n̅-N,k̅-l_0)(P(n̅,k̅,l_0)+Q̃)/R.

By the assumption we have |γ_n̅,k̅|=|1/RP(n̅, k̅,l_0)C_d(n̅-N,k̅-l_0)|⩽ 2. Therefore inequality (<ref>) can be written as |Q̃|⩽ Q. We get

    |γ_n,k|⩽3Q(n̅)C_d(n-N,k-l_0)/C_d(n̅-N,k̅-l_0).

Applying the estimate from  part  2 of Lemma <ref>  (n̅-n) times and using assumptions on the vertices  A and B,
we obtain that  C_d(n-N,k-l_0)/C_d(n̅-N,k̅-l_0)⩽ 3e^-C̃(δ)(n̅-n) for some C̃(δ)>0.
Finally we  get (an independent of the initial choice of l_0) estimate:

    |γ_n,k|⩽3Q(n̅)C_d(n-N,k-l_0)/C_d(n̅-N,k̅-l_0)⩽ 3e^-C(δ)(n̅-n)

for some  C(δ)>0.





 §.§ Examples of  limiting curves


Let q_1 and q_2 be two numbers (parameters) from (0, 1).
We consider the function S^p_q_1,q_2:[0,1]→ [0,1] that maps a number x with q_1-r-adic representation
x = ∑_j=1^∞ I_q_1(ω_j)q_1^j(t_q_1/q_1)^a̅_1·s̅_j+2a̅_2·s̅_j+…+d a̅_d·s̅_j   to 
    S^p_q_1,q_2(x) =  ∑_j=1^∞ I_q_2(ω_j)q_2^j(t_q_2/q_2)^a̅_1·s̅_j+2a̅_2·s̅_j+…+d a̅_d·s̅_j.

For any q_1-r-stationary point x_0=∑_j=1^m I_q_1(ω_j)q_1^j(t_q_1/q_1)^a̅_1·s̅_j+2a̅_2·s̅_j+…+d a̅_d·s̅_j
and any x ∈ [0, 1] the function S^p_q_1,q_2 satisfies the following self-affinity property:

    S^p_q_1,q_2(x_0+r_q_1x) =     S^p_q_1,q_2(x_0) + r_q_2S^p_q_1,q_2(x),

where r_q_i = q_i^m(t_q_i/q_i)^a̅_1·s̅_m+2a̅_2·s̅_m+…+d a̅_d·s̅_m, i=1,2.
Expression (<ref>) means that the graph of S^p_q_1,q_2 considered on the q-r-adic interval  [x_0, x_0+r_q_1] coincides after  renormalization with the graph of S^p_q_1,q_2 on the whole interval [0,1]. Also for q_1=1/r function S^p_1/r,q_2 is the distribution function of the measure μ̃_q_2.


Functions S^p_q_1,q_2(·) allow us to define new functions

    _p,q_1^k : = ∂^k S^p_q_1,q_2/∂ q_2^k|_q_2=q_1, k∈ℕ.

If k=0 we will assume that _p,q^0 (x) =x.
For q=1/2 and  k=1 function 1/2_1+x,1/2^1 is the Takagi function, see <cit.>. The function _p,q_1^k on the interval [x_0, x_0+r_q_1]  can be expressed by a linear combination of the functions _p,q_1^j,0⩽ j⩽ k. (Expression can be easily obtained  by differentiating  identity (<ref>) with respect to parameter q_2 and defining q_2 equal to q_1.)







 Functions  _p,q^k, q∈(0,1/a_0), k≥1, are continuous functions  on [0,1].




The proof is based on the fact that  any two points x and y from the same  q-r-adic interval of rank m have the same coordinates (ω_1, ω_2,…, ω_m) in q-r-adic expansion. This provides a straightforward estimate for the difference
|_p,q^1(x) - _p,q^1(y)|.


Let b=b_q denote  the ratio t_q/q. As shown in Section 3.3 above any  x in (0,1) can be coded by a  path ω=(ω_i)_i=1^∞, ω_i∈{0,1…,r-1}=𝒜,  in r-adic (perfectly balanced) tree ℳ_r. The function
_p,q^1 maps x=x_q∈[0,1] with q-r adic series representation
 
    x=∑_j=1^∞(∑_i=0^ω_j-1b^i)q^jb^a̅_1·s̅_j+2a̅_2·s̅_j+…+d a̅_d·s̅_j,

 to z=∂/∂ q x_q.
Let s̃_j denote the sum a̅_1·s̅_j+2a̅_2·s̅_j+…+d a̅_d·s̅_j.

Derivative ∂/∂ q(q^jb^l) equals to q^j-1b^l-1[(j-l)b+lt'_q],
where l =s̃_j-ω_j+i.
Using implicit function theorem we find that

    t'_q = -a_0d q^d-1+a_1(d-1)q^d-2t_q+…+a_d-1t_q^d-1-(d-1)q^d-2/a_1 q^d-1+2a_2q^d-2t_q+… a_dt_q^d-1d.

Let also  denote the maximum of the coefficients  {a_0,…,a_d} of the polynomial p(x).
We have  |t'_q|≤2d^2/q. Let p_max∈(0,1) denote the maximum of {q,t_q,t_q^2/q,…,t_q^d/q^d-1}.


Assume  y is  the left  boundary of some  q-r-adic interval of rank m, containing point x. Then the following inequality holds (we simply  write T for T_p,q^1):

    |T(y)-T(x)| ≤∑_j=m^∞∑_i=0^ω_j-1|∂/∂ q( q^jb^s̃_j+i)|.


Using estimate | q^jb^s̃_j+i |≤ (p_max)^j, 0≤ i≤ r-1, we see that the absolute value of ∂/∂ q(q^jb^l) for j>2 is estimated by expression P(j,q)(p_max)^j-2, where P(j,q)  is some polynomial. Define ε to be equal to 0.99. Then for  m large enough it holds:


    |T(y)-T(x)|≤∑_j=m^∞∑_i=0^ω_j-1P(j,q)(p_max)^j-2≤ C(p_max)^mε,

where C is some constant.

In general case we can assume that points x and x+δ are from some q-r-adic interval of rank m=m(δ), lim_δ→ 0m(δ)=+∞, and let  y be the  left boundary point of this interval. Then

    |T(x+δ)-T(x)| ≤ |T(y)-T(x)|+|T(y)-T(x+δ)|≤ 2Cp_max^mε

For k>1 we  can use  a similar argument based on the following  estimate for the k-th derivative: |∂^k/∂ q^kq^jb^l| ≤ P_k(j,q)(p_max)^j-k-1, where j>k and P_k(j,q) is some polynomial.


 
For a cylindrical function g=- ∑_j=0^dja_j1_{k^1(x_1)=j}∈ℱ_1 and  for  μ_q-a.e. x there is a stabilizing sequence l_n(x) such that  the limiting function is _p,q^1.





For simplicity we will present the proof for p(x)=1+x+x^2.
The general case  follows the same steps. Theorem <ref> implies that we can find the limiting function φ(x) as lim_n→∞φ_n,k, where (by the law of large numbers)  k_n/n→𝔼_μ_qk_1. Lemma <ref> imply that it is sufficient to show that the function φ(x) coincide with _p,q^1 at x=q^j and x=q^j-1(q+t_q), where j∈ℕ.



The function _p,q_1^1 maps point x =q^j to  ∂/∂ qq^j =jq^j-1  and point x=q^j-1(q+t_q) to q^j-2(jq+(j-1)t_q+t'_qq).
Using expression (<ref>) we see that
 t'_q =1-(2q+t_q)/2t_q+q.

Identity (<ref>) implies that  h_n,k^g=  k/nH_n,k. We need to find the following limits for  i∈ℕ, n→∞ and k_n/n→𝔼_μ_qk_1 = 2q+t_q (we write F for F_n,k):



  * lim1/R_n(F(H_n-i,k-2i) - H_n-i,k-2i/H_n,kF(H_n,k))

  * lim1/R_n(F(H_n-i,k-2i+H_n-i,k-2i+1) - H_n-i,k-2i+H_n-i,k-2i+1/H_n,kF(H_n,k))

We define the normalizing coefficient R_n by
R_n=qH_n,k/n(2-𝔼_μ_qk_1). After some computations we see that the first limit equals iq^i-1, and the second to q^i-2(iq+(i-1)t_q+t'_qq). These shows that that the limiting function φ coincides with the function _p,q_1^1 on a dense set of q-2-stationary points.  Therefore, by Theorem <ref> these functions coincide.






Numerical simulations show that limiting functions  _p,q^k, k⩾1,  and their linear combinations arise as limiting functions
 lim_n→∞φ_n,k_n^g for a general cylindrical function g∈ℱ_N.
We do not have any proof of this statement except for the case of the Pascal adic, see Theorem <ref> above. Expression (<ref>) shows  that for a cylindrical function  g∈ℱ_N the partial sum F^g_n,k  is defined by the coefficients h_N,k^g, 0≤ k≤ Nd. Its seems to be useful to define  h_N,k^g_m by the generating function h_N,k^g_m=coeff[v^m]  (h_0+h_1v+…+h_dv^d)^kp(v)^Nd-k, where functions g_m forms an orthogonal basis. (For the Pascal adic the function (1-av)^k(1+v)^n-k, a = 1-q/q, is the generating function of the Krawtchouk polynomials and the basis g_m is the basis of Walsh functions, see <cit.>).






§ LIMIT OF LIMITING CURVES


In this section we answer the question by  É. Janvresse, T. de la Rue and Y. Velenik from <cit.>, page 20, Section 4.3.1.



 

Let  q∈ (0,1) and t_q∈(0,1) be the unique solution in (0,1) of the equation

    q^d+q^d-1t+…+t^d=q^d-1.


As above, we denote by b=b_q the ratio t_q/q. Any  x in (0,1) has an almost unique (d+1)-adic representation:

    x = ∑_j=1^∞(∑_i=0^ω_j-1b^i)q^jb^s_j^1+2s_j^2+… ds_j^d-ω_j,

where ω=(ω_i)_i=1^∞, ω_i∈{0,1…,d}=𝒜, is a path in (d+1)-adic (perfectly balanced) tree ℳ_d+1 and s_j^k is the number of occurrences of letter k among (ω_1,ω_2, …, ω_j).

We denote by S_q(x) the  (anaclitic in parameter q) function defined by (a uniformly summable in x) series (<ref>). We put q_* equal to 1/(d+1)  (this is so called symmetric case t_q_*=q_*). If d=1  representation (<ref>) for  q^*=1/2 is a usual dyadic representation of x∈(0,1).


The authors of  <cit.> were interested in the limiting behavior of the graph of the function   
    _d: x ↦1/d+1∂ S_q/∂ q|_q=q_*
 for large values of  d (we also introduced vertical normalization by d+1, if  d=1 the graph of 1/2T_1 is the Takagi curve). On the basis of a series of numerical simulations they noticed that limiting curves for d→∞ seem to converge to a smooth curve. Below we will  show that the limiting curve for d=∞ is actually a parabola, see Fig. <ref>.

We are going split the unit interval into d+1   subintervals I_i=(i/d+1;i+1/d+1),0≤ i≤ d, of equal length and evaluate the function  _d at each of the (left) boundary points of these intervals. 
We also want to show that the function _d is uniformly in d bounded at these intervals. After that we  go to the limit in d.


 




Symmetry assumption  q = t_q = q_* and implicit function theorem (see (<ref>)) imply that
t'_q_* = - 2-d/d.
In its turn this implies b'_q = t'/q-b_q/q|_q=q_* = -2(d+1)/d.

Finally we find that ∂/∂ q (q^jb^r)  = jq^j-1b^r+rb^r-1q^jb'_q |_q=q_* =(q_*)^j-1(j+rq_*(-2(d+1)/d))= (q_*)^j-1(j - 2r/d)).




Note that the left boundary point a_d of I_ad, a∈[0,1],ad ≡ [ad], ([ ·  ] is an integer part) equals  a_d=ad/d+1 and is coded by the stationary path  ω = (ω_j)_j=1^∞∈ℳ_d+1 with ω_1 = ad and ω_j ≡ 0, j≥2.


We have

    _d(a_d) = 1/d+1∑_i=0^da-1(j-2(da(j-1)+i)/d) =a_d(1-a_d)d+1/d a_d(1-a_d).


This shows that the smooth curve (if exists) should be a parabola.

To complete the proof of the theorem it only remains to show that _d(x) is uniformly bounded in d at the intervals I_ad. Analogously to (<ref>) we see that for x∈ I_ad it holds

|_d(x)-_d(a_d)| = 1/d+1∑_j=2^∞ q_*^j-1∑_i=0^ω_j-1(j-2(s_j^1+2s_j^2+… + ds_j^d-ω_j+i)/d)≤100 d/d+1∑_j=2^∞ j q_*^j-1≤
≤400d/(d+1)^2. 
That finishes our proof.






 §.§ Question.


We may heuristically interpret results of Section 4 as existence of a limiting curve of a dynamical system defined by a diagram  with "infinite"  number of edges.
This leads us to the following questions: Does this system really exist? How to define it correctly? Which properties does it have?

 
















10

Ver81
A. M. Vershik, Uniform algebraic approximations of shift and multiplication operators,  Sov. Math. Dokl., 24:3 (1981), 97–100.

ver82
A. M. Vershik, A theorem on periodical Markov approximation in ergodic theory, J. Sov. Math., 28 (1982), 667–674.

VerLiv92 A. M. Vershik and  A. N. Livshits,
  Adic models of ergodic transformations, spectral theory, and related topics,
   Adv. in Soviet Math.  AMS Transl., 9, 1992, 185–204.

Vershik11
A. M. Vershik, The Pascal automorphism has a continuous spectrum,
 Funct. Anal. Appl., 45:3 (2011), 173–186.

Ver14  A. M. Vershik,
 The problem of describing central measures on the path spaces of graded graphs,
 Funct Anal Its Appl .,
 48:4 (2014), 256–271.

Ver15 A. M. Vershik, Several Remarks on Pascal Automorphism
and Infinite Ergodic Theory, Armenian Journal of Mathmatics, 7:2 (2015), 85–96.

Kac96 A. G. Kachurovskii,
The rate of convergence in ergodic theorems,
Russian Mathematical Surveys, 51, 4, (1996)
 653–703.

MinMan13
 I. E.  Manaev,  A. R. Minabutdinov,  The Kruskal-Katona Function, Conway Sequence, Takagi Curve, and Pascal Adic,   Transl: J. Math. Sci.(N.Y.), 196:2 (2014), 192–198.

Min14
A. R. Minabutdinov,  Random Deviations of Ergodic Sums for the Pascal Adic Transformation in the Case of the Lebesgue Measure,  Transl: J. Math. Sci.(N.Y.), 209:6, (2015), 953–978.

MinKraw
A. R. Minabutdinov,  A higher-order asymptotic expansion of the Krawtchouk polynomials, Transl.:  J. Math. Sci.(N.Y.) 215:6  (2016), 738–747.

LodMin15
A. A. Lodkin,A. R. Minabutdinov,
 Limiting Curves for the Pascal Adic Transformation,
   Transl: J. Math. Sci.(N.Y.),
216:1 (2016),
 94–119.


Bailey2006
S. Bailey,
Dynamical properties of some non-stationary, non-simple Bratteli-Vershik systems,
Ph.D. thesis, University of North Carolina, Chapel Hill,
2006.


HajanItoKakutani A. Hajan, Y. Ito, S. Kakutani,
 Invariant measure and orbits of dissipative transformations,
Adv. in Math., 9:1 (1972), 52–65.

Halasz1976
G. Halasz,
Remarks on the remainder in Birkhoff's ergodic theorem,
  Acta Mathematica Academiae Scientiarum Hungarica,
  28:3-4 (1976), 389–395.


PascalLooselyBernoulli
É. Janvresse,  T. de la Rue,
The Pascal adic transformation is loosely Bernoulli,
  Annales de l'Institut Henri Poincaré (B) Probability and
  Statistics, 40:2 (2004), 133 – 139.

DeLaRue
É. Janvresse, T. de la Rue, and Y. Velenik,
 Self-similar corrections to the ergodic theorem for the Pascal-adic
  transformation,  Stoch. Dyn., 5:1 (2005), 1–25.


Kakutani1976 S. Kakutani, A problem of equidistribution on the unit interval [0, 1], in: Lecture Notes in Math., vol. 541, Springer-Verlag, Berlin, (1976) 369–375.


Kruppel M. Krüppel, De Rham's singular function, its partial derivatives with respect to  the parameter and binary digital sums,
Rostocker Math. Kolloq., 64 (2009), 57–74.


Mela2006 X. Méla,
A class of nonstationary adic transformations,
       Ann. Inst. H. Poincaré Prob. and Stat.,
       42:1 (2006), 103–123.


MelaPetersen
X. Méla, K. Petersen, Dynamical properties of the Pascal adic transformation,
Ergodic Theory Dynam. Systems, 25:1 (2005), 227–256.


OldyzkoRichmond1985 A. M. Odlyzko,  L. B. Richmond, On the Unimodality of High Convolutions of Discrete Distributions,
Ann. Probab.,
13 (1985), 299–306.

Takagi1903 T. Takagi,
A simple example of the continuous function without derivative,
Proc. Phys.-Math. Soc., 1 (1903), 176–177.






