Criteria of ergodicity for p adic dynami

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 ofp-adic dynamical systems. We solved the problem of characterization of ergodicity and measure preserving for (discrete)p-adic dynamical systems for arbitrary primepfor iterations based on 1-Lipschitz functions. This problem was open since long time and only the casep¼2 was investigated in details. We formulated the criteria of ergodicity and measure preserving in terms of coordinate func-tions corresponding to digits in the canonical expansion ofp-adic numbers. (The coordinate representation can be useful, e.g., for applications to cryptography.) Moreover, by using this representation we can consider non-smoothp-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 devel-oped 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 andp-adic numbers are the only completions of the field of rational numbersQ. This is one of (purely mathematical) motiva-tions to study dynamical systems inQ_p, 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 ofp-adic

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 primep

for iterations based on 1-Lipschitz functions. This problem has been open since long time[6]and only the casep¼2 has been investigated in details. As is well known, inp-adic analysis the cases of even and odd prime numbers differ essentially, see, e.g., [46], and transition from p¼2 to

p>2 typically requires elaboration of novel mathematical technique.

We remind that anyp-adic integer (an element of the ringZ_p) can be expanded into the series:

x¼x0þpx1þ þpkxkþ ;xj2 f0;1;. . .;p1g:

Let functionsdkðxÞ;k¼0;1;2;. . .arekth digits in a base-p expansion of the number x2Z_p, i.e. dk:Zp! f0;1;. . .;

p1g;dkðxÞ ¼xk.

0960-0779/$ - see front matterÓ2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2014.01.001

⇑Corresponding author.

E-mail address:andrei.khrennikov@lnu.se(A. Khrennikov).

Contents lists available atScienceDirect

Chaos, Solitons & Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena

Any mapf:Z_p!Z_p can be represented in the coordi-nate form:

fðxÞ ¼d0ðfðxÞÞ þpd1ðfðxÞÞ þ þpk_d

kðfðxÞÞ þ : ð1:1Þ

Our aim is to find criteria of measure-preserving and ergodicity in terms of the coordinate functions dkðfðxÞÞ. 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 Section2we shall explain this technique in details.)

The representation of the function by the van der Put

seriesis actively used inp-adic analysis, see e.g.[40,46].

Marius van der Put introduced this series in his disserta-tion ‘‘Algèbres de foncdisserta-tions 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 ob-tained 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 thep-adic analysis and p-adic dynamical systems, as well as introduce the necessary notations.

For any prime numberpthep-adic normj jpis defined in the following way. For every nonzero integer n let

ordpðnÞ be the highest power of p which divides n, i.e.

n0ðmodpordpðnÞ_Þ;nX0ðmodpordpðnÞþ1_{Þ. Then we define} jnjp¼pordpðnÞ;j0jp¼0. For rationals mn2Q we set

jn

mjp¼pordpðnÞþordpðmÞ.

The completion ofQwith respect to the p-adic metric

q

_pðx;yÞ ¼ jxyj_p is called the field of p-adic numbersQ_p.

The metric

q

p satisfies the so-called strong triangle inequalityjxyj_p6maxðjxj_p;jyj_pÞwhere equality holds if

jxj_p–jyj_p. The set fx2Q_p:jxj_p61g is called the set of p-adic integers. It is denoted by Zp. In some sense, the set ofp-adic integers is an analogue of the interval½0;1

for real numbers.

Hereinafter, we will consider only thep-adic integersZ_p.

Everyx2Z_pcan be expanded in canonical form, namely in the form of a series which converges byp-adic norm:

x¼x0þpx1þ þpkxkþ ; xk2 f0;1;. . .;p1g; kP0:

If necessary, we can identify every p-adic integer with a sequence of digitsðx0;x1;. . .;xk;. . .Þ. As½xkwe denote the firstkelements of this sequence, i.e.

½xk¼ ðx0;x1;. . .;xk1Þ; kP1: ð1:2Þ

Let a2Z_p and r be a positive integer. The set

BprðaÞ ¼ fx2Z_p:jxaj

p6prg ¼aþprZp is a ball of radiuspr _{with centera.}

1.2. P-adic functions

In this article will be considered functionsf:Z_p!Z_p, which satisfy the Lipschitz condition with constant 1 (1-Lipschitz functions). Recall that f:Z_p!Z_p is 1-Lips-chitz function if

jfðxÞ fðyÞjp6jxyjp; for allx;y2Zp:

This condition is equivalent to that from xyðmodpk_Þ followsfðxÞ fðyÞðmodpk_{Þfor allkP1.}

For allkP1 a 1-Lipschitz transformation f:Z_p!Z_p

the reduced mapping modulo pk _{is f}

k1:Z=pkZ!Z=pkZ, z#fðzÞðmodpk_{Þ. Mappingf}

k1 is a well defined (thefk1

does not depend on the choice of representative in the ball

zþpk_Zp).

The property of a function to be 1-Lipschitz is signifi-cant. For example, let f¼x2_þx

2 be a 2-adic function. This

function is not 1-Lipschitz since jfð2Þ fð0Þj2¼1 but j20j2¼2

1

. Now let us define a function f1 as f1ðxðmod 2ÞÞ f1ðxÞðmod 2Þ. Thenf1ð0Þ ¼0 andf1ð2Þ ¼1.

However, 20ðmod 2Þ.

A 1-Lipschitz transformation f:Z_p!Z_p is called bijective modulopk_{if the reduced mappingf}

k1is a

permu-tation onZ₌pk_{Z. Andfis called transitive modulop}k_iff k1

is a permutation that is a cycle of lengthpk_.

We introduce two representations ofp-adic functions which we use to describe ergodic properties, namely the van der Put series and the coordinate representation of 1-Lipschitzp-adic functions.

1.2.1. Van der Put series

The van der Put series are defined in the following way. Letf:Z_p!Z_pbe a continuous function. Then there exists a unique sequence ofp-adic coefficientsB0;B1;B2;. . .such

that

fðxÞ ¼X

1

m¼0

Bm

v

ðm;xÞ ð1:3Þ

for all x2Z_p. Here the characteristic function

v

ðm;xÞ is given by

v

ðm;xÞ ¼ 1; if jxmjp6pn;

0; otherwise;

book[46]for detailed presentation of theory of van der Put series). The numbernin the definition of

v

ðm;xÞhas a very natural meaning. It is just the number of digits in a base-p

expansion ofm2N₀. Then

blogpmc ¼

ðthe number of digits in a base-pexpansion formÞ 1;

therefore n¼ blog_pmc þ1 for all m2N₀ and blog_p0c ¼0 (recall thatb

a

cdenotes the integral part of

a

).

The coefficientsBmare related to values of the functionf in the following way. Letm¼m0þ þmn2pn2þmn1 pn1_;m

j2 f0;. . .;p1g;j¼0;1;. . .;n1 and mn1–0,

then

Bm¼ fðmÞ fðmmn1p

n1_{Þ; ifmPp;}

fðmÞ; otherwise: (

We consider a simple example of the van der Put series for the functiongðxÞ ¼x2_{withp¼2. Then}

gðxÞ ¼0

v

ðx;0Þ þ1

v

ðx;1Þ þ4

v

ðx;2Þ þ8

v

ðx;3Þ

þ16

v

ðx;4Þ þ24

v

ðx;5Þ þ32

v

ðx;6Þ þ40

v

ðx;7Þ þ

Letx¼5. The characteristic function

v

ðx;mÞis equal 1 for

m¼1 andm¼5, thengð5Þ ¼1

v

ð5;1Þ þ24

v

ð5;5Þ ¼25. 1-Lipschitz functionsf:Z_p!Z_pin 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 functionfpresented via van der Put series(1.3)is 1-Lips-chitz if and only ifjBmjp6pblogpmcfor allmP0. Assuming

Bm¼pblogpmcbm, we find that the functionfis 1-Lipschitz if and only if it can be represented as

fðxÞ ¼X

1

m¼0

pblogpmc_bm

v

_ðm;xÞ

for suitablebm2Zp;mP0.

1.2.2. Coordinate representation of 1-Lipschitz functions

Let the functionf:Z_p!Z_pbe represented in the coor-dinate form(1.1)(functiondkdefined onZp). According to Proposition 3.33 in [6],fis a 1-Lipschitz function if and only if for every kP1 the k-th coordinate function

dkðfðxÞÞ does not depend on dkþsðxÞ for all sP1, i.e.

dkðfðxþpkþ1ZpÞÞ ¼dkðfðxÞÞfor allx2 f0;1;. . .;pkþ11g. Taking into account Notation (1.2) for kP0, we consider the functions ofp-valued logic

u

k:f0;. . .;p1g f0;. . .;p1g |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

kþ1

! f0;. . .;p1g

and

u

k:½xkþ1#dkðfðxÞÞ.

Using these functions ofp-valued logic, any 1-Lipschitz functionf:Z_p!Z_pcan be represented as:

fðxÞ ¼fðx0þ þpkxkþ Þ ¼

X1

k¼0

pk

u

kðx0;. . .;xkÞ

¼X

1

k¼0

pk

u

kð½xkþ1Þ ð1:4Þ

For example, in this notation, the functions fk;kP0 (defined for 1-Lipschitz functionf) can be represented as

fkðxÞ ¼Pki¼0pi

u

ið½xiþ1Þ.

Hereinafter, we will also use the following method of defining the functions

u

_kðx0;. . .;xkÞ;kP1 (as a function ofp-valued logic). The function

u

kðx0;. . .;xkÞcan be de-fined by its sub-functions obtained by fixing the first k

variablesðx0;. . .xk1Þ. Taking into account (1.2) the

sub-function of the sub-function

u

kðx0;. . .;xkÞ, obtained by fixing the variables x0¼a0;. . .;xk1¼ak1;ai2 f0;. . .;p1g, is denoted by

u

_k;½a

k, where a¼a0þpa1þ. . .þp k1_a

k1. In

this notations, the function

u

_kðx0;. . .;xkÞ can be repre-sented as

u

kðx0;. . .;xkÞ ¼

X pk₁

a¼0

I½akðx0;. . .;xk1Þ

u

k;½akðxkÞ; ð1:5Þ

whereI½ak is a characteristic function, i.e.

I½akðx0;. . .;xk1Þ ¼

1; if ðx0;. . .;xk1Þ ¼ ½ak;

0; otherwise:

:

In this notation, 1-Lipschitz function f:Z_p!Z_p

becomes:

fðxÞ ¼X

1

k¼0

pk

u

kðx0;. . .;xkÞ

¼

u

0þ

X1

k¼1

pkX pk₁

a¼0

I½akð½xk1Þ

u

k;½akðxkÞ ð1:6Þ

Relation of the form(1.6)we call the coordinate repre-sentation of 1-Lipschitz functionf. For example, letp¼3 and we need to find the value offmodulo 33given in the form(1.6)at the point 19. Then, using the notation(1.2)

fð19Þ

u

0ð1Þ þ3

u

1;ð0Þð0Þ þ3 2

u

2;ð1;0Þð2Þ ðmod 3 3

Þ:

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

u

k;½ak.

Any function

u

_k can be given by a polynomial of the ring ofðkþ1Þ-variate polynomialsðZ=pZÞ½x0;. . .;xk, with coefficients from the residue ringZ₌pZ, whose degree in each variable is at mostp1. In other words,

u

_kis defined as an element of a factor-ringðZ=pZÞ½x0;. . .;xkmodulo an ideal generated by all polynomialsxp

i xi;i¼0;1;. . .;k. In particular, the function

u

_kis represented in the form (expansion by the leading variablexk):

u

kðx0;. . .;xkÞ ¼xpk1

a

p1ðx0;. . .;xk1Þ þ. . .

þ

a

0ðx0;. . .;xk1Þ;

where

a

iðx0;. . .;xk1Þ 2 ðZ=pZÞ½x0;. . .;xk1;i¼0;1;. . .; p1. In particular,

u

k;½ak¼x p1

k

a

p1ð½akÞ þ þ

a

0ð½akÞ

is a polynomial fromðZ=pZÞ½xk(

a

ið½akÞÞ 2Z=pZ;i¼ ð0;1;

. . .;p1).

polynomial where the leading coefficient depends onSfirst terms of the½a_k, (S6k).

Thus, coordinate functions

u

k;½ak;

u

0can be considered as the functions ofp-valued logic and as transformation of the ringZ=pZ. In each particular case we will determine the way of representation by setting the domain of defini-tion of these funcdefini-tions. For example, if we say that

u

₀is gi-ven onf0;. . .;p1g, then it means that the function

u

0is

regarded as a function ofp-valued logic.

Conditions on ergodicity are formulated in terms of compositions of functions

u

_k;½a

k. The composition of func-tions we denote as ‘‘’’ i.e.

u

k;½akðxkÞ

u

k;½bkðxkÞ ¼

u

k;½akð

u

k;½bkðxkÞÞ; a;b2 f0;1;. . .;p k_1g:

As will be shown inTheorem 2.1: 1-Lipschitz function preserves the measure as soon as all the functions

u

k;½ak;a2 f0;1;. . .;p

k_{1g;kP0 are bijective on Z=pZ,} i.e. define permutations on the set ofpelements. Therefore, in some cases, we use the terminology from the theory of groups of permutations. For example, if

u

_k;½a

k is transitive onZ=pZ, then it means that this function defines a transi-tive permutation on the set ofpelements, or of symmetric group of permutationsSp.

1.2.3. Some classes of 1-Lipschitz functions

This article will cover the following classes of 1-Lips-chitz functions. Letf:Z_p!Z_p be a 1-Lipschitz function.

The function f belongs to ‘0 if all the functions

u

kðx0;x1;. . .;xkÞ;kP1 (the function

u

0 is not under

restrictions) from(1.4)offdefined by polynomials over a fieldZ=pZof the form:

u

kðx0;x1;. . .;xkÞ ¼xkþ

a

kðx0;x1;. . .;xk1Þ; ð1:7Þ

where

a

kðx0;x1;. . .;xk1Þ 2 ðZ=pZÞ½x0;x1;. . .;xk1. In terms

of the coordinate representation (1.6) the belonging to the class ‘0 determined by the conditions

u

k;½ak ¼xkþ

a

kð½akÞ;a2 f0;1;. . .;pk1g;kP0;

a

kð½akÞ 2Z=pZ.

LetSbe a positive integer. The functionfbelongs to‘S, if all the functions

u

ðx0;x1;. . .xkÞ;kP1 (the function

u

₀is not under restrictions) are defined by polynomials over

Z₌pZof the form:

u

ðx0;x1;. . .xkÞ ¼xkAkðx0;x1;. . .;xSÞ þ

a

kðx0;x1;. . .;xk1Þ;

ð1:8Þ

i.e. polynomialsAkdepend on no more thanSvariables. In terms of the coordinate representation(1.6)the condition of a functionfto belong to the class‘Sbecomes:

u

k;½ak¼

xkAkð½akÞ þ

a

kð½akÞ; ifk<S; xkAkð½aSÞ þ

a

kð½akÞ; ifkPS;

for alla2 f0;. . .;pk_{1gkP0 (hereA}

kð½aSÞ 2Z=pZ). We also will use the definition of uniformly differentia-ble functions modulop(see, for example,[6, Defintion 3.27 and 3.28 on p.58, 60]).

A functionf:Z_p!Z_pis said to be uniformly differen-tiable modulo ps _{on Z}

p if there exists positive integer

Nand@sfðxÞ 2Qpsuch that for anyk>Nthe congruence

fðxþpk_{hÞ fðxÞ þp}k_h@

sfðxÞðmodpkþsÞ holds

simulta-neously for allx;h2Zp. The smallest of theseNis denoted

by NsðfÞ. For example, functions defined by polynomials fromZ_p½xare uniformly differentiable modulop.

1.3. P-adic dynamics

Dynamical system theory study trajectories (orbits), i.e. sequences of iterations:

x0; x1¼fðx0Þ;. . .;xiþ1¼fðxiÞ ¼fðiþ1Þðx0Þ;. . .;

wherefðsÞ_{ðxÞ ¼fðfð. . .fðxÞÞÞ. . .Þ} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

s .

We consider ap-adic autonomous dynamical system

hZ_p;

l

p;fi. The spaceZpis equipped with a natural proba-bility measure, namely, the Haar measure

l

p normalized so that

l

pðZpÞ ¼1. BallsBprðaÞof nonzero radii constitute the base of the corresponding

r

-algebra of measurable subsets,

l

pðBprðaÞÞ ¼pr.

Note that the 1-Lipschitz functions f:Z_p!Z_p are continuous.

As usual, a measurable mappingf:Z_p!Z_p is called measure-preserving if

l

pðf1ðUÞÞ ¼

l

pðUÞfor each measur-able subsetU Zp.

A measure-preserving mapping f:Z_p!Z_p is called ergodic iff1_{ðUÞ ¼Uimplies either}

l

pðUÞ ¼0 or

l

pðUÞ ¼1. In Section 2 we give a criterion of ergodicity ( Theo-rem 2.2) of 1-Lipschitz functionsf:Z_p!Z_pfor anyp–2. A description of ergodic 1-Lipschitz 2-adic functions is known (see, for example,[3,4,6]– in terms of Mahler’s ser-ies;[7,9,52]– in terms of van der Put’s series). To check the ergodicity offit is necessary for everykP0 to check the transitivity of all permutations

u

k;½ak;a2 f0;1;. . .;p

k_1g;

kP0 in especially order of their appearance in the result-ing product (as an ergodic function is measure-preservresult-ing, then the functions

u

_k;½a

kare bijective onZ=pZ, i.e. are per-mutations on this set). The order of these perper-mutations in the resulting product is determined by the order of the elements fromZ₌pk_{Zin the sequence}

s

k¼ 0;fk1ð0Þ;. . .;fð

pk_2Þ k1 ð0Þ;f

ðpk_1Þ k1 ð0Þ

n o

:

The order of the elements in the sequence

s

k is signifi-cantly important. This is due to the fact that the symmetric groupSpis nonabelian. In some special cases, the permuta-tions

u

_k;½a

k;a2 f0;1;. . .;p

k_{1g are commuting. In this}

case, the conditions of ergodicity can be formulated in a more compact way (as an example, see theCorollary 2.5). In theCorollary 2.6we answer the following question. Let fbe 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 de-gree of conditionality, is surprisingly small.

It turned out that forkP1 of allpk permutations of

u

_k;½a

k;a2 f0;1;. . .;p

k_{1gone should in a special way pick}

just one permutation, and the remainingpk₁ permuta-tions can be chosen arbitrarily. This means, in particular, that usingTheorem 2.2andCorollary 2.6, we can induc-tively (by the parameterk) build any ergodic 1-Lipschitz functionf:Z_p!Zp.

In Section3we give a criterion on ergodicity of 1-Lips-chitz functionsfof class‘0and ‘S(forp–2). In general, the functions of these classes are characterised by the fact that all functions

u

k;½ak;a2 f0;1;. . .;p

k_{1g;kP0 in the}

coordinate representation(1.6)are given by linear polyno-mials over a field Z=pZ. Ergodic functions of the class‘0

described inTheorem 3.1. Functions from‘0admit the

fol-lowing functional representation (seeProposition 3.1)

fðxÞ ¼fðx0þpx1þ Þ ¼

u

0ðx0Þ þ ðxx0Þ þpgðxÞ;

where

u

0:Z=pZ!Z=pZ;gðxÞ:Zp!Zp is 1-Lipschitz function. The class ‘0 consists of ergodic functions

de-scribed in terms of the Mahler’s series (see, for example, Theorem 4.40 from [6]). Since permutations defined by the functions

u

k;½ak¼xkþ

a

kð½akÞ;a2 f0;1;. . .;p

k_1g

commute in the symmetric group Sp for allkP1, then the conditions of ergodicity are formulated in terms of

integral sums for the Volkenborn integral 1

pk Ppk₁

i¼0 fðiÞ

, see, e.g.,[46].

Ergodicity criterion for the functions from the class‘Sis presented inTheorem 3.5(or in equivalent form in(3.6)). As a corollary from this theorem we formulate a criterion of ergodicity of uniformly differentiable modulop 1-Lips-chitz functions. Thus, open question 4.60 [6, p.132]are fully resolved. Note that 1-Lipschitz uniformly differentia-ble modulop2_{ergodic functions are described by V.}

Ana-shin, see, for example, Theorem 4.55[6, p.126].

In Section3we also present new proofs of ergodicity for functions of the form fðxÞ ¼cþxþpðhðxþ1Þ hðxÞÞ, whereh:Z_p!Z_pis 1-Lipschitz function andc2Z_p(such functions have been described in, for example,[6, Lemma 4.41, p. 112]). We generalize ergodicity criterion for func-tions of the form fðx0þpx1þ Þ ¼cþa0x0þ þ akpkxkþ , wherec;ak2Zp;kP0 to the case of an arbi-traryp(forp¼2 ergodic such functions are described in

[6, Theorem 9.20, p.286]).

In Section4we consider the casep¼3. The main char-acterization of the casep¼3 is that all the transformations of the field of residuesZ₌3Zcan be set by linear polynomi-als onðZ₌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

p¼3 (Theorem 4.2). InTheorem 4.4we give a 3-adic ana-logue ofCorollary 2.6. Forp¼3 conditions on construction of the ergodic function from measure-preserving function are simplified in comparison with the general case (for

p¼3 requires to determine in a special way the value of the function only at point 0).

In Section4we present the criteria on measure-preser-vation (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-preserv-ing 1-Lipschitz function is the sum of an arbitrary 1-Lips-chitz function, and some special function which provides fulfiling the property of measure-preservation. It turned out that for p¼3 in this special function we can distin-guish a ‘‘constant’’ term (the form of representation of this

term can vary). InTheorems 4.7 and 4.9as a ‘‘constant’’ term is considered a functionx. As a result, the criteria on measure-preservation and ergodicity are obtained in a new form of representation. Using such a representation, theExamples 4.8 and 4.10illustrate 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 Section5we give a criterion on ergodicity of 1-Lips-chitz 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 anyp.

In the section, we prove a criterion on ergodicity of 1-Lipschitz functionsf:Z_p!Z_p (Theorem 2.2) in terms of their coordinate representation(1.4). There is an example of a class of functions where conditions on ergodicity rep-resented in the ‘‘compact’’ form (Corollary 2.5). InCorollary 2.6we answer the following question ‘‘How much should one change measure-preserving 1-Lipschitz function to get an ergodic function?’’ In addition, we present a crite-rion on measure-preservation for 1-Lipschitz function rep-resented in the coordinate form (1.4). This theorem is needed for the proof of the ergodicity criterion.

Theorem 2.1. Let f:Z_p!Z_pbe 1-Lipschitz function in the

coordinate representation (1.6). The function f preserves

measure if and only if all functions

u

0 and

u

k;½ak; a2

f0;1. . .;pk_{1g and kP1 is bijective (permutations) on}

the setf0;. . .;p1g.

Proof. Let fðxÞ ¼P1m¼0Bm

v

ðm;xÞ ¼ P1

m¼0pblogpmcbm

v

ðm;xÞ be the van der Put representation of the functionf and

a2 f0;. . .;pk_{1g. Let us find values of the coefficientsbm.} ThenbmðfÞ ¼BmðfÞ ¼fðmÞform2 f0;. . .;p1gand for

m¼aþpk_x

k;a2 f0;. . .;pk1g;xk–0;kP1

baþpk_x kðfÞ ¼

1

pkBaþpk_x kðfÞ ¼

fðaþpk_{xkÞ fðaÞ} pk

¼

u

kð½ak;xkÞ

u

kð½ak;0Þ þpð

u

kþ1ð½ak;xk;0Þ

u

kþ1ð½ak;0;0Þ þ. . .Þ

Thus

bmðfÞ

u

0ðmÞðmodpÞ; m2 f0;. . .p1g;

bmðfÞ ¼baþpk_x

kðfÞ

u

kð½ak;xkÞ

u

kð½ak;0Þ

u

k;½akðxkÞ

u

k;½akð0ÞðmodpÞ; m¼aþp

k_{xk; xk–0}

By Theorem 2.1[38]the functionfpreserves the measure if and only if

1. b0ðfÞ;b1ðfÞ;. . .;bp1ðfÞ establish a complete set of

residues modulop;

2. baþpkðfÞ;. . .;baþpkðp1ÞðfÞare all nonzero residues modulop.

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:Z_p!Z_pbe 1-Lipschitz function in the

coordinate representation(1.6),where all functions

u

₀ and

u

k;½ak, a2 f0;1;. . .;p

k_{1g;kP1are permutations on the}

setf0;. . .;p1g(i.e. f preserves the measure).

The function f is ergodic if and only if all permutations

u

₀

and

Fk¼

u

k; fðpk1Þ

k1 ð0Þ

k

u

k;fðpk2Þ k1 ð0Þ

k

u

k;½0k; kP1

are transitive on thef0;. . .;p1g.

Proof. Let a¼a0þpa1þ. . .þpk1ak12 f0;1;. . .;pk1g.

And let us find the coordinate representation forf_kðpk1ÞðaÞ,

kP1. Then we have:

f_kðpk1Þ h i

k¼ ðF1;½a1ða0Þ;. . .;Fk1;½ak1ðak2Þ;Fk;½akðak1ÞÞ;

where

F0;½a1¼

u

0

u

0

u

0

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} pk1

;

F1;½a1¼

u

_2;fðpk11Þ 1 ð½a1Þ

u

2;f₁ðpk12Þð½a1Þ

u

1;½a1;

. . .

Fk1;½ak1¼

u

_k1;f_kðpk₁11Þð½ak1Þ

u

k1;f_kðpk₁12Þð½ak1Þ

u

k1;½ak1;

Fk;½ak¼

u

_k;fðpk11Þ k1 ð½akÞ

u

k;fðpk12Þ

k1 ð½akÞ

u

k;½ak;

ð2:1Þ

Assume thatfis ergodic. By Theorem 4.23[6, p.99]if the functionfis ergodic, then, for anysP0, the functionfsis transitive on the set of residues modulopsþ1_{and, in fact,} ps _{is the minimal integer such thatf}ðps_Þ

s ðaÞ aðmodpsþ1Þ. The function fk:Z=pkþ1Z!Z=pkþ1Z induces a map on

Z_=pkZ. This map coincides withfk1(fis 1-Lipschitz

func-tion by initial condifunc-tions).

It means that the permutationsF0;½a1;F1;½a1;. . .;Fk1;½ak1

from the coordinate representation of thef_kðpk1Þ are iden-tical. Indeed, in this case each permutation is a degree of the identical permutation

, i.e.

F0;½a1¼

;F1;½a1¼

;. . .;Fk1;½ak1¼

:

As we know,f_kðpkÞð½a_kÞ ½a_kðmodpkþ1_Þandpk _{is minimal} (transitivity offkis by Theorem 4.23[6, p.99]).

Then from the representation and

f_kðpkÞ h i

k¼ f

ðpk1_ÞðpÞ

k

k

¼

;

;. . .;

;Fk;½ak Fk;½ak |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

p 0

B @

1 C A

it follows that

Fk;½ak Fk;½ak |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

p

¼

ð2:2Þ

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

transitive. Since f0¼

u

0, then transitivity of the map

u

0

follows from Theorem 4.23[6, p.99].

We now present the proof that transitivity all functions

u

₀and

u

k;½ak;a2 f0;1;. . .;p

k_{1g;kP1 are sufficient for}

ergodicity. We shall use induction with respect to the parameterk¼0;1;. . .to show transitivity of the functions

fk.

For k¼0 transitivity of f0 follows from the relation

f0¼

u

0and transitivity of

u

0follows from the condition of Theorem.

Letfk1 be transitive on the set of residues modulopk. Sincefkinduces a map onZ=pk1, which coincides withfk1 (f is 1-Lipschitz function), then f_kðpkÞ¼ ð;

;. . .;

;Fk;½akÞ, where Fk;½ak is defined in (2.1) and

is the identity permutation on the setf0;. . .;p1g. By condition of the Theorem, the mapFk;½ak is the transitive permutation on the setf0;. . .;p1g. Therefore,

fkðpkÞ h i

k¼ f

ðpk1_ÞðpÞ k

k

¼

;

;. . .;

;Fk;½ak. . .Fk;½ak |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

p 0

B @

1 C A

¼ ð

;

;. . .;

Þ

and suchpk_{is minimal. It means thatf}

kis transitive on the set of residues modulopk_{. Thus we proved that the} func-tions fk;k¼0;1;. . . are transitive. Then by Proposition 4.35[6, p.105]the functionfis ergodic.

For the proof of the Theorem it is sufficient to show that the conditions of ergodicity criterion offdoes not depend on the choice of the parametera2 f0;. . .;pk_{1g;kP1. It}

is enough to prove the following statement. Let

a–b2 f0;1;. . .;pk_{1g and kP1. If function f}

k1 is transitive modulo pk _{(i.e. on f0;1;. . .;p}k_{1g), then}

permutations onf0;1;. . .;p1g

ga¼

u

k;fðpk1Þ

k1 ðaÞ

k

u

k;fðpk2Þ k1 ðaÞ

k

u

k;½ak;

gb¼

u

k; fðpk1Þ

k1 ðbÞ

k

u

k;fðpk2Þ k1 ðbÞ

k

u

k;½bk

have the same cyclic structure. Indeed, fk1 is transitive

modulopk_{. Then there exist an integerr6p}k1_{1 such}

thatf_kðrÞ₁ðaÞ ¼band, therefore,f_kðtÞ₁ðbÞ ¼f_kðtþ₁rÞðaÞ. Then

ga¼

u

_k;fðpk1Þ

k1 ðaÞ

k

u

_k;fðrþ1Þ

k1ðaÞ ½ _k

u

k;fðrÞ

k1ðaÞ

½ _k

u

k;½ak ð2:3Þ

¼

u

k;fðpk11rÞ

k1 ðbÞ

k

u

_k;fð2Þ

k1ðbÞ ½ k

u

k;½fk1ðbÞk

u

k;fðr1Þ

k1 ðaÞ ½ k

u

k;½ak ð2:4Þ

Consistently making conjugation of the permutation

(2.3)by permutations

u

k;½ak;. . .;

u

k;fðr1Þ

k1ðaÞ ½ k

, we obtain

u

_k;½fðr1Þ k1ðaÞk

u

k;½akga

u

k;½ak 1

u

_k;½fðr1Þ k1 ðaÞk

1

¼

u

_k;½fðr1Þ k1 ðaÞk

u

k;½ak

u

_k;½fðpk11rÞk k1 ðbÞk

. . .

u

k;½fk1ðbÞk

¼

u

k;½fðpk1Þ

k1 ðbÞk

u

k;½fðpkrÞ k1 ðbÞk

u

k;½fðpk1rÞ

k1 ðbÞk

u

k;½fk1ðbÞk

¼gb:

Let G¼

u

_k;fðr1Þ

k1 ðaÞ ½ k

u

_k;½a

k. Then GgaG

1

¼gb,

i.e., the permutationsg_a;g_bare conjugate permutations. It is well known that cyclic structure of conjugate permuta-tions coincide.

To complete the proof it suffices to choosea¼0 (for all

kP1) and setFk;½0k¼Fk. h

Remark 2.3. Theorem 2.2can be considered as generaliza-tion to the casep–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 mathe-matical proof before the one presented in this book).

Comment 2.4. To check a function on ergodicity by using conditions ofTheorem 2.2, one should check transitivity of the permutations Fk, k¼1;2;. . .. Each permutation is a product of permutations

u

_k;½a

k;a2 f0;. . .;p

k_{1g. The}

order of their appearance in the resulting product (i.e. in

Fk) is defined by the sequence of residues modulopk

s

k¼ f0;fk1ð0Þ;. . .;fð

pk_2Þ k1 ð0Þ;f

ðpk_1Þ k1 ð0Þg:

Note that the order of residues modulo pk _{in the} sequence

s

k is significantly important. This is due to the fact that the symmetric groupSp (permutations onZ=pZ) is nonabelian. Therefore, in general by determiningFkwe cannot change the order of the permutations

u

k;½ak.

If we suppose that the permutations

u

k;½ak2Sp;a2

f0;1;. . .;pk_{1gcommute, then the conditions of}

ergodic-ity offare simplified. In theCorollary 2.5we considered the case such that, for each k1 the permutations

u

k;½ak2Sp belong to the cyclic group generated by a permutation gk. In this case, all

u

k;½ak2Sp commute,Fk does not depend on the order of elements in the sequence

s

k, and, moreover,Fk¼gakfor some

a

depending only onk.

Therefore, to verify the transitivity ofFk, there is no need to

build the sequence

s

k. This simplifies essentially the

verification of ergodicity off.

Corollary 2.5. Let f:Z_p!Z_p be 1-Lipschitz function in the

coordinate representation(1.6).

Suppose that the subfunctions have the form:

u

k;½ak ¼g nkðaÞ

k ; a2 f0;. . .;p

k_1g

; kP1_; ð2_:5Þ

where gk is a permutation onZ=pZand nkðaÞis a positive

integer (g0

k- identity permutation).

Then the function f is ergodic if and only if

1.

u

0;gkare transitive permutations;

2. Pp_ak_¼01nkðaÞX0 modpÞ; kP1.

Proof. Let f be an ergodic function. By Theorem 2.2

permutations

u

0 and Fk¼

u

_k ;fðpk1Þ

k1 ð0Þ

k

u

k;½fk1ð0Þk

u

k;½0k;kP1 are transitive on

Z_=pZ. Then

Fk¼

u

k;½fðpk1Þ

k1 ð0Þk

u

k;½fk1ð0Þk

u

k;½0k

¼gnkðfkpk11ð0ÞÞ

k g

nkð0Þ

k ¼g

nkðpk1Þþþnkð0Þ

k ¼g

Rk k ;

whereRk¼nkðpk1Þ þ þnkð0Þ.

AsFk is transitive onZ=pZ, thenord Fk¼p.

Letord gk¼

a

. Thenp¼ord Fk¼ord g

Rk

k ¼

a

gcdða;RkÞ, i.e.p divides

a

. If the permutationgkhasr1cycles of the length 1,r2cycles of the length 2;. . .rpcycles of the lengthp, and r1þ2r2þ. . .þprp¼p, then

a

¼ord gk¼lcmðijri–0Þ.

As p is a prime number and p devides

a

, then the permutationgkconsists of one cycle of lengthp, i.e.

a

¼p

and gk is transitive permutation. From p¼gcdðaa;RkÞ;

a

¼p follows that gcdð

a

;RkÞ ¼gcdðp;RkÞ ¼1, i.e. Rk¼Pp

k₁ a¼0nk ðaÞX0ðmodpÞ.

And vice versa, let us find an order of each permutation

Fk,kP1. By one of the conditions of this corollary

u

₀¼f₀ is a transitive permutation and, in particular,

ff₀ðp1Þð0Þ;. . .;f0ð0Þ;0g ¼ f0;1;. . .;p1g. Then F1¼

u

1; ½f₀ðp1Þð0Þ₁

u

₁;½f0ð0Þ1

u

1;½01 ¼g

n1ðp1Þþþn1ð0Þ

1 ¼g

R1

1, whereR1 ¼n1ðp1Þ þ þn1ð0Þ. By initial conditions

g₁ is transitive on Z₌pZ, then ord g₁¼p. As

R1X0ðmod pÞ, then gcdðp;R1Þ ¼1 and ord F1¼ ord

gR1

1 ¼

p

gcdðp;R1Þ¼ p. Thus F1 is transitive permutation on Z_=pZ. Moreover, the functionf1is transitive onZ=p2Z.

Computation of the valuesord Fk;kP2 is performed in a similar way, taking into account that at each step we prove transitivity of the functionsfk1. Finally, we obtain that the permutationsFk;kP1 are transitive onZ=pZ. As

u

₀is transitive onZ=pZ, then the functionfis ergodic by

Theorem 2.2. h

In theCorollary 2.6, in particular, we answer the follow-ing 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 allkP1 each ofpk1 per-mutations

u

k;½ak;a2 f1;2;. . .;p

k_1g(aX0ðmodpk_{Þ) can} be chosen arbitrarily and only permutation

u

k;½0kis chosen in a special way.

Corollary 2.6. Let f:Z_p!Z_pbe 1-Lipschitz function in the

coordinate representation(1.6),where the subfuctions of the

coordinate functions,

u

0 and

u

k;½ak;a2 f0;1;. . .;p k_1g;

kP1are permutations on the setZ₌pZand

u

0is transitive.

Then it is always possible to construct permutations g₁;g₂;. . .;gk;. . . such that by setting

u

~k;½0k¼gk;k¼1;

2;. . ., the corresponding function~f which is defined with the

aid of subfunctions

u

k;½ak; a–0, and

u

~k;½0k is ergodic.

Proof. Let us find permutationsg1;g2;. . .;gk;. . .by

induc-tion. By conditions of theorem, permutation

u

₀¼f0 is

transitive on Z₌pZ, and, in particular, f₀ðp1Þð0Þ;. . .;f0

n

ð0Þ;0g ¼ f0;1;. . .;p1g. By using notations from Theo-rem 2.2, we setG1¼

u

₁;½fðp1Þ

0 ð0Þ1

u

1;½f ðp2Þ

0 ð0Þ1

u

1;½f0ð0Þ1.

We also choose some transitive permutationH1onZ=pZ. Letg₁¼G1

1 H1. Then set

u

~1;0¼g1, and we obtain that permutation Fe1¼

u

1;½f₀ðp1Þð0Þ1

u

_1;½fðp2Þ 0 ð0Þ1

u

_1;½f0ð0Þ1

~

u

1;½0k¼H1 is transitive onZ=pZ. Moreover, the function

~

Now let permutations g₁;g₂;. . .;gk1;. . .have already been constructed, and the function~fk1 is transitive on

Z=pk_{Z. Let us construct a suitable permutationg} k. Let Gk¼

u

k;½~fð_kpk₁1Þð0Þ

k

u

_k;½~_f k1ð0Þk

andHkbe some transitive permutation onZ=pZ. And set

gk¼G

1

k Hk. Then, for

u

~k;½0k¼gk, we obtain that the permutation

e Fk¼

u

k;½~fð_kpk₁1Þð0Þ

k

u

k;½~_f k1ð0Þk

~

u

k;½0k¼Hk

is transitive onZ=pZ. And the function~fk is transitive on

Z_=pkþ1_Z.

Thus we have shown existence of permutations

g1;g2;. . .;gk;. . . such that for

u

~k;½0k¼gk;kP1 permuta-tionseFkare transitive onZ=pZ. As

u

0is transitive onZ=pZ, then the function~f is ergodic byTheorem 2.2. h

Comment 2.7. Using the results ofCorollary 2.6and the criterion of Theorem 2.2 we can inductively construct any 1-Lipschitz ergodic function in the representation

(1.6)as follows. Permutation

u

0 we choose transitive on Z₌pZ. Let the permutations

u

_k1;½ak1;a2 f0;1;. . .; pk1_{1g are constructed, i.e. all of the functions}

u

s;s¼1;2;. . .;k1 are choosen. Let us construct

u

k. To

do this, the permutations

u

_k;½a

k;a2 f1;. . .;p

k_{1g (i.e.}

aX0ðmodpk_{ÞÞwe choose arbitrarily and find}

Gk¼

u

k;fðpk1Þ

k1 ð0Þ

k

u

k;½fk1ð0Þk

(the function fk1 defined via

u

k1;½ak1;a2 f0;1;. . .; pk1_{1g). Then we choose}

u

k;½0k in such a way that Gk

u

k;½0k be a transitive permutation onZ=pZ. Aspis a prime number, then one can write this condition analyti-cally. Namely,

u

_k;½0

k is a solution in permutations ðGkXÞp¼

, where

is the identical permutation.

As a result, the function

u

kis constructed. The number

of suitable choice equals to ðp!Þk1ðp1Þ! (or 1

p of all

suitable functions

u

kfor construction a

measure-preserv-ing functions).

Proceeding in this way to select the functions

u

k;k1,

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 forp 6¼ 2

In this section, we prove a criterion on ergodicity of 1-Lipschitz functionsfof class‘0and‘S(Theorems 3.1 and

3.5). In general, classes‘0and‘S are characterized by the fact that the functions

u

k;½ak;a2 f0;1;. . .;p

k_1g;kP0

in the coordinate representation of(1.6)fdefined by linear polynomials over a fieldZ₌pZ(for the class‘0these

poly-nomials are unitary, which allow us to represent functions from‘0explicitly, i.e. without using the coordinate

repre-sentation(1.6), seeLemma 3.1).

Examples of the application of the criteria on ergodicity for the classes‘0and‘Sare:

a new proof of the ergodicity for the functions of the

form cþxþpðhðxþ1Þ hðxÞÞ, where h:Z_p!Z_p is

1-Lipschitz function,cX0ðmodpÞ(Corollary 3.3);

ergodicity criterion (Theorem 3.4) for the functions of the form

fðx0þpx1þ Þ ¼cþa0x0þ þakpkxkþ ;

c;ak2Zp;kP0 forp–2 (the casep¼2 is considered in, for example,[6, Theorem 9.20 (part 2), p.286]);

description of uniformly differentiable modulo p 1-Lipschitz 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 functionsfof class‘0(Theorem 3.1). The

func-tions 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 polyno-mials commute in the symmetric group of permutations. This property allow us to formulate a criterion on ergodic-ity in terms of integral sums for the Volkenborn integral, see, e.g., Schikhof[46]. Functions from‘0admits an explicit

representation (Lemma 3.1). As an example of the use of

Theorem 3.1we give a new proof of(3.3)known criterion on ergodicity of functions of the form cþxþpðhðxþ1Þ 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 ofTheorem 3.1is a generalization of criterion on ergodicity for 2-adic functions (p¼2) of the form

fðx0þpx1þ þpkxkþ Þ ¼cþa0x0þ þakpkxkþ ;

whereak2Zp;kP1;c2Zpup to the casep–2.

The explicit form of the functions from the class‘0is

presented in the following statement.

Lemma 3.1. Let f:Zp!Zpbe 1-Lipschitz function. Function

f belongs to‘0if and only if f can be represented as

fðxÞ ¼fðx0þpx1þ þpkxkþ Þ

¼

u

0ðx0Þ þ ðxx0Þ þpgðxÞ;

where g:Z_p!Z_p is 1-Lipschitz function and

u

0:Z=pZ!

Z=pZ.

Proof. Let fðxÞ ¼P1

m¼0Bm

v

ðm;xÞ ¼P1m¼0pblogpmcbm

v

ðm;xÞ be the van der Put representation of the 1-Lipschitz func-tion f2‘0. Let us find values of the coefficients bm for

mPp. Denote m¼m0þpm1þ þpkm k¼mþpkmk;

m2 f0;. . .;pk_1g;m

k2 f1;. . .;p1g (i.e. mk–0). Then

(under notations from(1.4) and (1.6))

Bm¼Bmþpk_m

k¼fðmþp

k_{mkÞ fðmÞ} ¼pk_ð

u

kð½mk;mkÞ

u

kð½mk;0Þ þp kþ1_{ð. . .Þ}

;

i.e.

bm¼bmþpk_x

k

u

kð½mk;mkÞ

u

kð½mk;0Þ mkþ

a

kð½mkÞ

a

kð½mkÞ mk ðmodpÞ:

In other words, bm¼bmþpkmk¼mkþp

bmþpkmk¼mk þpbmfor suitablep-adic integerbm.

We set fðiÞ ¼Bi¼bi¼biðmodpÞ þpbi;bi2Zp;i¼

f0;. . .;p1g. Then the function

u

0:Z=pZ!Z=pZ from

representation(1.6)takes the form

u

0ðxÞ ¼b0ðmodpÞ

v

ð0;xÞ þ þbp1ðmodpÞ

v

ðp1;xÞ

and

X1

m¼p

bmðmodpÞ pblogpmc

v

_{ðm;xÞ ¼}X

1

m¼p

qðmÞ

v

ðm;xÞ ¼xx0;

whereqðmÞ ¼qðmþpk_m

kÞ ¼pkmk;mk–0.

And let us consider the 1-Lipschitz function

gðxÞ ¼P1

m¼0bmpblogpmc

v

ðm;xÞ. Thus, fðxÞ ¼X

1

m¼0

bmpblogpmc

v

_ðm;xÞ

¼X

1

m¼0

bmðmodpÞ pblogpmc

v

_ðm;xÞ

þpX

1

m¼0

bmpblogpmc

v

_ðm;xÞ

¼X p1

m¼0

bmðmodpÞpblogpmc

v

_{ðm;xÞ þ}X

1

m¼p

bmðmodpÞ

pblogpmc

v

_{ðm;xÞ þpgðxÞ} ¼

u

0ðx0Þ þ ðxx0Þ þpgðxÞ:

Vice versa, let

fðxÞ ¼fðx0þpx1þ þpkxkþ Þ

¼

u

0ðx0Þ þ ðxx0Þ þpgðxÞ

andfðxÞ ¼P1_m¼0bmpblogpmc

v

ðm;xÞ.

Set m¼m0þpm1þ þpkmk¼mþpkmk;mk–0 we

obtain that formPp

bm¼bmþpk_m k

1

pkðfðmþp

k_{mkÞ fðmÞÞ}

1

pkð

u

0ðm0Þ þ ðmþpkmkm0Þ þpgðm þpkmkÞ

ð

u

0ðm0Þ þ ðmm0Þ þpgðmÞÞÞ

1 pkðp

k_{mkþp ðgðmþp}k_{mkÞ gðmÞÞÞ}

mkþ 1

pk1ðgðmþp

k_{mkÞ gðmÞÞ ðmodpÞ:}

As the function g is 1-Lipschitz, then gðmþpk_m kÞ

gðmÞð modpk_Þand

1

pk1ðgðmþp

k_{mkÞ gðmÞÞ 0 modp:}

ThenbmmkðmodpÞ.

On the other hand (under notations from(1.4)),

bm

u

kð½mk;mkÞ

u

kð½mk;0Þ modp:

or

u

kð½mk;mkÞ ¼mkþ

u

kð½mk;0Þ ¼mkþ

a

kð½mkÞ:

Thusf2‘0. h

Ergodicity criterion functions of the class‘0is presented

in the following theorem.

Theorem 3.1.Let p–2 and f:Z_p!Z_p be 1-Lipschitz function and f 2‘0, i.e.

fðxÞ ¼fðx0þpx1þ þpkxkþ Þ

¼

u

0ðx0Þ þ ðxx0Þ þpgðxÞ;

where g:Z_p!Z_{p is 1-Lipschitz function and}

u

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

1.

u

0is a transitive (monocycle) permutation onZ=pZ;

2. 2p2 þ1

pk Ppk₁

a¼0fðaÞX0ðmodpÞ, or in the equivalent

form 1

pk1

Ppk₁ a¼0gðaÞX2

p2

ðmodpÞ; kP1.

Proof. Note that the functionfpreserves measure as soon as

u

0is bijective onZ=pZ. Indeed,f2‘0implies that all

functions

u

k;½ak;a2 f0;. . .;p

k_{1g;kP1 in the coordinate}

representation off(1.6)are bijective as linear polynomials (see(1.7)) over the field of residues modulop. 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 a2 f0;1;. . .;

pk_{1gthe function}

u

k;½akis a permutation on

Z_=pZ_;kP1. As inTheorem 2.2, let us consider the permutations

Fk¼

u

k;fðpk1Þ

k1 ð0Þ

k

u

k;fðpk2Þ k1 ð0Þ

k

u

k;½0k;

wherekP1.

By the assumption the permutations

u

k;½akare given by linear polynomials of the form

u

k;½akðxkÞ ¼xkþ

a

kð½akÞ. Therefore

Fk¼xkþ

a

k fðp k_Þ k1ð0Þ

h i

k

þ

a

k fðp k_2Þ k1 ð0Þ

h i

k

þ þ

a

kð½0kÞ

¼xkþX pk₁

i¼0

a

k fkðiÞ1ð0Þ

h i

k

:

Let us show by induction with respect tokP0 that the functionsfkare transitive onZ=pkþ1Z. Fork¼0, the func-tionf0is transitive becausef0¼

u

0, where

u

0is transitive

onZ_=pZby the assumption. Suppose now thatfk1is

tran-sitive onZ₌pk_{Z. Then}

Fk¼xkþX pk₁

i¼0

a

k fkðiÞ1ð0Þ

h i

k

¼xkþX pk₁

i¼0

a

kð½ikÞ:

On the other hand, seta¼a0þpa1þ þpk1ak1(under

notations in(1.2)), then we obtain

X pk₁

a¼0

fðaÞ X pk₁

a¼0

pk

u

kða0;. . .;ak1;0Þ þ

Xk1

i¼0

pi

u

iða0;a1;. . .;aiÞ

!

X pk₁

a¼0

pk

a

kð½akÞ þ

Xk1

i¼0

pi

u

iðx0;x1;. . .;xiÞ

!

pkX pk₁

a¼0

a

kð½akÞ þ X pk₁

a¼0

As fk1 is bijective on Z=pk1Z, then Pp k₁ a¼0fk1ðaÞ ¼pkðp2k1Þ, i.e.

X pk₁

a¼0

fðaÞ pkX pk₁

a¼0

a

kð½akÞ þ

pk_ðpk_1Þ

2 ðmodp

kþ1_Þ;

or

1

pk X pk₁

a¼0

fðaÞ X

pk₁

a¼0

a

kð½akÞ þ

pk₁

2

X pk₁

a¼0

a

kð½akÞ 2

1

ðmodpÞ;

where 212p2ðmodpÞ.

Then by the second condition of this Theorem

X pk₁

a¼0

a

kð½akÞ

1

pk X pk₁

a¼0

fðaÞ þ2p2

X0 ðmodpÞ:

And, therefore, the permutationFk is transitive onZ=pZ. This means that the functionfk is transitive onZ=pkZ. By Theorem 4.23 [6, p.99] from transitivity of fk on

Z=pk_{Z;kP0 follows that the functionfis ergodic.} Vice versa, let the functionfbe ergodic function. Then

f0¼

u

0 and functionsFk;kP1 are transitive onZ=pZby Theorem 2.2. As

Fk¼xkþX pk₁

i¼0

a

k fkðiÞ1ð0Þ

h i

k

¼xkþX pk₁

a¼0

a

kð½akÞ;

then transitivity of Fk means that Ppk₁

a¼0

a

kð½akÞX0

ðmodpÞ. Then

1

pk X pk₁

a¼0

fðaÞ X

pk₁

a¼0

a

kð½akÞ þ

pk₁

2

X pk₁

a¼0

a

kð½akÞ 2

1

ðmodpÞ;

i.e. 2p2þ1

pk Ppk₁

a¼0fðaÞX0ðmodpÞ;kP1.

To complete the proof, it remains to note that

2p2

þ1 pk

X pk₁

a¼0

fðaÞ

2p2þ1 pk p

k1X

p1

½a1¼0

u

0ð½a1Þ þ

X pk₁

a¼0

ða ½a1Þ þp

X pk₁

a¼0

gðaÞ 0

@

1 A

2p2

þ p1

2 þ

pk1₁ 2

þ 1 pk1

X pk₁

x¼0

gðxÞ

2p2

þ 1 pk1

X pk₁

a¼0

gðaÞ ðmodpÞ:

h

Comment 3.2.In theComment 2.4was stated that if for each k1 all permutations

u

_k;½a

k;a2 f0;. . .;p k1_1g

belong to a cyclic subgroup of the symmetric groupSp, then the conditions of ergodicity off are simplified. In Theo-rem 3.1we considered the case such that the cyclic group is generated by permutation onZ=pZ, which is given by a linear polynomialxþc;c–0 over the fieldZ₌pZ. In this case, the ergodicity of f is determined by the value of

1

pk Ppk₁

a¼0fðaÞ ðmodpÞ;k1.

Note that such sums appear in the definition of Volkenborn’s integral, see, for example, Definition 55.1

[46]. However, real coupling between theory of Volken-born integration and ergodicity of p-adic dynamical sys-tems has not yet been clarified.

As an example, the criterion ofTheorem 3.1we give a new proof of the known (for example, see Lemma 4.41, p.112 [6]) ergodicity criterion for functions of the form

fðxÞ ¼cþxþpðhðxþ1Þ hðxÞÞ, where h:Z_p!Z_p is 1-Lipschitz function. Note that the functions fromCorollary 3.3 belong to the class of functions considered in Theo-rem 3.1. In other words,Theorem 3.1generalize the results of Lemma 4.41,[6].

Corollary 3.3 (see Lemma 4.41, p.112[6]). Let f :Zp!Zp; fðxÞ ¼cþxþpðhðxþ1Þ hðxÞÞ, where h:Z_p!Z_p is

1-Lipschitz function and p–2. If cX0 modp, then f is an

ergodic function.

Proof. As

u

0ðxÞ f0ðxÞ fðxÞmodpÞ x0þcmodpÞ,

wherex¼x0þpx1þ , then

u

0ðxÞis transitive

permuta-tion oncecX0ðmodpÞ. Ashis 1-Lipschitz function, then

hðpk_{Þ hð0Þ 0ðmodp}k_{Þ. Then forkP2} 1

pk1

X pk₁

i¼0

ðhðiþ1Þ hðiÞÞ hðp k_{Þ hð0Þ}

pk1

0X2p2

ðmodpÞ:

Thus the functionfis ergodic byTheorem 3.1. h

Another example of the use of the Theorem 3.1 is

Theorem 3.4. This generalizes the results of Theorem 9.20

[6, part 2, p.286]to the casep–2.

Theorem 3.4. Let f :Z_p!Z_pðp–2Þ be a 1-Lipschitz function of the form

fðx0þpx1þ þpkxkþ Þ ¼cþa0x0þ þakpkxkþ ;

where ak2Zp;kP1;c2Z_p. The function f is ergodic if and

only if cX0ðmodpÞand ak1ðmodpÞ;kP1.

Proof. Let a function f:Z_p!Z_p be represented in the coordinate form. Then for suitable p-valued function

lkðx0;x1;. . .;xk1Þ

u

kðx0;x1;. . .;xkÞ 1

pkðfðx0þ þp

k_{xkÞ fðx}

0þ

þpk1_xk

1ÞðmodpkÞÞðmodpÞ

akxkþlkðx0;x1;. . .;xk1ÞðmodpÞ:

Letfbe an ergodic function. ByTheorem 2.2 permuta-tions

u

₀and

Fk¼

u

k;fðpk1Þ

k1 ð0Þ

k

u

k;½fk1ð0Þk

u

k;½0k; kP1

are transitive on Z=pZ. As for suitable bk2Z=pZ;

k¼1;2;. . .

Fk¼ akxkþlk fðpk_1Þ k1 ð0Þ

ðakxkþlkð0ÞÞ

¼apkkxkþbk;

then apk

k ak1ðmodpÞ. From transitivity of

u

0 cþa0x0ðmodpÞ follows that cX0ðmodpÞ and a0

1ðmodpÞ.

Vice versa, by assumptionak1ðmodpÞ;kP1. Then

u

kðx0;x1;. . .;xkÞ ¼xkþlkðx0;x1;. . .;xk1Þ

for suitable p-valued function lkðx0;x1;. . .;xk1Þ. This, in

particular, means that the function fsatisfies the condi-tions ofTheorem 3.1.

Let us check the function f on ergodicity. As

cX0ðmodpÞ and a01ðmodpÞ, then

u

0cþx0

ðmodpÞis transitive onZ=pZ. Note that,

1

pk X pk1

i¼0 fðiÞ 1

pk X

x0;...;x_k1

ðcþa0x0þ þak1pk1xk1Þ

1

pk p

k_cþa

0pk1

pðp1Þ

2 þ. . .þak1p2k2

pðp1Þ

2

cþa0

p1

2 ca02 p2

c2p2

ðmodpÞ:

Therefore, 2p2 þ1

pk Ppk₁

i¼0 fðiÞ cX0ðmodpÞ. Then by Theorem 3.1, the functionfis ergodic. h

3.2. Ergodic functions of the classes‘S

In this subsection we prove the ergodicity criterion for functions from the class‘S, where the parameterSis a fixed positive integer (Theorem 3.5). These functions are charac-terized by the fact that their coordinate functions

u

k; kP0 are given by linear by the variablexk polynomi-als inðZ₌pZÞ½x0;. . .;xk, in which the coefficients atxkare polynomials that depend on no more thanSvariables (see the representation (1.8)). In contrast to the general criterion of ergodicity (see Theorem 2.2), where it is required to calculate the values of all iterations

fðiÞ_{ð0Þ ðmodp}k_{Þ;i2 f0;1;. . .;p}k_{1g;kP1, to check} ergodicity of functions from the class‘Swe need to calcu-late the iterations for thelimited number of values,namely,

fðiÞ_{ð0Þ ðmodp}S_{Þ;i2 f0;1;. . .;p}S_{1g, for everykPS.} By the criterion of Theorem 3.5 the Corollary 3.7

describes the ergodic 1-Lipschitz uniformly differentiable function modulo p. The problem of description of these functions has been put in the open question 4.60 [6, p.132]. As a result, taking into account the criterion of ergodicity by V. Anashin for 1-Lipschitz uniformly differen-tiable function modulo p2 _{(see[3,4]or Theorem 4.55 [6,} p.126]), a complete description of ergodic 1-Lipschitz uniformly differentiable function modulo pa _{for any}

a

P1 was obtained.

Ergodicity criterion for functions from‘Sis presented in the following theorem.

Theorem 3.5. Let p-adic (p–2) measure-preserving

1-Lipschitz function f:Z_p!Z_p be presented in the

coordi-nate form (1.4), where

u

kðx0;x1;. . .;xkÞ are such p-valued functions that

u

kðx0;. . .;xkÞ ¼xkAkðx0;. . .;xS1Þ þ

a

kðx0;. . .;xk1Þ; kPS for some fixed integer S.

The function f is ergodic if and only if holds simultaneously

1. f is transitive modulo pS_{(i.e. f}

S1is transitive onZ=pSZ);

2. QpS₁

i¼0 Akð½iSÞ 1ðmodpÞ; kPS;

3. for kPSþ1

2p2X

pS₁

i¼0

s

ðiÞ

k

X pS₁

i¼0

s

ðiÞ

k pk

fðiþ1Þ_ð0ÞðmodpS_Þ pSk

X

pkS₁

b¼0

f fðiÞ_ð0ÞðmodpS_{Þ þp}S

b

#

X0ðmodpÞ;

and for k¼S

0ðmodpÞXX

pS₁

i¼0

s

ðiÞ

S pS½fðf

ðiÞ_ð0ÞðmodpS_ÞÞ

fðiþ1Þ_ð0ÞðmodpS_Þ; where

s

ðiÞ

k ¼ QpS₁

j¼iþ1Akð½fðjÞð0ÞSÞ;i¼0;1;. . .;pk2and

s

ðiÞ k ¼

1for i¼pk_1.

Proof. Sincefpreserves the measure, then byTheorem 2.1

functions

u

k;½xk and

u

0are permutations onZ=pZ. Note that under the Notation(1.2)fork¼S;Sþ1;. . .

Fk¼

u

k;fðpk1Þ

k1 ð0Þ

k

u

k;fðpk2Þ k1 ð0Þ

k

u

k;½0k

¼ Ak fkðpk11Þð0Þ

h i

k

xkþ

a

k fðp k_1Þ k1 ð0Þ

h i

k

ðAkð½0kÞ xkþ

a

kð½0kÞÞ ¼xk Y pk₁

i¼0

Akð½fðiÞ

k1ð0ÞkÞ !

þ

a

k ½fð pk1_1Þ

k1 ð0Þk

þ X pk1₂

i¼0

a

kð½fð iÞ

k1ð0ÞkÞ

Y pk₁

j¼iþ1

Akð½fðjÞ

k1ð0ÞkÞ !!

¼xkQkþLk:

By induction on thek¼S;Sþ1;. . .we show that

Fk;0¼

u

k; fðpk1Þ k1 ð0Þ

k

u

k;½0k

are transitive permutations onZ₌pZ.

From transitivity of the function fmodulopS _follows

that f_SðpS₁Þð0Þ 0ðmodpS_{Þ and f}ðpmþS_Þ

Sþm1ð0Þ 0ðmodpSÞ;

m¼1;2;. . .. As by assumption p-valued functions Ak

depend only on variablesx0;x1;. . .;xS1, then any sequence

Akð½0kÞ;Akð½fk1ð0ÞkÞ;. . .;Ak fðp k_1Þ k1 ð0Þ

h i

k

n o

;k¼S;Sþ1;. . .

has period of the lengthpS_{. Then from the second condition}

of theorem follows that

Qk¼ Y pk₁

i¼0

Ak f_kðiÞ1ð0Þ

h i

k

¼Y pk₁

i¼0

AkðiÞ ¼ Y pS₁

i¼0

Akð½ikÞ !pkS

¼1

Lk¼

a

k f

ðpk_1Þ k1 ð0Þ

h i

k

þX pk₂

i¼0

a

k f

ðiÞ

k1ð0Þ

h i

k

Y

pk₁

j¼iþ1

Ak f_kðjÞ1ð0Þ

h i

k

!

¼ X pkS₁

a¼0

a

k fðp S_1þPS_aÞ

k1 ð0Þ

h i

k

þX pS₂

i¼0

Y pS₁

j¼iþ1

Ak fðjÞ

k1ð0Þ

h i

k

! pXkS₁

a¼0

a

k fð iþPS_a

Þ

k1 ð0Þ

h i

k

!

:

Let

s

ðiÞ k ¼

QpS₁

j¼iþ1Ak fkðjÞ1ð0Þ

h i

k

;i¼0;1;. . .;pk1_2;kPS

then

Lk¼ X pkS₁

a¼0

a

k f

ðpS_1þPS_aÞ

k1 ð0Þ

h i

k

þX pS₂

i¼0

s

ðiÞ

k X pkS₁

a¼0

a

k fðiþP S_aÞ k1 ð0Þ

h i

k

!

:

Thus,Fk¼xkþLk;kPS.

Suppose we have already shown thatFk1is transitive onZ₌pZ, andfk1is transitive onZ=pkZ. Let us check that

Fk¼xkþLkis transitive onZ=pZ. It is enough to show that Lk–0. Asfis a 1-Lipschitz function and by the induction fk1is transitive onZ=pkZ, then fori2 f0;1;. . .;pS1g

f_kðiþ1PSaÞð0Þðmodp

k_Þ:

a

_{¼0;1;. . .;p}kS₁

n o

¼ fkðiÞ1ð0Þðmodp

S_{Þ þp}S_{b:b¼0;1;. . .;p}kS₁

n o

ð3:1Þ

andf_kðiÞ₁ð0Þ f_SðiÞ1ð0ÞmodpSÞ.

Then

X pkS₁

a¼0

a

k fðiþP S_aÞ k1 ð0Þ

h i

k

¼ X pkS₁

b¼0

a

k fSðiÞ1ð0Þðmodp

S_{Þ þp}S

b

h i

k

¼ X pkS₁

b¼0

a

kð½fðiÞð0ÞðmodpSÞ þpSbkÞ

and

Lk¼ X pkS₁

a¼0

a

kð½fðp S_1Þ

ð0ÞðmodpS_{Þ þp}S

a

kÞ

þX pS₂

i¼0

s

ðiÞ

k X pkS₁

a¼0

a

kð½fðiÞð0ÞðmodpSÞ þpS

a

kÞ !

:

As by assumptionAk;kPS depend only on variables

x0;x1;. . .;xS1, then

Ak fðjÞ

k1ð0Þ

h i

k

¼Ak fðjÞ

k1ð0Þðmodp

S_Þ

h i

k

¼Ak fðjÞ

S1ð0Þ

h i

S

¼Akð½fðjÞ_ð0Þ

SÞ:

And, therefore, fori2 f0;1;. . .;pk1_2g

s

ðiÞ

k ¼

Y pS₁

j¼iþ1

Ak f_SðjÞ1ð0Þ

h i

k

¼ Y pS₁

j¼iþ1

Akð½fðjÞ_ð0Þ

SÞ:

Let D2 f0;1;. . .;pS_{1g and b2 f0;1;. . .;p}kS1_1g.

Fork>S, using coordinate representation of the function

fand relation(3.1)we obtain (under Notations(1.2))

X pkS₁

b¼0

fð

D

þpS_{bÞ ¼} X pkS₁

b¼0

Xk1

i¼0

pi

u

iþpk

u

kð½

D

S;½bkS;0Þ

!

¼ X pkS₁

b¼0

fð

D

þpS

bÞðmodpk_Þ

þpk X pkS₁

b¼0

a

kð½

D

þpSbkÞ

pkS_fð

D

ÞðmodpS_{Þ þp}SpkSðpkS1Þ

2

þpk X pkS₁

b¼0

a

kð½

D

þpSbkÞðmodp kþ1_Þ:

Then, taking into account thatp–2 and that the element 2 is invertible inZ₌pZ(21

2p2

ðmodpÞ), we obtain

X pkS₁

b¼0

a

kð½

D

þpSbkÞ 2 p2

1 pk p

kS_ðfð

D

ÞðmodpS_ÞÞ

X

pkS₁

b¼0

fð

D

þpS_bÞ !

ðmodpÞ: ð3:2Þ

In the casek¼S,

fð

D

Þ ¼X S1

i¼0

pi

u

iþpS

u

Sð½

D

S;0Þ

fð

D

ÞðmodpS_{Þ þp}S

a

Sð½

D

ÞSÞðmodpSþ

1

Þ;

or, respectively,

a

Sð½

D