Criteria of ergodicity for p adic dynami

Chaos, Solitons & Fractals 60 (2014) 11–30

Contents lists available at ScienceDirect

Chaos, Solitons & Fractals
Nonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier.com/locate/chaos

Criteria of ergodicity for p-adic dynamical systems in terms
of coordinate functions
Andrei Khrennikov ⇑, Ekaterina Yurova
International Center for Mathematical Modelling in Physics and Cognitive Sciences, Linnaeus University, Växjö S-35195, Sweden

a r t i c l e

i n f o

Article history:
Received 25 April 2013
Accepted 6 January 2014
Available online 3 February 2014

a b s t r a c t
This paper is devoted to the problem of ergodicity of p-adic dynamical systems. We solved
the problem of characterization of ergodicity and measure preserving for (discrete) p-adic
dynamical systems for arbitrary prime p for iterations based on 1-Lipschitz functions. This
problem was open since long time and only the case p ¼ 2 was investigated in details. We
formulated the criteria of ergodicity and measure preserving in terms of coordinate functions corresponding to digits in the canonical expansion of p-adic numbers. (The coordinate
representation can be useful, e.g., for applications to cryptography.) Moreover, by using this
representation we can consider non-smooth p-adic transformations. The basic technical
tools are van der Put series and usage of algebraic structure (permutations) induced by
coordinate functions with partially frozen variables. We illustrate the basic theorems by
presenting concrete classes of ergodic functions. As is well known, p-adic spaces have
the fractal (although very special) structure. Hence, our study covers a large class of
dynamical systems on fractals. Dynamical systems under investigation combine simplicity
of the algebraic dynamical structure with very high complexity of behavior.
Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction
Algebraic and arithmetic dynamics are actively developed fields of general theory of dynamical systems. The
bibliography collected by Franco Vivaldi [48] contains

216 articles and books; extended bibliography also can
be found in books of Silverman [47] and Anashin and
Khrennikov [6]. Such studies are based on combination of
number theory and theory of dynamical systems. And, as
it often happens in mathematics, combination can induce
novel constructions interesting for both areas of research.
By Ostrowsky theorem [46] the fields of real and p-adic
numbers are the only completions of the field of rational
numbers Q. This is one of (purely mathematical) motivations to study dynamical systems in Qp , see. e.g., [1–50]
(the complete list of reference would be very long; hence,
we refer to [6,31,48]). We point to applications of p-adic
⇑ Corresponding author.
E-mail address: andrei.khrennikov@lnu.se (A. Khrennikov).
0960-0779/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.chaos.2014.01.001

dynamical system in physics (see [19] for a recent review),
cognitive science and genetics [1,2,20,26,27,35–37,41].
This paper is devoted to the problem of ergodicity of
p-adic dynamical systems. We solved the problem of

characterization of ergodicity and measure preserving for
(discrete) p-adic dynamical systems for arbitrary prime p
for iterations based on 1-Lipschitz functions. This problem
has been open since long time [6] and only the case p ¼ 2
has been investigated in details. As is well known, in p-adic
analysis the cases of even and odd prime numbers differ
essentially, see, e.g., [46], and transition from p ¼ 2 to
p > 2 typically requires elaboration of novel mathematical
technique.
We remind that any p-adic integer (an element of the
ring Zp ) can be expanded into the series:

x ¼ x0 þ px1 þ    þ pk xk þ    ; xj 2 f0; 1; . . . ; p  1g:
Let functions dk ðxÞ; k ¼ 0; 1; 2; . . . are kth digits in a base-p
expansion of the number x 2 Zp , i.e. dk : Zp ! f0; 1; . . . ;
p  1g; dk ðxÞ ¼ xk .

12

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

Any map f : Zp ! Zp can be represented in the coordinate form:

Every x 2 Zp can be expanded in canonical form, namely
in the form of a series which converges by p-adic norm:

f ðxÞ ¼ d0 ðf ðxÞÞ þ pd1 ðf ðxÞÞ þ    þ pk dk ðf ðxÞÞ þ    :

x ¼ x0 þ px1 þ    þ pk xk þ    ;

ð1:1Þ

Our aim is to find criteria of measure-preserving and
ergodicity in terms of the coordinate functions dk ðf ðxÞÞ.
We restrict our study to the class of 1-Lipschitz functions.
The coordinate representation can be useful, e.g., for
applications to cryptography. Moreover, by using this
representation we can consider non-smooth p-adic
transformations.
The basic technical tools are van der Put series and

usage of algebraic structure (permutations) induced by
coordinate functions with partially frozen variables. (In
Section 2 we shall explain this technique in details.)
The representation of the function by the van der Put
series is actively used in p-adic analysis, see e.g. [40,46].
Marius van der Put introduced this series in his dissertation ‘‘Algèbres de fonctions continues p-adiques’’ at
Utrecht Universiteit in 1967, [42]. There are numerous
results in studies of functions with zero derivatives,
antiderivation [46] obtained using van der Put series. Later
van der Put basis was adapted to the case of n-times
continuously differentiable functions in one and several
variables [18]. First results on applications of the van der
Put series in theory of p-adic dynamical systems, the
problems of ergodicity and measure preserving, were obtained in [7], see also [8,38,51,52]. In this paper we apply
this technique to find criteria of ergodicity in terms of
the coordinate representation.
In some Theorems (3.1 and 3.4) conditions of ergodicity
are formulated in terms of integral sums for the
Volkenborn integral, see, e.g., [46]. One can expect to find
formulations of ergodicity in terms of this integral (playing

an important role in number theory). However, at the
moment this is an open problem.
We illustrate the basic theorems by presenting concrete
classes of ergodic functions which satisfy conditions of
these theorems.
1.1. P-adic numbers
We recall some definitions related to the p-adic analysis
and p-adic dynamical systems, as well as introduce the
necessary notations.
For any prime number p the p-adic norm j  jp is defined
in the following way. For every nonzero integer n let
ordp ðnÞ be the highest power of p which divides n, i.e.
n  0ðmod pordp ðnÞ Þ; n X 0ðmod pordp ðnÞþ1 Þ. Then we define
jnjp ¼ pordp ðnÞ ; j0jp ¼ 0. For rationals mn 2 Q we set
j mn jp ¼ pordp ðnÞþordp ðmÞ .
The completion of Q with respect to the p-adic metric
qp ðx; yÞ ¼ jx  yjp is called the field of p-adic numbers Qp .
The metric qp satisfies the so-called strong triangle
inequality jx  yjp 6 maxðjxjp ; jyjp Þ where equality holds if
jxjp – jyjp . The set fx 2 Qp : jxjp 6 1g is called the set of

p-adic integers. It is denoted by Zp . In some sense, the
set of p-adic integers is an analogue of the interval ½0; 1
for real numbers.
Hereinafter, we will consider only the p-adic integers Zp .

xk 2 f0; 1; . . . ; p  1g; k P 0:

If necessary, we can identify every p-adic integer with a
sequence of digits ðx0 ; x1 ; . . . ; xk ; . . .Þ. As ½xk we denote the
first k elements of this sequence, i.e.

½xk ¼ ðx0 ; x1 ; . . . ; xk1 Þ;

k P 1:

ð1:2Þ

Let a 2 Zp and r be a positive integer. The set
Bpr ðaÞ ¼ fx 2 Zp : jx  ajp 6 pr g ¼ a þ pr Zp is a ball of
radius pr with center a.

1.2. P-adic functions
In this article will be considered functions f : Zp ! Zp ,
which satisfy the Lipschitz condition with constant 1
(1-Lipschitz functions). Recall that f : Zp ! Zp is 1-Lipschitz function if

jf ðxÞ  f ðyÞjp 6 jx  yjp ;

for all x; y 2 Zp :

This condition is equivalent to that from x  yðmod pk Þ
follows f ðxÞ  f ðyÞðmod pk Þ for all k P 1.
For all k P 1 a 1-Lipschitz transformation f : Zp ! Zp
the reduced mapping modulo pk is fk1 : Z=pk Z ! Z=pk Z,
z # f ðzÞðmod pk Þ. Mapping fk1 is a well defined (the fk1
does not depend on the choice of representative in the ball
z þ pk Zp ).
The property of a function to be 1-Lipschitz is signifi2

cant. For example, let f ¼ x 2þx be a 2-adic function. This
function is not 1-Lipschitz since jf ð2Þ  f ð0Þj2 ¼ 1 but

j2  0j2 ¼ 21 . Now let us define a function f1 as
f1 ðx ðmod 2ÞÞ  f1 ðxÞðmod 2Þ. Then f1 ð0Þ ¼ 0 and f1 ð2Þ ¼ 1.
However, 2  0 ðmod 2Þ.
A 1-Lipschitz transformation f : Zp ! Zp is called
bijective modulo pk if the reduced mapping fk1 is a permutation on Z=pk Z. And f is called transitive modulo pk if fk1
is a permutation that is a cycle of length pk .
We introduce two representations of p-adic functions
which we use to describe ergodic properties, namely the
van der Put series and the coordinate representation of
1-Lipschitz p-adic functions.
1.2.1. Van der Put series
The van der Put series are defined in the following way.
Let f : Zp ! Zp be a continuous function. Then there exists
a unique sequence of p-adic coefficients B0 ; B1 ; B2 ; . . . such
that

f ðxÞ ¼

1
X

Bm vðm; xÞ

ð1:3Þ

m¼0

for all x 2 Zp . Here the characteristic function
given by

vðm; xÞ ¼



vðm; xÞ is

1; if jx  mjp 6 pn ;
0;

otherwise;

where n ¼ 0 if m ¼ 0, and n is uniquely defined by the
inequality pn1 6 m 6 pn  1 otherwise (see Schikhof’s

13

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

book [46] for detailed presentation of theory of van der Put
series). The number n in the definition of vðm; xÞ has a very
natural meaning. It is just the number of digits in a base-p
expansion of m 2 N0 . Then

blogp mc ¼
ðthe number of digits in a base-p expansion for mÞ  1;
therefore n ¼ blogp mc þ 1 for all m 2 N0 and blogp 0c ¼ 0
(recall that bac denotes the integral part of a).
The coefficients Bm are related to values of the function f
in the following way. Let m ¼ m0 þ    þ mn2 pn2 þ mn1
pn1 ; mj 2 f0; . . . ; p  1g; j ¼ 0; 1; . . . ; n  1 and mn1 – 0,
then

Bm ¼

(

f ðmÞ  f ðm  mn1 pn1 Þ; if m P p;

uk ðx0 ; . . . ; xk Þ ¼

otherwise:

f ðmÞ;

gðxÞ ¼ 0  vðx; 0Þ þ 1  vðx;1Þ þ 4  vðx; 2Þ þ 8  vðx; 3Þ
þ 16  vðx;4Þ þ 24  vðx;5Þ þ 32  vðx;6Þ þ 40  vðx; 7Þ þ  
Let x ¼ 5. The characteristic function vðx; mÞ is equal 1 for
m ¼ 1 and m ¼ 5, then gð5Þ ¼ 1  vð5; 1Þ þ 24  vð5; 5Þ ¼ 25.
1-Lipschitz functions f : Zp ! Zp in terms of the van der
Put series were described in [46]. We follow Theorem 3.1
[9] for convenience for further study. In this theorem, the
function f presented via van der Put series (1.3) is 1-Lipschitz if and only if jBm jp 6 pblogp mc for all m P 0. Assuming
Bm ¼ pblogp mc bm , we find that the function f is 1-Lipschitz if
and only if it can be represented as
1
X
pblogp mc bm vðm; xÞ

for suitable bm 2 Zp ; m P 0.
1.2.2. Coordinate representation of 1-Lipschitz functions
Let the function f : Zp ! Zp be represented in the coordinate form (1.1) (function dk defined on Zp ). According to
Proposition 3.33 in [6], f is a 1-Lipschitz function if and
only if for every k P 1 the k-th coordinate function
dk ðf ðxÞÞ does not depend on dkþs ðxÞ for all s P 1, i.e.
dk ðf ðx þ pkþ1 Zp ÞÞ ¼ dk ðf ðxÞÞ for all x 2 f0; 1; . . . ; pkþ1  1g.
Taking into account Notation (1.2) for k P 0, we
consider the functions of p-valued logic

uk : f0; . . . ; p  1g      f0; . . . ; p  1g ! f0; . . . ; p  1g
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
kþ1

and uk : ½xkþ1 # dk ðf ðxÞÞ.
Using these functions of p-valued logic, any 1-Lipschitz
function f : Zp ! Zp can be represented as:
1
X
pk uk ðx0 ; . . . ; xk Þ
k¼0

1
X
¼
pk uk ð½xkþ1 Þ
k¼0

I½ak ðx0 ; . . . ; xk1 Þuk;½ak ðxk Þ;

ð1:5Þ

where I½ak is a characteristic function, i.e.

I½ak ðx0 ; . . . ; xk1 Þ ¼



1; if ðx0 ; . . . ; xk1 Þ ¼ ½ak ;
0; otherwise:

:

In this notation, 1-Lipschitz function f : Zp ! Zp
becomes:

f ðxÞ ¼

1
X
pk uk ðx0 ; . . . ; xk Þ
k¼0

¼ u0 þ

k 1
pX
1
X
pk
I½ak ð½xk1 Þuk;½ak ðxk Þ

k¼1

ð1:6Þ

a¼0

Relation of the form (1.6) we call the coordinate representation of 1-Lipschitz function f. For example, let p ¼ 3
and we need to find the value of f modulo 33 given in the
form (1.6) at the point 19. Then, using the notation (1.2)

f ð19Þ  u0 ð1Þ þ 3u1;ð0Þ ð0Þ þ 32 u2;ð1;0Þ ð2Þ ðmod 33 Þ:

m¼0

f ðxÞ ¼ f ðx0 þ    þ pk xk þ   Þ ¼

k 1
pX

a¼0

We consider a simple example of the van der Put series
for the function gðxÞ ¼ x2 with p ¼ 2. Then

f ðxÞ ¼

For example, in this notation, the functions fk ; k P 0
(defined for 1-Lipschitz function f) can be represented as
P
fk ðxÞ ¼ ki¼0 pi ui ð½xiþ1 Þ.
Hereinafter, we will also use the following method of
defining the functions uk ðx0 ; . . . ; xk Þ; k P 1 (as a function
of p-valued logic). The function uk ðx0 ; . . . ; xk Þ can be defined by its sub-functions obtained by fixing the first k
variables ðx0 ; . . . xk1 Þ. Taking into account (1.2) the subfunction of the function uk ðx0 ; . . . ; xk Þ, obtained by fixing
the variables x0 ¼ a0 ; . . . ; xk1 ¼ ak1 ; ai 2 f0; . . . ; p  1g, is
denoted by uk;½ak , where a ¼ a0 þ pa1 þ . . . þ pk1 ak1 . In
this notations, the function uk ðx0 ; . . . ; xk Þ can be represented as

ð1:4Þ

The need to use such a representation is due to the fact
that the main results on ergodicity of 1-Lipschitz functions
will be formulated in terms of functions uk;½ak .
Any function uk can be given by a polynomial of the
ring of ðk þ 1Þ-variate polynomials ðZ=pZÞ½x0 ; . . . ; xk , with
coefficients from the residue ring Z=pZ, whose degree in
each variable is at most p  1. In other words, uk is defined
as an element of a factor-ring ðZ=pZÞ½x0 ; . . . ; xk  modulo an
ideal generated by all polynomials xpi  xi ; i ¼ 0; 1; . . . ; k.
In particular, the function uk is represented in the form
(expansion by the leading variable xk ):

uk ðx0 ; . . . ; xk Þ ¼ xp1
k ap1 ðx0 ; . . . ; xk1 Þ þ . . .
þ a0 ðx0 ; . . . ; xk1 Þ;
where
ai ðx0 ; . . . ; xk1 Þ 2 ðZ=pZÞ½x0 ; . . . ; xk1 ; i ¼ 0; 1; . . . ;
p  1. In particular,

uk;½ak ¼ xp1
k ap1 ð½ak Þ þ    þ a0 ð½ak Þ
is a polynomial from ðZ=pZÞ½xk  (ai ð½ak ÞÞ 2 Z=pZ; i ¼ ð0; 1;
. . . ; p  1).
Representation of functions uk;½ak by polynomials over a
field Z=pZ will be used to describe the families of ergodic
functions. For example, uk;½ak ¼ xk þ a0 ð½ak Þ is a linear unitary polynomial or uk;½ak ¼ xk Að½aS Þ þ a0 ð½ak Þ is a linear

14

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

polynomial where the leading coefficient depends on S first
terms of the ½ak , (S 6 k).
Thus, coordinate functions uk;½ak ; u0 can be considered
as the functions of p-valued logic and as transformation
of the ring Z=pZ. In each particular case we will determine
the way of representation by setting the domain of definition of these functions. For example, if we say that u0 is given on f0; . . . ; p  1g, then it means that the function u0 is
regarded as a function of p-valued logic.
Conditions on ergodicity are formulated in terms of
compositions of functions uk;½ak . The composition of functions we denote as ‘‘’’ i.e.
k

uk;½ak ðxk Þ  uk;½bk ðxk Þ ¼ uk;½ak ðuk;½bk ðxk ÞÞ; a; b 2 f0; 1; . . . ; p  1g:
As will be shown in Theorem 2.1: 1-Lipschitz function
preserves the measure as soon as all the functions
uk;½ak ; a 2 f0; 1; . . . ; pk  1g; k P 0 are bijective on Z=pZ,
i.e. define permutations on the set of p elements. Therefore,
in some cases, we use the terminology from the theory of
groups of permutations. For example, if uk;½ak is transitive
on Z=pZ, then it means that this function defines a transitive permutation on the set of pelements, or of symmetric
group of permutations Sp .
1.2.3. Some classes of 1-Lipschitz functions
This article will cover the following classes of 1-Lipschitz functions. Let f : Zp ! Zp be a 1-Lipschitz function.
The function f belongs to ‘0 if all the functions
uk ðx0 ; x1 ; . . . ; xk Þ; k P 1 (the function u0 is not under
restrictions) from (1.4) of f defined by polynomials over a
field Z=pZ of the form:

uk ðx0 ; x1 ; . . . ; xk Þ ¼ xk þ ak ðx0 ; x1 ; . . . ; xk1 Þ;

ð1:7Þ

where ak ðx0 ; x1 ; . . . ; xk1 Þ 2 ðZ=pZÞ½x0 ; x1 ; . . . ; xk1 . In terms
of the coordinate representation (1.6) the belonging to
the class ‘0 determined by the conditions uk;½ak
¼ xk þ ak ð½ak Þ; a 2 f0; 1; . . . ; pk  1g; k P 0; ak ð½ak Þ 2 Z=pZ.
Let S be a positive integer. The function f belongs to ‘S , if
all the functions uðx0 ; x1 ; . . . xk Þ; k P 1 (the function u0 is
not under restrictions) are defined by polynomials over
Z=pZ of the form:

uðx0 ; x1 ; . . . xk Þ ¼ xk Ak ðx0 ; x1 ; . . . ; xS Þ þ ak ðx0 ; x1 ; . . . ; xk1 Þ;
ð1:8Þ
i.e. polynomials Ak depend on no more than S variables. In
terms of the coordinate representation (1.6) the condition
of a function f to belong to the class ‘S becomes:

uk;½ak ¼



xk Ak ð½ak Þ þ ak ð½ak Þ; if k < S;
xk Ak ð½aS Þ þ ak ð½ak Þ;
k

if k P S;

for all a 2 f0; . . . ; p  1gk P 0 (here Ak ð½aS Þ 2 Z=pZ).
We also will use the definition of uniformly differentiable functions modulo p (see, for example, [6, Defintion 3.27
and 3.28 on p.58, 60]).
A function f : Zp ! Zp is said to be uniformly differentiable modulo ps on Zp if there exists positive integer
N and @ s f ðxÞ 2 Qp such that for any k > N the congruence
f ðx þ pk hÞ  f ðxÞ þ pk h@ s f ðxÞðmod pkþs Þ
holds
simultaneously for all x; h 2 Zp . The smallest of these N is denoted

by N s ðf Þ. For example, functions defined by polynomials
from Zp ½x are uniformly differentiable modulo p.
1.3. P-adic dynamics
Dynamical system theory study trajectories (orbits), i.e.
sequences of iterations:

x0 ; x1 ¼ f ðx0 Þ; . . . ; xiþ1 ¼ f ðxi Þ ¼ f ðiþ1Þ ðx0 Þ; . . . ;
where f ðsÞ ðxÞ ¼ f ðf ð. . . f ðxÞÞÞ . . .Þ.
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
s

We consider a p-adic autonomous dynamical system
hZp ; lp ; f i. The space Zp is equipped with a natural probability measure, namely, the Haar measure lp normalized
so that lp ðZp Þ ¼ 1. Balls Bpr ðaÞ of nonzero radii constitute
the base of the corresponding r-algebra of measurable
subsets, lp ðBpr ðaÞÞ ¼ pr .
Note that the 1-Lipschitz functions f : Zp ! Zp are
continuous.
As usual, a measurable mapping f : Zp ! Zp is called
measure-preserving if lp ðf 1 ðUÞÞ ¼ lp ðUÞ for each measurable subset U Zp .
A measure-preserving mapping f : Zp ! Zp is called
ergodic if f 1 ðUÞ ¼ U implies either lp ðUÞ ¼ 0 or lp ðUÞ ¼ 1.
In Section 2 we give a criterion of ergodicity (Theorem 2.2) of 1-Lipschitz functions f : Zp ! Zp for any p – 2.
A description of ergodic 1-Lipschitz 2-adic functions is
known (see, for example, [3,4,6] – in terms of Mahler’s series; [7,9,52] – in terms of van der Put’s series). To check the
ergodicity of fit is necessary for every k P 0 to check the
transitivity of all permutations uk;½ak ; a 2 f0; 1; . . . ; pk  1g;
k P 0 in especially order of their appearance in the resulting product (as an ergodic function is measure-preserving,
then the functions uk;½ak are bijective on Z=pZ, i.e. are permutations on this set). The order of these permutations in
the resulting product is determined by the order of the
elements from Z=pk Z in the sequence

n

k

k

o

ðp 2Þ
ðp 1Þ
sk ¼ 0; fk1 ð0Þ; . . . ; fk1
ð0Þ; fk1 ð0Þ :

The order of the elements in the sequence sk is significantly important. This is due to the fact that the symmetric
group Sp is nonabelian. In some special cases, the permutations uk;½ak ; a 2 f0; 1; . . . ; pk  1g are commuting. In this
case, the conditions of ergodicity can be formulated in a
more compact way (as an example, see the Corollary 2.5).
In the Corollary 2.6 we answer the following question.
Let f be a measure-preserving 1-Lipschitz function. How
much should one change such function to get an ergodic
function? It turns out that such change, with a certain degree of conditionality, is surprisingly small.
It turned out that for k P 1 of all pk permutations of
uk;½ak ; a 2 f0; 1; . . . ; pk  1g one should in a special way pick
just one permutation, and the remaining pk  1 permutations can be chosen arbitrarily. This means, in particular,
that using Theorem 2.2 and Corollary 2.6, we can inductively (by the parameter k) build any ergodic 1-Lipschitz
function f : Zp ! Zp .

15

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

In Section 3 we give a criterion on ergodicity of 1-Lipschitz functions f of class ‘0 and ‘S (for p – 2). In general,
the functions of these classes are characterised by the fact
that all functions uk;½ak ; a 2 f0; 1; . . . ; pk  1g; k P 0 in the
coordinate representation (1.6) are given by linear polynomials over a field Z=pZ. Ergodic functions of the class ‘0
described in Theorem 3.1. Functions from ‘0 admit the following functional representation (see Proposition 3.1)

f ðxÞ ¼ f ðx0 þ px1 þ   Þ ¼ u0 ðx0 Þ þ ðx  x0 Þ þ p  gðxÞ;
where u0 : Z=pZ ! Z=pZ; gðxÞ : Zp ! Zp is 1-Lipschitz
function. The class ‘0 consists of ergodic functions described in terms of the Mahler’s series (see, for example,
Theorem 4.40 from [6]). Since permutations defined by
the functions uk;½ak ¼ xk þ ak ð½ak Þ; a 2 f0; 1; . . . ; pk  1g
commute in the symmetric group Sp for all k P 1, then
the conditions of ergodicity are formulated in terms of
 Pk

integral sums for the Volkenborn integral p1k pi¼01 f ðiÞ ,

see, e.g., [46].
Ergodicity criterion for the functions from the class ‘S is
presented in Theorem 3.5 (or in equivalent form in (3.6)).
As a corollary from this theorem we formulate a criterion
of ergodicity of uniformly differentiable modulo p 1-Lipschitz functions. Thus, open question 4.60 [6, p.132] are
fully resolved. Note that 1-Lipschitz uniformly differentiable modulo p2 ergodic functions are described by V. Anashin, see, for example, Theorem 4.55 [6, p.126].
In Section 3 we also present new proofs of ergodicity for
functions of the form f ðxÞ ¼ c þ x þ pðhðx þ 1Þ  hðxÞÞ,
where h : Zp ! Zp is 1-Lipschitz function and c 2 Zp (such
functions have been described in, for example, [6, Lemma
4.41, p. 112]). We generalize ergodicity criterion for functions of the form f ðx0 þ px1 þ   Þ ¼ c þ a0 x0 þ    þ
ak pk xk þ   , where c; ak 2 Zp ; k P 0 to the case of an arbitrary p(for p ¼ 2 ergodic such functions are described in
[6, Theorem 9.20, p.286]).
In Section 4 we consider the case p ¼ 3. The main characterization of the case p ¼ 3 is that all the transformations
of the field of residues Z=3Z can be set by linear polynomials on ðZ=3ZÞ½x. As a corollary from the general criterion of
ergodicity (Theorem 2.2) we describe ergodic 1-Lipschitz
3-adic functions using such characterization of the case
p ¼ 3 (Theorem 4.2). In Theorem 4.4 we give a 3-adic analogue of Corollary 2.6. For p ¼ 3 conditions on construction
of the ergodic function from measure-preserving function
are simplified in comparison with the general case (for
p ¼ 3 requires to determine in a special way the value of
the function only at point 0).
In Section 4 we present the criteria on measure-preservation (Theorem 4.7) and ergodicity (Theorem 4.9) of
1-Lipschitz 3-adic functions with the use of the additive
form of representation. Earlier additive representation
was used to describe the measure-preserving 1-Lipschitz
p-adic functions (see [38]). In this form, measure-preserving 1-Lipschitz function is the sum of an arbitrary 1-Lipschitz function, and some special function which provides
fulfiling the property of measure-preservation. It turned
out that for p ¼ 3 in this special function we can distinguish a ‘‘constant’’ term (the form of representation of this

term can vary). In Theorems 4.7 and 4.9 as a ‘‘constant’’
term is considered a function x. As a result, the criteria
on measure-preservation and ergodicity are obtained in a
new form of representation. Using such a representation,
the Examples 4.8 and 4.10 illustrate how wide is the class
of measure-preserving ergodic 1-Lipschitz 3-adic functions
compared with the known classes of such functions (see,
for example, [6]).
In Section 5 we give a criterion on ergodicity of 1-Lipschitz 2-adic functions (Theorem 5.1), which is formulated
in terms of integral sums for the Volkenborn integral,
see, e.g., Schikhof [46].
2. General criteria on ergodicity for any p.
In the section, we prove a criterion on ergodicity of 1Lipschitz functions f : Zp ! Zp (Theorem 2.2) in terms of
their coordinate representation (1.4). There is an example
of a class of functions where conditions on ergodicity represented in the ‘‘compact’’ form (Corollary 2.5). In Corollary
2.6 we answer the following question ‘‘How much should
one change measure-preserving 1-Lipschitz function to
get an ergodic function?’’ In addition, we present a criterion on measure-preservation for 1-Lipschitz function represented in the coordinate form (1.4). This theorem is
needed for the proof of the ergodicity criterion.
Theorem 2.1. Let f : Zp ! Zp be 1-Lipschitz function in the
coordinate representation (1.6). The function f preserves
measure if and only if all functions u0 and uk;½ak ; a 2
f0; 1 . . . ; pk  1g and k P 1 is bijective (permutations) on
the set f0; . . . ; p  1g.
P1 blog mc
P
p
bm vðm; xÞ
Proof. Let f ðxÞ ¼ 1
m¼0 Bm vðm; xÞ ¼
m¼0 p
be the van der Put representation of the function f and
a 2 f0; . . . ; pk  1g. Let us find values of the coefficients bm .
Then bm ðf Þ ¼ Bm ðf Þ ¼ f ðmÞ for m 2 f0; . . . ; p  1g and for
m ¼ a þ pk xk ; a 2 f0; . . . ; pk  1g; xk – 0; k P 1

baþpk xk ðf Þ ¼

1
f ða þ pk xk Þ  f ðaÞ
Baþpk xk ðf Þ ¼
k
p
pk

¼ uk ð½ak ; xk Þ  uk ð½ak ; 0Þ þ pðukþ1 ð½ak ; xk ; 0Þ
 ukþ1 ð½ak ; 0; 0Þ þ . . .Þ
Thus

bm ðf Þ  u0 ðmÞðmod pÞ;

m 2 f0; . . . p  1g;

bm ðf Þ ¼ baþpk xk ðf Þ  uk ð½ak ; xk Þ  uk ð½ak ; 0Þ  uk;½ak ðxk Þ
 uk;½ak ð0Þðmod pÞ;

m ¼ a þ pk xk ; xk – 0

By Theorem 2.1 [38] the function f preserves the measure if
and only if
1. b0 ðf Þ; b1 ðf Þ; . . . ; bp1 ðf Þ establish a complete set of
residues modulo p;
2. baþpk ðf Þ; . . . ; baþpk ðp1Þ ðf Þ are all nonzero residues modulo p.
Then the first condition is equivalent to bijectivity of u0
and the second condition to bijectivity of uk;½ak . h

16

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

Now let us state a general criterion on ergodicity for a
1-Lipschitz function in terms of the coordinate functions
(1.6).
Theorem 2.2. Let f : Zp ! Zp be 1-Lipschitz function in the
coordinate representation (1.6), where all functions u0 and
uk;½ak , a 2 f0; 1; . . . ; pk  1g; k P 1 are permutations on the
set f0; . . . ; p  1g (i.e. f preserves the measure).
The function f is ergodic if and only if all permutations u0
and

F k ¼ u  ðpk 1Þ   u  ðpk 2Þ       uk;½0k ;
k; fk1

ð0Þ

k; fk1

k

ð0Þ

kP1

k

are transitive on the f0; . . . ; p  1g.

Proof. Let a ¼ a0 þ pa1 þ . . . þ pk1 ak1 2 f0; 1; . . . ; pk  1g.
ðpk1 Þ
And let us find the coordinate representation for fk
ðaÞ,
k P 1. Then we have:

h

ðpk1 Þ

fk

i

k

¼ ðF 1;½a1 ða0 Þ; . . . ; F k1;½ak1 ðak2 Þ; F k;½ak ðak1 ÞÞ;

transitive. Since f0 ¼ u0 , then transitivity of the map u0
follows from Theorem 4.23 [6, p.99].
We now present the proof that transitivity all functions
u0 and uk;½ak ; a 2 f0; 1; . . . ; pk  1g; k P 1 are sufficient for
ergodicity. We shall use induction with respect to the
parameter k ¼ 0; 1; . . . to show transitivity of the functions
fk .
For k ¼ 0 transitivity of f0 follows from the relation
f0 ¼ u0 and transitivity of u0 follows from the condition of
Theorem.
Let fk1 be transitive on the set of residues modulo pk .
Since fk induces a map on Z=pk1 , which coincides with fk1
ðpk Þ
(f is 1-Lipschitz function), then fk ¼ ð; ; . . . ; ; F k;½ak Þ,
where F k;½ak is defined in (2.1) and  is the identity
permutation on the set f0; . . . ; p  1g. By condition of the
Theorem, the map F k;½ak is the transitive permutation on
the set f0; . . . ; p  1g. Therefore,

h

ðpk Þ

fk

i

k

F 0;½a1 ¼ u0  u0     u0 ;
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
pk1

F 1;½a1 ¼ u

ðpk1 1Þ

2; f1

u

ð½a1 Þ

ðpk1 2Þ

2; f1

ð½a1 Þ

    u1;½a1 ;

. ..
F k1; ½ak1 ¼ u
F k;½ak ¼ u

k;f

k1; f

ðpk1 1Þ
ð½ak1 Þ
k1

ðpk1 1Þ
ð½ak Þ
k1

u

k; f

u

k1;f

ðpk1 2Þ
ð½ak Þ
k1

ðpk1 2Þ
ð½ak1 Þ
k1

    uk1;½ak1 ;

    uk;½ak ;
ð2:1Þ

Assume that f is ergodic. By Theorem 4.23 [6, p.99] if the
function f is ergodic, then, for any s P 0, the function fs is
transitive on the set of residues modulo psþ1 and, in fact,
ðps Þ
ps is the minimal integer such that fs ðaÞ  aðmod psþ1 Þ.
kþ1
kþ1
The function fk : Z=p Z ! Z=p Z induces a map on
Z=pk Z. This map coincides with fk1 (f is 1-Lipschitz function by initial conditions).
It means that the permutations F 0;½a1 ; F 1;½a1 ; . . . ; F k1;½ak1
ðpk1 Þ

from the coordinate representation of the fk
are identical. Indeed, in this case each permutation is a degree of
the identical permutation , i.e.

ðpk Þ

k

k

1

0

C
B
¼ @; ; . . . ; ; F k;½ak  . . .  F k;½ak A
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
p

g a ¼ u  ðpk 1Þ   u  ðpk 2Þ       uk;½ak ;
k; fk1

¼



ðpÞ

ðpk1 Þ
fk

it follows that



k

1

0

k

k; fk1

ðaÞ

k; fk1

ðbÞ

k

k; fk1

ðbÞ

k

k

have the same cyclic structure. Indeed, fk1 is transitive
modulo pk . Then there exist an integer r 6 pk1  1 such
ðrÞ
ðtÞ
ðtþrÞ
that fk1 ðaÞ ¼ b and, therefore, fk1 ðbÞ ¼ fk1 ðaÞ. Then
g a ¼ u  ðpk 1Þ       uk;½f ðrþ1Þ ðaÞ  uk;½f ðrÞ

     uk;½ak
ð2:3Þ
ðaÞ
ðaÞ
k1
k1
k
k
k1
k
 ðpk1 1rÞ       u ð2Þ
 uk;½fk1 ðbÞk  uk;½f ðr1Þ ðaÞ 
k;½f
ðbÞ

k; f

¼u

k; fk1

ðbÞ

k1

k

k1

k

k

ð2:4Þ

Consistently making conjugation of the permutation
(2.3) by permutations uk;½ak ; . . . ; uk;½f ðr1Þ ðaÞ , we obtain
k1



uk;½f ðr1Þ ðaÞ     uk;½ak  g a  uk;½ak
k1

k

¼ uk;½f ðr1Þ ðaÞ     uk;½ak  u

C
B
¼ @; ; . . . ; ; F k;½ak      F k;½ak A
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

F k;½ak      F k;½ak ¼ 
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ðaÞ

g b ¼ u  ðpk 1Þ   u  ðpk 2Þ       uk;½bk

     uk;½ak

As we know, fk ð½ak Þ  ½ak ðmod pkþ1 Þ and pk is minimal
(transitivity of fk is by Theorem 4.23 [6, p.99]).
Then from the representation and

i

¼ fk



and such pk is minimal. It means that fk is transitive on the
set of residues modulo pk . Thus we proved that the functions fk ; k ¼ 0; 1; . . . are transitive. Then by Proposition
4.35 [6, p.105] the function f is ergodic.
For the proof of the Theorem it is sufficient to show that
the conditions of ergodicity criterion of fdoes not depend
on the choice of the parameter a 2 f0; . . . ; pk  1g; k P 1. It
is enough to prove the following statement. Let
a – b 2 f0; 1; . . . ; pk  1g and k P 1. If function fk1 is
transitive modulo pk (i.e. on f0; 1; . . . ; pk  1g), then
permutations on f0; 1; . . . ; p  1g

F 0;½a1 ¼ ; F 1;½a1 ¼ ; . . . ; F k1;½ak1 ¼ :

ðpk Þ
fk

ðpÞ
ðpk1 Þ

¼ ð; ; . . . ; Þ

where

h



k1

¼u

p

k

ðpk 1Þ
k;½f
ðbÞ
k1
k

    u

k;½f

1

ðpk1 1rÞk
k1

ðpk rÞ
k;½f
ðbÞ
k1
k

u

k




    uk;½f ðr1Þ ðaÞ
ðbÞ

k1

k

1

 .. .  uk;½fk1 ðbÞk
k

ðpk 1rÞ
k;½f
ðbÞ
k1
k

    uk;½fk1 ðbÞk

¼ gb :
ð2:2Þ

p

and, moreover, p is minimal natural number for holding
the condition (2.2). Hence, the permutation F k;½ak is

Let G ¼ uk;½f ðr1Þ ðaÞ      uk;½ak . Then G  g a  G1 ¼ g b ,
k1

k

i.e., the permutations g a ; g b are conjugate permutations. It
is well known that cyclic structure of conjugate permutations coincide.

17

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

To complete the proof it suffices to choose a ¼ 0 (for all
k P 1) and set F k;½0k ¼ F k . h
Remark 2.3. Theorem 2.2 can be considered as generalization to the case p – 2 of Theorem 4.39 (Folklore) from the
book [6]. The latter was ‘‘known’’ by people working in
cryptography, but without formal 2-adic presentation
(and the authors of [6] do not know any rigorous mathematical proof before the one presented in this book).
Comment 2.4. To check a function on ergodicity by using
conditions of Theorem 2.2, one should check transitivity of
the permutations F k , k ¼ 1; 2; . . .. Each permutation is a
product of permutations uk;½ak ; a 2 f0; . . . ; pk  1g. The
order of their appearance in the resulting product (i.e. in
F k ) is defined by the sequence of residues modulo pk

sk ¼

ðpk 2Þ
ðpk 1Þ
f0; fk1 ð0Þ; . . . ; fk1 ð0Þ; fk1 ð0Þg:

Note that the order of residues modulo pk in the
sequence sk is significantly important. This is due to the
fact that the symmetric group Sp (permutations on Z=pZ)
is nonabelian. Therefore, in general by determining F k we
cannot change the order of the permutations uk;½ak .
If we suppose that the permutations uk;½ak 2 Sp ; a 2
f0; 1; . . . ; pk  1g commute, then the conditions of ergodicity of f are simplified. In the Corollary 2.5 we considered
the case such that, for each k
1 the permutations
uk;½ak 2 Sp belong to the cyclic group generated by a
permutation g k . In this case, all uk;½ak 2 Sp commute, F k
does not depend on the order of elements in the sequence
sk , and, moreover, F k ¼ g ak for some a depending only on k.
Therefore, to verify the transitivity of F k , there is no need to
build the sequence sk . This simplifies essentially the
verification of ergodicity of f.
Corollary 2.5. Let f : Zp ! Zp be 1-Lipschitz function in the
coordinate representation (1.6).
Suppose that the subfunctions have the form:

uk;½ak ¼ g knk ðaÞ ; a 2 f0; . . . ; pk  1g; k P 1;

ð2:5Þ

where g k is a permutation on Z=pZ and nk ðaÞ is a positive
integer (g 0k - identity permutation).
Then the function f is ergodic if and only if

Proof. Let f be an ergodic function. By Theorem 2.2
permutations u0 and F k ¼ u  ðpk 1Þ       uk;½fk1 ð0Þk
k1

ð0Þ

k

uk;½0k ; k P 1 are transitive on Z=pZ. Then

Fk ¼ u

ðpk 1Þ
k; ½fk1 ð0Þ
k

     uk;½fk1 ð0Þk  uk;½0k

pk 1

nk ðfk1 ð0ÞÞ

¼ gk

n ð0Þ

     gkk
k

n ðpk 1Þþþnk ð0Þ

¼ gkk

k

divides a. If the permutation g k has r1 cycles of the length
1, r2 cycles of the length 2; . . . rp cycles of the length p, and
r1 þ 2r 2 þ . . . þ p  rp ¼ p, then a ¼ ord g k ¼ lcmðijri – 0Þ.
As p is a prime number and p devides a, then the
permutation g k consists of one cycle of length p, i.e. a ¼ p
and g k is transitive permutation. From p ¼ gcdðaa;Rk Þ ; a ¼ p
P k 1
nk
follows that gcdða; Rk Þ ¼ gcdðp; Rk Þ ¼ 1, i.e. Rk ¼ pa¼0
ðaÞ X 0 ðmod pÞ.
And vice versa, let us find an order of each permutation
F k , k P 1. By one of the conditions of this corollary u0 ¼ f0
is a transitive permutation and, in particular,
ðp1Þ

ff0

ð0Þ; . . . ; f0 ð0Þ; 0g ¼ f0; 1; . . . ; p  1g.

ðp1Þ
½f0
ð0Þ1

Then

F 1 ¼ u1;

n ðp1Þþþn1 ð0Þ
g 11

     u1;½f0 ð0Þ1  u1;½01 ¼

¼ g R11 ,

where R1 ¼ n1 ðp  1Þ þ    þ n1 ð0Þ. By initial conditions
g 1 is transitive on Z=pZ, then ord g 1 ¼ p. As
R1 X 0ðmod pÞ, then gcdðp; R1 Þ ¼ 1 and ord F 1 ¼ ord
g R11 ¼

p
gcdðp;R1 Þ

¼ p. Thus F 1 is transitive permutation on

Z=pZ. Moreover, the function f1 is transitive on Z=p2 Z.
Computation of the values ord F k ; k P 2 is performed in
a similar way, taking into account that at each step we
prove transitivity of the functions fk1 . Finally, we obtain
that the permutations F k ; k P 1 are transitive on Z=pZ. As
u0 is transitive on Z=pZ, then the function f is ergodic by
Theorem 2.2. h
In the Corollary 2.6, in particular, we answer the following question. Let f be a measure-preserving 1-Lipschitz
function. How much should one change such function to
get an ergodic function? It turns out that such change, with
a certain degree of conditionality, is surprisingly small.
In representation (1.6) for all k P 1 each of pk  1 permutations uk;½ak ; a 2 f1; 2; . . . ; pk  1g (a X 0ðmod pk Þ) can
be chosen arbitrarily and only permutation uk;½0k is chosen
in a special way.
Corollary 2.6. Let f : Zp ! Zp be 1-Lipschitz function in the
coordinate representation (1.6), where the subfuctions of the
coordinate functions, u0 and uk;½ak ; a 2 f0; 1; . . . ; pk  1g;
k P 1 are permutations on the set Z=pZ and u0 is transitive.
Then it is always possible to construct permutations
~ k;½0 ¼ g k ; k ¼ 1;
g 1 ; g 2 ; . . . ; g k ; . . . such that by setting u
k
~
2; . . ., the corresponding function f which is defined with the
~ k;½0 is ergodic.
aid of subfunctions uk;½ak ; a – 0, and u
k

1. u0 ; g k are transitive permutations;
Ppk 1
k P 1.
2.
a¼0 nk ðaÞ X 0 mod pÞ;

k; f

Let ord g k ¼ a. Then p ¼ ord F k ¼ ord g Rk k ¼ gcdðaa;R Þ, i.e. p

R

¼ gkk ;

where Rk ¼ nk ðp  1Þ þ    þ nk ð0Þ.
As F k is transitive on Z=pZ, then ord F k ¼ p.

Proof. Let us find permutations g 1 ; g 2 ; . . . ; g k ; . . . by induction. By conditions of theorem, permutation u0 ¼ f0 is
n
ðp1Þ
transitive on Z=pZ, and, in particular, f0
ð0Þ; . . . ; f0

ð0Þ; 0g ¼ f0; 1; . . . ; p  1g. By using notations from Theorem 2.2, we set G1 ¼ u1;½f ðp1Þ ð0Þ  u1;½f ðp2Þ ð0Þ      u1;½f0 ð0Þ1 .
0

1

0

1

We also choose some transitive permutation H1 on Z=pZ.
~
Let g 1 ¼ G1
1  H 1 . Then set u1;0 ¼ g 1 , and we obtain that
F 1 ¼ u ðp1Þ
permutation e
 u ðp2Þ
   u

1;½f0

ð0Þ1

1;½f0

ð0Þ1

1;½f0 ð0Þ1

~ 1;½0k ¼ H1 is transitive on Z=pZ. Moreover, the function
u

~f is transitive on Z=p2 Z.
1

18

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

Now let permutations g 1 ; g 2 ; . . . ; g k1 ; . . . have already
been constructed, and the function ~f k1 is transitive on
Z=pk Z. Let us construct a suitable permutation g k . Let

Gk ¼ u

ðpk 1Þ
k;½~f
ð0Þ
k1

k

     uk;½~f k1 ð0Þ

k

and Hk be some transitive permutation on Z=pZ. And set
~
g k ¼ G1
k  H k . Then, for uk;½0k ¼ g k , we obtain that the
permutation

e
Fk ¼ u

ðpk 1Þ
k;½~f k1 ð0Þ

k

~ k;½0k ¼ Hk
     uk;½~f k1 ð0Þ  u
k

is transitive on Z=pZ. And the function ~f k is transitive on
Z=pkþ1 Z.
Thus we have shown existence of permutations
~ k;½0 ¼ g k ; k P 1 permutag 1 ; g 2 ; . . . ; g k ; . . . such that for u
k
tions e
F k are transitive on Z=pZ. As u0 is transitive on Z=pZ,
~
then the function f is ergodic by Theorem 2.2. h
Comment 2.7. Using the results of Corollary 2.6 and the
criterion of Theorem 2.2 we can inductively construct
any 1-Lipschitz ergodic function in the representation
(1.6) as follows. Permutation u0 we choose transitive on
Z=pZ. Let the permutations uk1;½ak1 ; a 2 f0; 1; . . . ;
pk1  1g are constructed, i.e. all of the functions
us ; s ¼ 1; 2; . . . ; k  1 are choosen. Let us construct uk . To
do this, the permutations uk;½ak ; a 2 f1; . . . ; pk  1g (i.e.
a X 0 ðmod pk ÞÞ we choose arbitrarily and find

Gk ¼ u  ðpk 1Þ       uk;½fk1 ð0Þk
k; fk1

ð0Þ

k

(the function fk1 defined via uk1;½ak1 ; a 2 f0; 1; . . . ;
pk1  1g). Then we choose uk;½0k in such a way that
Gk  uk;½0k be a transitive permutation on Z=pZ. As p is a
prime number, then one can write this condition analytically. Namely, uk;½0k is a solution in permutations
ðGk  XÞp ¼ , where  is the identical permutation.
As a result, the function uk is constructed. The number
of suitable choice equals to ðp!Þk1 ðp  1Þ! (or 1p of all
suitable functions uk for construction a measure-preserving functions).
Proceeding in this way to select the functions uk ; k
1,
then we construct a 1-Lipschitz ergodic function. It is clear
that any 1-Lipschitz ergodic function can be constructed
using this method.

3. Examples of classes of ergodic functions for p 6¼ 2
In this section, we prove a criterion on ergodicity of 1Lipschitz functions f of class ‘0 and ‘S (Theorems 3.1 and
3.5). In general, classes ‘0 and ‘S are characterized by the
fact that the functions uk;½ak ; a 2 f0; 1; . . . ; pk  1g; k P 0
in the coordinate representation of (1.6) f defined by linear
polynomials over a field Z=pZ (for the class ‘0 these polynomials are unitary, which allow us to represent functions
from ‘0 explicitly, i.e. without using the coordinate representation (1.6), see Lemma 3.1).
Examples of the application of the criteria on ergodicity
for the classes ‘0 and ‘S are:

 a new proof of the ergodicity for the functions of the
form c þ x þ pðhðx þ 1Þ  hðxÞÞ, where h : Zp ! Zp is
1-Lipschitz function, c X 0ðmod pÞ (Corollary 3.3);
 ergodicity criterion (Theorem 3.4) for the functions of
the form

f ðx0 þ px1 þ   Þ ¼ c þ a0 x0 þ    þ ak pk xk þ    ;
c; ak 2 Zp ; k P 0 for p – 2 (the case p ¼ 2 is considered
in, for example, [6, Theorem 9.20 (part 2), p.286]);
 description of uniformly differentiable modulo p 1Lipschitz functions (see, for example, [6]), i.e. open
question 4.60 [6, p.132] are fully resolved.
3.1. Ergodic functions of the class ‘0
In the subsection we prove a criterion on ergodicity of
1-Lipschitz functions f of class ‘0 (Theorem 3.1). The functions in this class are characterized by the fact that the
sub-functions of the coordinate functions are of the form
(1.7), i.e. defined by linear unitary polynomials over a field
Z=pZ. The permutations that are defined by these polynomials commute in the symmetric group of permutations.
This property allow us to formulate a criterion on ergodicity in terms of integral sums for the Volkenborn integral,
see, e.g., Schikhof [46]. Functions from ‘0 admits an explicit
representation (Lemma 3.1). As an example of the use of
Theorem 3.1 we give a new proof of (3.3) known criterion
on ergodicity of functions of the form c þ x þ pðhðx þ 1Þ
hðxÞÞ (see [3,4] and [6, Lemma 4.41, p.112]). Note that
these type of functions belong to the class ‘0 . Another
example of application of Theorem 3.1 is a generalization
of criterion on ergodicity for 2-adic functions (p ¼ 2) of
the form

f ðx0 þ px1 þ    þ pk xk þ   Þ ¼ c þ a0 x0 þ    þ ak pk xk þ    ;
where ak 2 Zp ; k P 1; c 2 Zp up to the case p – 2.
The explicit form of the functions from the class ‘0 is
presented in the following statement.
Lemma 3.1. Let f : Zp ! Zp be 1-Lipschitz function. Function
f belongs to ‘0 if and only if f can be represented as

f ðxÞ ¼ f ðx0 þ px1 þ    þ pk xk þ   Þ
¼ u0 ðx0 Þ þ ðx  x0 Þ þ p  gðxÞ;
where g : Zp ! Zp is 1-Lipschitz function and u0 : Z=pZ !
Z=pZ.
P
P1 blog mc
p
Proof. Let f ðxÞ ¼ 1
bm vðm; xÞ
m¼0 Bm vðm; xÞ ¼
m¼0 p
be the van der Put representation of the 1-Lipschitz function f 2 ‘0 . Let us find values of the coefficients bm for
 þ pk mk ;
m P p. Denote m ¼ m0 þ pm1 þ    þ pk m k ¼ m
 2 f0; . . . ; pk  1g; mk 2 f1; . . . ; p  1g (i.e. mk – 0). Then
m
(under notations from (1.4) and (1.6))

 þ pk mk Þ  f ðmÞ

Bm ¼ Bmþp
 k mk ¼ f ðm
¼ pk ðuk ð½mk ; mk Þ  uk ð½mk ; 0Þ þ pkþ1 ð. . .Þ;
i.e.

19

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

uk ð½mk ; mk Þ ¼ mk þ uk ð½mk ; 0Þ ¼ mk þ ak ð½mk Þ:

bm ¼ bmþp
 k xk  uk ð½mk ; mk Þ  uk ð½mk ; 0Þ
 mk þ ak ð½mk Þ  ak ð½mk Þ  mk

ðmod pÞ:

Thus f 2 ‘0 .

 k ¼m
In other words, bm ¼ bmþp
k m ¼ mk þ pbmþp


k
mk
k
m for suitable p-adic integer b
m .
þpb
 ;b
 2 Zp ; i ¼
We
set
f ðiÞ ¼ Bi ¼ bi ¼ bi ðmod pÞ þ pb
i
i
f0; . . . ; p  1g. Then the function u0 : Z=pZ ! Z=pZ from
representation (1.6) takes the form

u0 ðxÞ ¼ b0 ðmod pÞvð0; xÞ þ    þ bp1 ðmod pÞvðp  1; xÞ

h

Ergodicity criterion functions of the class ‘0 is presented
in the following theorem.
Theorem 3.1. Let p – 2 and f : Zp ! Zp be 1-Lipschitz
function and f 2 ‘0 , i.e.

f ðxÞ ¼ f ðx0 þ px1 þ    þ pk xk þ   Þ
¼ u0 ðx0 Þ þ ðx  x0 Þ þ p  gðxÞ;

and
1
1
X
X
bm ðmod pÞ  pblogp mc  vðm; xÞ ¼
qðmÞvðm; xÞ ¼ x  x0 ;

u0 : Z=pZ ! Z=pZ. The function f is ergodic if and only if

 þ pk mk Þ ¼ pk mk ; mk – 0.
where qðmÞ ¼ qðm
And let us consider the 1-Lipschitz
P
 blogp mc vðm; xÞ. Thus,
gðxÞ ¼ 1
m¼0 bm p

1. u0 is a transitive (monocycle) permutation on Z=pZ;
P k 1
2. 2p2 þ p1k pa¼0
f ðaÞ X 0 ðmod pÞ, or in the equivalent
Ppk 1
1
gðaÞ
X 2p2 ðmod pÞ; k P 1.
form pk1
a¼0

m¼p

m¼p

f ðxÞ ¼

function

1
X
bm pblogp mc vðm; xÞ

where

g : Zp ! Zp

is

1-Lipschitz

function

and

m¼0

¼

1
X
bm ðmod pÞ  pblogp mc  vðm; xÞ

m¼0

1
X
m pblogp mc vðm; xÞ
þp b
m¼0

p1
1
X
X
bm ðmod pÞpblogp mc vðm; xÞ þ
bm ðmod pÞ
¼
m¼p

m¼0

blogp mc

p

 vðm; xÞ þ pgðxÞ

¼ u0 ðx0 Þ þ ðx  x0 Þ þ pgðxÞ:
Vice versa, let

f ðxÞ ¼ f ðx0 þ px1 þ    þ pk xk þ   Þ
¼ u0 ðx0 Þ þ ðx  x0 Þ þ pgðxÞ
P
blogp mc
and f ðxÞ ¼ 1
vðm; xÞ.
m¼0 bm p
 þ pk mk ; mk – 0 we
Set m ¼ m0 þ pm1 þ    þ pk mk ¼ m
obtain that for m P p

bm ¼ bmþp
 k mk 


1
 þ pk mk Þ  f ðmÞÞ

ðf ðm
pk

1
 þ pk mk  m0 Þ þ p  gðm
 þ pk mk Þ
ðu ðm0 Þ þ ðm
pk 0
  m0 Þ þ p  gðmÞÞÞ

 ðu0 ðm0 Þ þ ðm

1 k
 þ pk mk Þ  gðmÞÞÞ

ðp mk þ p  ðgðm
pk
1
 þ pk mk Þ  gðmÞÞ

ðmod pÞ:
 mk þ k1 ðgðm
p


 þ pk mk Þ 
As the function g is 1-Lipschitz, then gðm

gðmÞðmod
pk Þ and

1
 þ pk mk Þ  gðmÞÞ
  0 mod p:
ðgðm
pk1
Then bm  mk ðmod pÞ.
On the other hand (under notations from (1.4)),

bm  uk ð½mk ; mk Þ  uk ð½mk ; 0Þ mod p:
or

Proof. Note that the function f preserves measure as soon
as u0 is bijective on Z=pZ. Indeed, f 2 ‘0 implies that all
functions uk;½ak ; a 2 f0; . . . ; pk  1g; k P 1 in the coordinate
representation of f (1.6) are bijective as linear polynomials
(see (1.7)) over the field of residues modulo p. Then by the
criterion on measure-preserving functions in terms of
coordinate functions, see Theorem 2.1 in [38], the function
f is measure-preserving and for all fixed a 2 f0; 1; . . . ;
pk  1g the function uk;½ak is a permutation on Z=pZ; k P 1.
As in Theorem 2.2, let us consider the permutations

F k ¼ u  ðpk 1Þ   u  ðpk 2Þ       uk;½0k ;
k; fk1

ð0Þ

k

k; fk1

ð0Þ

k

where k P 1.
By the assumption the permutations uk;½ak are given by
linear polynomials of the form uk;½ak ðxk Þ ¼ xk þ ak ð½ak Þ.
Therefore

F k ¼ xk þ ak
¼ xk þ

h

k 1
pX

i
i
h k
ðpk Þ
ðp 2Þ
fk1 ð0Þ
þ ak fk1 ð0Þ
þ    þ ak ð½0k Þ
k

ak

i¼0

h

k

i
ðiÞ
fk1 ð0Þ :
k

Let us show by induction with respect to k P 0 that the
functions fk are transitive on Z=pkþ1 Z. For k ¼ 0, the function f0 is transitive because f0 ¼ u0 , where u0 is transitive
on Z=pZ by the assumption. Suppose now that fk1 is transitive on Z=pk Z. Then

F k ¼ xk þ

k 1
pX

h

ðiÞ
ak fk1
ð0Þ

i¼0

i
k

¼ xk þ

k 1
pX

ak ð½ik Þ:

i¼0

On the other hand, set a ¼ a0 þ pa1 þ    þ pk1 ak1 (under
notations in (1.2)), then we obtain
k 1
pX

f ðaÞ 

a¼0

k 1
pX

a¼0



k 1
pX

a¼0

 pk 

k1
X
p uk ða0 ;. .. ;ak1 ; 0Þ þ
pi ui ða0 ; a1 ;. .. ;ai Þ
k

i¼0

!
k1
X
k
i
p ak ð½ak Þ þ
p ui ðx0 ; x1 ; .. .; xi Þ
i¼0

k 1
pX

a¼0

ak ð½ak Þ þ

k 1
pX

fk1 ðaÞ ðmod pkþ1 Þ:

a¼0

!

20

A. Khrennikov, E. Yurova / Chaos, Solitons & Fractals 60 (2014) 11–30

As fk1 is bijective on Z=pk1 Z, then
¼p

k ðpk 1Þ

2

k 1
pX

, i.e.

f ðaÞ  pk 

a¼0

k 1
pX

ak ð½ak Þ þ

a¼0

pk ðpk  1Þ
2

Ppk 1
a¼0

fk1 ðaÞ

ðmod pkþ1 Þ;

or
k 1
pk 1
pX
pk 1
1 X
pk  1 X

f
ðaÞ

a
ð½a
Þ
þ
ak ð½ak Þ  21

k
k
pk a¼0
2
a¼0
a¼0

ðmod pÞ;

where 21  2p2 ðmod pÞ.

Then by the second condition of this Theorem
k 1
pX

pk 1

ak ð½ak Þ 

a¼0

1 X

f ðaÞ þ 2p2 X 0 ðmod pÞ:
pk a¼0

And, therefore, the permutation F k is transitive on Z=pZ.
This means that the function fk is transitive on Z=pk Z. By
Theorem 4.23 [6, p.99] from transitivity of fk on
Z=pk Z; k P 0 follows that the function f is ergodic.
Vice versa, let the function f be ergodic function. Then
f0 ¼ u0 and functions