do do trong kh p_adic don gian

8/14/2019 Do Do Trong Kh P_adic Don Gian

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MEASURABLE DYNAMICS OF SIMPLE p-ADICPOLYNOMIALS

JOHN BRYK AND CESAR E. SILVA

1. INTRODUCTION.

The p-adic numbers have many fascinating properties that are differentfrom those of the real numbers. These properties are a consequence of thefact that the distance in the p-adics is measured using a non-Archimedianabsolute value or norm. In this article we study the dynamics of algebraicallydened transformations on the p-adics and see that there is a strong connec-tion between the topological property of minimality, which is easy to checkfor such transformations, and the measure-theoretic property of ergodicity.

We start with a section that introduces the p-adic numbers, includingtheir topology and the relevant measures on them, and then dene the basicnotions from dynamics that we require. In section 4 we show that minimalisometries on subsets of the p-adics are dened on nite unions of ballsand are never totally ergodic. In Theorem 4.4 we give a new short proof that for an invertible isometry of a compact open subset of the p-adicsminimality implies unique ergodicity and that ergodicity implies minimality.(As we discuss, this theorem is known in a more general context.) Thus

for isometries on compact open subsets of the p-adics, the properties of minimality, ergodicity, and unique ergodicity are equivalent.In later sections we study classes of transformations dened by multipli-

cation, translation, and monomial mappings. We demonstrate that they areminimal (hence uniquely ergodic) on balls or, in some examples, spheres inQ p. Many of these results are known, but we present a unied and differenttreatment of them.

Dynamics on the p-adics has been the focus of several researchers recently.While most of this work deals with topological or complex dynamical prop-erties of the p-adics, for which the reader may refer to [1], [4], [13], [17], andthe references cited in these articles, measurable dynamics on the p-adicshas also received some exposure, particularly in [14], [2], [3] [12], [7], and [8].

2. THE p-ADIC NUMBERS.

Let p be a prime number. The construction of the eld Q p of p-adicnumbers is analogous to that of the real number eld R . However, whereasR is constructed by completing the eld Q of rational numbers with respectto the usual absolute value, we construct the p-adic numbers by using the p-adic norm . We start with an informal description that illustrates the

To appear in Amer. Math. Monthly March 2005 .1


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main ideas. If n is a positive integer, then we can write n in the formn = N i=0 ci p

i , where the ci are unique integers in

{0, . . . p

1

}and N

0.

We call this the p-adic representation of n . Suppose now that we wishto extend this representation to any rational number. We start with anexample, considering the case of p = 5 and the rational number 3/ 20.Then we can write

320

=15

3

4=

15

31 5

=35

11 5

=35

(50 + 5 1 + 5 2 + )

= 3 5 1 + 3 50 + 3 51 + 3 52 + =

i= 13 5i .

At the moment we are not going to worry about issues of convergence but just consider the sum in a formal way. (We used this example becausethe formal expansion for 1 / (1 5) is well known, but similar expansionscan be obtained for any rational number.) There are a couple of thingswe notice from this example. First, to obtain p-adic representations of rational numbers it seems that we need to allow innite series of powersof p, possibly starting at a negative power such as i = 1. Second, wehave to make sense of the innite sum. In other words, we need to considersome metric under which we can show convergence of the partial sums of the innite series, or more precisely, a metric under which the partial sumsform Cauchy sequences.

To continue our informal description we consider a set X consisting of allformal sums x of the type

x =

i= k

ci pi ,

where k is an integer, ci = ci (x) belongs to {0, 1, . . . p 1}for i = k, k +1 , . . . ,and ck = 0. Add to this set the point 0 represented as i=0 cix i where ci = 0for i 0. Using the formal manipulations illustrated by our example (witha general prime p replacing 5) we can identity each rational number witha unique element of X . A standard diagonalization argument shows thatthe set X is uncountable, so it necessarily contains elements that do notcome from members of Q . To make sense of the innite expansions weintroduce a notion of convergence in X motivated by the fact that we desiresequences of partial sums of the form we have discussed to form Cauchysequences. We dene a distance so that two innite formal sums are closeif a large number of their initial terms are the same. More precisely,dene the distance between the formal sums i= k ci p

i and i= di pi to be

p min {k, } if k = , and otherwise dene it to be p m , where m is the rstinteger such that m k = and cm = dm (of course, the distance is 0if cm = dm for all m). Then the distance between the partial sums S nand S n + m (m > 0) is p (n +1) (identifying the partial sum S n = ni= 1 ci p

i

with the series i= 1 ci pi in which ci = 0 when i > n and assuming that


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cn +1 = 0). Therefore, under this distance the partial sums of our examplewould form a Cauchy sequence.

We now change gears and formally dene the p-adic metric on Q . Theobvious place to start is to dene the p-adic norm |x| p of a rational numberx. To do this when x is nonzero, we write x in the form x = p (r/s ), where p divides neither r nor s, and dene

|x| p = p .(And, of course, we set |0| p = 0.) In particular, if we express a naturalnumber n using its p-adic representation, n = N i= k ci p

i where ck = 0, then

|n| p = p k , which agrees with the distance we used for our example. Onecan easily verify that |x| p = |x| p and |xy| p = |x| p|y| p. The rst interestingproperty we notice about this norm is that it satises the strong triangleinequality :

(1) |x + y| p max{|x| p, |y| p},with equality holding if |x| p = |y| p. Normed elds that satisfy inequality (1)are called non-Archimedean . (It is not hard to see that the strong triangleinequality implies that for any integer n different from 1 and any nonzero xin Q it is the case that |nx | p |x| p, clearly contradicting the Archimedeanproperty of the reals.) An interesting consequence of the non-Archimedeanproperty is that a series of rational numbers i=0 a i converges in Q p if andonly if limi |a i | p = 0 (for an explanation of this property and otherconsequences the reader is referred to [6]). The p-adic metric is the metricdened on Q by d p(x, y ) = |x y| p. This metric also satises the strongtriangle inequality. Given p, we can complete Q in the standard way toarrive at a normed eld ( Q p, | | p) such that Q is dense in Q p, | | p extendsthe p-adic norm from Q to Q p, and every Cauchy sequence in Q p (withrespect to | | p) is convergent.One can show that each nonzero x in Q p is uniquely representable in themanner

x =

n = k

cn pn ,

where k is an integer, cn = cn (x) belongs to {0, 1, . . . , p 1}for n = k, k +1, . . . , ck = 0, and convergence is with respect to the p-adic norm. Infact it is easy to see that |x| p = p k . (Of course x = 0 admits a trivialrepresentation of this type, say with k = 0 and cn = 0 for n = 0 , 1, . . . . )Addition in Q p is dened termwise modulo p with a carry to the right. Forexample, 1 + ( i=0 ( p1) pi) = i=0 p pi = 0, so the additive inverse of 1 is i=0 ( p 1) pi . Multiplication is dened in a similar way.One special subset of Q p that we shall consider quite often is the ring of p-adic integers Z p = {x Q p : |x| p 1}. It is not difficult to see that Z pis the closure of Z in Q p and consists of precisely those x in Q p admitting


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representations of the type

x =

n =0cn pn

with cn in {0, . . . , p 1}. The ring Z p is not a eld ( p does not have amultiplicative inverse in Z p), but it is an integral domain with a uniqueprime ideal, namely, pZ p. For each integer k we can talk about the factorring Z p/p k Z p, a ring that is isomorphic to Z /p kZ . If n satises gcd( n, p ) = 1,then [n] is a unit in Z /p kZ (here and later [ x] signies the equivalenceclass in the relevant quotient structure of an element x). The set of allunits in Z p/p k Z p is a multiplicative group ( Z p/p k Z p) that is isomorphic to(Z /p kZ ) .

The topology of Q

p is generated by the metric induced by the p-adic norm.Given y in Q p and r 0, we dene the closed ball B r (y) of radius r centeredat y by B r (y) = {xQ p : |x y| p r}, the corresponding open ball B r (y)by B r (y) = {xQ p : |x y| p < r }, and the sphere of radius r centered aty by S r (y) = {xQ p : |x y| p = r}.Note that B1(0) = Z p. Moreover, observing that the image of Q p \ {0}under the mapping x |x| p is the discrete subset { pn : nZ}of R , we seethat closed and open balls of positive radius are equivalent, in the sense thatB r (y) = B r + (y) for all sufficiently small > 0. Thus, for the remainder of the paper, whenever we say ball we mean a closed ball of positive radius.Furthermore, we assume that any radius r that occurs has the form r = pfor some integer , which we may do because B r (y) = B p (y) when p

r < p +1 . For the same reasons, S r (y) is nonempty if and only if r = pfor some . One can verify that spheres, as well as closed and open balls,are both compact and open sets in the p-adic topology. (As in any metricspace, a set S is compact if every open cover of S has a nite subcover,and S is locally compact if each point of S has a compact neighborhood.)Finally, every point inside of a ball is a center of that ballthat is, if Bis a ball of radius r , then B r (x) = B for all x in B . It follows that if twoballs B and B of equal radius have a nonempty intersection, then B = B ,a fact that we use in several of our proofs.

We conclude this section with a fascinating property of spheres that doesnot hold in the case of the reals: any sphere in Q p can be written as a

nite union of pairwise disjoint balls! In fact, let x be in Q p and considerthe sphere S = S r (x) of radius r = pn for some integer n. Clearly, S is contained in B r (x). Let s = pn 1. We rst show that B r (x) can bewritten as the union of p disjoint balls of radius s. To see this note rst thatZ p(= B1(0)) is the disjoint union of the p cosets that constitute Z p/p Z p,and it is easy to see that each of these cosets is a subball of Z p of radius1/p. Since B r (x) = x + p n Z p, we nd that B r (x) is the disjoint union of p subballs of radius snamely, the images of the cosets of pZ p under thesame dilation and translation. Let the centers of these p disjoint balls be y i ,


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with y1 = x. Consider the set B= B r (x) \ B s (x). Then B is the union of the p

1 pairwise disjoint balls B s (yi ) with yi not in B s (x). Now we claim

that B= S r (x). Indeed, if y is in B, then |x y| p r but |x y| p > s .Therefore |x y| p = r and y is in S r (x). For the converse, let y be a pointin S r (x). Then y is in B r (x), but since |x y| p = r > s , y is not in B s (x).Therefore y is in Band S r (x) = B, showing that S r (x) is the nite unionof pairwise disjoint balls.

The p-adic numbers were introduced by Kurt Hensel in the late nine-teenth century (1897) in order to study the properties of algebraic numbers.Hensels original description of the p-adic numbers involved an analogy be-tween the ring of integers and the ring of polynomials over the complexnumbers, the crux of which was the development of a representation of ratio-nal numbers analogous to that of Laurent expansions of rational functionsnamely, the p-adic expansion. The p-adic numbers have played a fundamen-tal role in number theory, and are now nding wider applications in science(see, for example, [13]).

3. TOPOLOGICAL AND MEASURABLE DYNAMICS.

In its simplest form a (discrete time, abstract) dynamical system consistsof a set X and a map or transformation T : X X . We will mainly beinterested in the case when the transformation T is invertible. In fact, wemention only one example that is noninvertible (in section 4) and one lemma(Lemma 4.5) that treats the noninvertible case. Thus absent an explicitstatement to the contrary, the reader may assume that any transformationencountered in this paper is invertible.

One of the rst objects of interest is the orbit of point a point x in X under T , dened as {T n (x)}n = and denoted by O(x) (here T n denotesthe n-fold iterate of T , T n = T T , with n factors, T 0 is the identitymapping of X , and T n = ( T 1)n for a positive integer n).

To study properties of such iterations we impose some structure on thespace and the transformations. In topological dynamics we require X tobe a topological space and T to be a continuous transformation; in theinteresting cases we assume that X is compact. In the invertible case T isassumed to be a homeomorphism. The important notion from topological

dynamics for us is minimality. A transformation T : X X is minimal if the orbit O(x) of each x in X is dense in X . It is clear from the denitionthat a homeomorphism T is minimal if and only if n = T n (U ) = X foreach nonempty open set U in X . Using Zorns lemma, it can be shown thatfor any homeomorphism T of a compact space X there exists a nonemptyclosed subset E of X such that E is T-invariant (i.e., T (E ) = E ) and therestriction of T to E is minimal. Minimal sets such as E were introduced in1912 by Birkhoff in his study of the stability of dynamical systems [5, vol.1, p. 660]. It is clear that a minimal homeomorphism is indecomposable


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. The transformation T is ergodic with respect to if (A) is 0 or 1 for anyT -invariant set A in

L(X ). (This notion was rst called metrical transitivity

and was introduced in 1928 by Birkhoff and Smith [5, vol. 2, p. 380].)From the point of view of measurable dynamics the ergodic transformationsare the indecomposable transformations. The reason that this notion isimportant stems from the ergodic theorem, proved in 1931, rst in the caseof mean convergence by von Neumann and then in the stronger case of pointwise convergence by Birkhoff. (The reader may refer to [5, vol. 2, pp.462-465] for a short history, to [22] for a modern account, and to [20] or[21] for the proofs of these theorems.) The pointwise convergence version of the ergodic theorem asserts that ergodicity is equivalent to the the propertythat

limN 1

N

N 1

n =0

IA(T

n

(x)) = (A)(3)holds for each measurable set A in X and for each point x of X outside of a set of measure zero (in general depending on A). (Here I A denotes thecharacteristic function of the set A.) In the full statement of the theoremIA is replaced by an integrable function f and (A) is replaced by A fd ;however, this more general statement follows from (3) by standard approx-imation results. The quantity on the left of (3) is clearly (the limit of) theaverage number of times that images of x under iterates of T land in A. Theergodic theorem then asserts that for any measurable set A and for any pointx outside some set E = E (A) of measure zero, the average number of timesthat the forward orbit of x visits A tends to the measure of A or, in otherwords, that the time-average is asymptotically equal to the space-averagefor all points outside some set of measure zero. This theorem solved an im-portant problem that came from statistical mechanics: to give a conditionunder which the time-average and the space-average of a dynamical systemwould agree. While verication of the ergodicity property for actual phys-ical systems has proved to be difficult, the notion of ergodicity has foundnumerous applications in mathematics. This is partly due to the fact thatevery nite-measure-preserving transformation admits a decomposition intoergodic components. Roughly stated, if is an invariant measure for atransformation T , then can be written as an integral of measures each of which is dened on some invariant subset of X and is ergodic with respect

to T . (For the case of some of our examples this decomposition is illustratedin Figures 1, 2, and 3.) This theorem has a long history, starting with workof von Neumann in 1932 and culminating with independent developmentsby Maharam and Rohlin, for which the reader may refer to [15, sec. 8] and[20, chap. 3], respectively.

We often use another condition equivalent to ergodicity that is easier tocheck. Assume that T is invertible. Clearly the set i= T

i(A) is T -invariant for any given subset A of X . Therefore, if A has positive measureand T is ergodic, then (X \ i= T i (A)) = 0. It follows that if T is


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ergodic, then for each pair of sets A and B of positive measure there existsan integer n such that (T n (A)

B ) > 0 (i.e., sets of positive measure hit

other sets of positive measure under iteration by T ). Without too muchadditional work the reader can show that the converse is also true and thatin fact n can be assumed to be positive. In general it does not follow fromthe ergodicity of T that T 2 is also ergodic. If it happens that T n is ergodicfor all n different from 0 we say that T is totally ergodic .

One last notion of signicance for what follows is that of unique ergodic-ity, which we dene only when X is a compact subset of Q p. A homeomor-phism T : X X is uniquely ergodic if there exists only one T -invariantprobability measure dened on L(X ). (In all of our examples this measure will be normalized Lebesgue measure on L(X ).) Interestingly, this alreadyimplies that T is ergodic with respect to . For if T were not ergodic withrespect to , one could construct another T -invariant probability measure different from . Indeed, suppose that Y were a T -invariant set in L(X )with (Y ) (Y c) > 0. Then the probability measure dened on L(X ) by (A) = (A Y )/ (Y ) would be a T -invariant probability measure differ-ent from , a contradiction. Therefore T must be ergodic with respect to . In the uniquely ergodic case one has a renement of the ergodic theo-rem. In fact, T is uniquely ergodic if and only for all continuous functionsf : X R ,

limN

1N

N 1

n =0f (T n (x)) = fd,(4)

where the convergence is uniform for all x in X (see [15, sec. 5.3]).4. MINIMAL ISOMETRIES AND PERMUTATIONS ON BALLS.

All of the transformations T we consider have the property that

|T (x) T (y)| p = |x y| pfor all x and y in some compact subspace X of Q p with (X ) > 0. (Anysuch T is said to be an isometry with respect to | | p.) For example, thetransformation T : Z p Z p given by T (x) = x + 1 is clearly an isometry.Consider B r (x) for any x in Q p. If T is an invertible isometry, then ybelongs to B r (T (x)) if and only if T 1(y) lies in B r (x). It follows that

T (B r (x)) = B r (T (x)), so the image of a ball is a ball of equal size. Thisimplies that T is measure-preserving for Lebesgue measure.An important property that will surface in all of our examples is mini-

mality. We warn the reader that there is another commonly used dynamicalconcept that is related to but weaker than minimalityit is called topolog-ical transitivitywhere one requires only that the orbit of some point bedense. However, as one might expect, an invertible isometry that is topo-logically transitive turns out to be minimal [21, p. 131]. Since all of ourexamples are invertible isometries, and since showing that they are minimal


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is in general no harder than showing that they are topologically transitive,we discuss only minimality. The following has a simple proof, but we state

it as a theorem for future reference.Theorem 4.1. If T : Z p Z p is dened by T (x) = x + 1 , then T isminimal.Proof. Since the class [1] generates each of the nite groups Z p/p k Z p, givenany > 0 and any points x and y of Z p we can nd n such that |T n (x)y| p =|x + n y| p p k < .

We remark that while we proved that the forward orbit {T n (x)}n =0 (oftendenoted O+ (x)) of every point x is dense, this is no stronger than minimalitysince for a homeomorphism T of a compact space X every orbit of T is denseif and only if every forward orbit of T is dense [21, p. 129].

Now minimality turns out to be an extremely useful property for ourpurposes: it is easy to verify for many algebraic transformations. In thissection we rst show in Lemma 4.2 that, if T : X X is a minimalisometry of a subset X of Q p that contains a nonempty open set, then X must be a nite union of balls, hence a compact open set. In particular, thisrules out minimal isometries T : X X for subsets X of Q p that containopen sets and have (X ) = . Because the transformations of interest tous are algebraic, requiring that their domains of denition contain open setsis not an unreasonable restriction.

Theorem 4.4 is the real engine of this paper. It states that for an invertibleisometry of a compact open subset of Q p minimality is equivalent to uniqueergodicity. Now, it is already well known that minimal isometries of compactmetric spaces must be uniquely ergodic (see [15, sec. 5.8]). However, ourproof of this fact for compact open subsets of Q p is much simpler than thestandard proof. Also, in our context, we give a short proof that ergodicityimplies minimality, hence unique ergodicitya theorem also known in amore general setting [15].

We note in passing that minimality and unique ergodicity are distinctproperties. For example, as shown by Furstenberg [9], there are minimalhomeomorphisms of the two-dimensional torus that are not uniquely ergodic.Also, while a uniquely ergodic map is minimal on the support of its invariantmeasure [15, sec. 5.2], there exist uniquely ergodic homeomorphisms (onthe unit circle in the complex plane for example) that are not minimal (see

[21]). Furthermore, it is easy to construct ergodic isometries that are neitherminimal nor uniquely ergodic. For example, let X = S p(0) {0}, and letT : X X be dened so that the restriction of T to S p(0) is any ergodicisometry on S p(0) (such as the maps of Theorem 7.2) and T (0) = 0. Let be the probability measure dened on measurable subsets A of S p(0) by (A) = (A)/ (S p(0)) and extend to a measure on X by setting ({0}) = 0(i.e., (A) = (AS p(0)). Then T is an ergodic isometry of X with respectto that is not minimal and is not uniquely ergodic. (Note, however, that is not positive on open sets of X , so does not satisfy Theorem 4.4(3).)


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We start with a lemma that is of independent interest.

Lemma 4.2. Let X be a subset of Q p

, let T : X

X be an invertible

isometry, and let x be a point of X . If O+ (x) is dense in X , then X iscontained in a nite union of balls, hence X is of nite -measure. Fur-thermore, if X contains some ball B around x of positive radius r , then X = n 1i=0 T

i (B ) for some n > 0; in particular, X is a compact open subset of Q p.

Proof. Let r > 0. Then there exists a smallest n > 0 such that d(T n (x), x) r would hold for all nonnegative i, so z would not be a limitpoint of the forward orbit of x. This would be a contradiction. We concludethat X = A.

Now if B = B r (x) lies in X , then we have X = n 1i=0 B r (T i(x)). To

complete the proof we note that any two balls of equal radius must beeither disjoint or equal and that B r (T i (x)) = T i (B ).

The reader may want to verify that every compact open subset of Q p is anite unions of balls. We do not need this fact explicitly. However, in whatfollows one could substitute compact open set for nite union of balls.We also obtain the following interesting corollary:

Corollary 4.3. Let X be a nonempty compact open set in Q p and let T :X

X be a minimal invertible isometry. Then for any ball B of positive

radius contained in X there exists an integer n > 0 such that T n (B ) = B .Furthermore, if X = B p (a) for some a in Q p and in Z , then for each x in B p (a) and each j with j > it is the case that T p

j (B p j (x)) = B p j (x).

In particular, T is not totally ergodic for Lebesgue measure.

Proof. We prove only the second part. From the proof of Lemma 4.2 welearn that there is an integer n such that

B p (x) = B p j (x) B p j (T n 1(x))(5)and such that the balls in the union are pairwise disjoint. We infer thatT n (B p j (x)) = B p j (x), so T n is not ergodic. Using the fact that the setson each side of equation (5) have the same measure, we obtain p = np jor, equivalently, n = p j .

We are now ready for the main theorem of this section.

Theorem 4.4. Let X be a nonempty compact open open set in Q p, and let T : X X be an invertible isometry. The following statements areequivalent:

(1) T : X X is minimal.(2) T : X X is uniquely ergodic.


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(3) T is ergodic for any T -invariant probability measure on L(X ) that is positive on nonempty open sets (such as normalized Lebesgue mea-sure).

Proof. (1)(2): Let be the normalized Lebesgue measure on X . Since T is an isometry, is a T -invariant measure. Suppose now that T is minimal onX , and let be any T -invariant probability measure on X . Consider a ballB of positive radius r contained in X . By Lemma 4.2, X can be written inthe form X = n 1i=0 T

i(B ), where the sets in the union are pairwise disjointand n > 0 is an integer. Since is a T -invariant probability measure, thisyields 1 = n(B ). Similarly, 1 = n (B ). Then (B ) = 1 /n = (B ). Thus and agree on all balls contained in X and therefore = , showing thatT is uniquely ergodic. (The fact that = can be seen by noting that anysuch measure satises an equality similar to (2).)

(2) (3): As we have already seen, if T is uniquely ergodic, then T isergodic with respect to its unique invariant probability measure on L(X ).(3)(1): Suppose that T is ergodic on X with respect to a probabilitymeasure that is positive on nonempty open sets. Let > 0, and considerballs B1 = B r (x) and B2 = B r (y) in X with r < / 2. As T is ergodic withrespect to , there exists n > 0 such that (T n (B1) B2) > 0. Since T isan isometry, T n (B1) is a ball of radius r centered at T n (x). Thus by thetriangle inequality, |T n (x) y| p 2r < . As this holds for all > 0, y is alimit point of O+ (x). Because x and y in X are arbitrary, this shows thatT is minimal.

Theorem 3.2 holds in the more general setting where X is a compactmetric space (say with metric d) and T : X X is an equicontinuousmap, as was pointed out to the authors by J. Auslander. (A map T issaid to be equicontinuous if for each > 0 there exists > 0 such thatd(T n (x), T n (y)) < holds for all n and for all x and y in X satisfyingd(x, y ) < .) In this case the proof that (3) (1) is the same as theone given for Theorem 4.4, (2) (3) is immediate, and the implication(1)(2) was demonstrated by Oxtoby [15, sec. 5.8]. Isometries are clearlyequicontinuous; furthermore, it can be shown that any equicontinuous mapT : X X is an isometry for some metric on X that is topologicallyequivalent to the given metric.

Once we have deduced that a map T is an invertible isometry on Q p,

it is simply a matter of nding the sets on which it is minimal (providedthey exist). Various criteria help us decide where to look. If it happensthat d(x, T (x)) = r for all x in Q p, then we focus on balls of radius r .This is the case, for example, for the map T (x) = x + 1, where clearly

|x T (x)| p = 1. Therefore we study this map on balls of radius 1, suchas Z p. On the other hand, if T has a xed point y (i.e., T (y) = y), thenwe have that |x y| p = |T (x) T (y)| p = |T (x) y| p. In this case all thepoints of the orbit of x are equidistant from y, so we focus our attention onspheres around y. An example is given by the map M a : Z p Z p dened


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by M a (x) = ax , which has a xed point at 0. So if M a is an isometry (whichis only possible if

|a

| p = 1), then we look at spheres around 0. In the end,

showing that T is minimal on any of these spaces relies on the algebraic andnumber theoretic qualities of the p-adics. We intend to establish minimalityonly for a few distinguished maps and then deduce minimality for othermaps using the notion of isomorphism that we study in section 5.

It is well known that isometries T : X X on compact spaces X cannotbe weakly mixing [20, p. 54]. Recall that a transformation T preserving aprobability measure is weakly mixing if the product transformation T T :X X X X is ergodic with respect to the product measure . Thisimplies, in particular, that T is totally ergodic. It is interesting to note that,in view of Corollary 4.3, a minimal isometry in Q p is not even totally ergodic(with respect to its unique invariant measure). This leads to the question:Can any probability-measure-preserving transformations on Q

pdened by

rational polynomial functions be weakly mixing (or even mixing) on aninvariant subset of Q p having positive measure? In this context we note thatif we are willing to consider noninvertible transformations, one can easilyconstruct mixing, hence weakly mixing, examples on Z p. (A transformationT that preserves a nite measure is mixing if for all measurable setsA and B with (A) > 0 it is true that lim n (T n (A) B )/ (A) = (B ); in other words, the relative proportion of space that T n (A) occupiesin an arbitrary set B is in the limit the measure of B .) These mixingtransformations are the analogues of the Bernoulli transformations and aredened on Z p by T (0.c0c1c2 . . . ) = 0 .c1c2 . . . . (alternatively, by T (x) = px

c0(x) p 1, where c0(x) is the coefficient of p0 in the representation of x.)

Then T is a noninvertible nite-measure-preserving transformation that canbe shown to be mixing by standard methods similar to those used to establishmixing for the Bernoulli transformations on [0 , 1). (The main part of theargument is to verify that for any balls B 1 = B p (c) and B2 = B p k (d) andfor any n > k it is the case that (T n (B1) B2) = (B1)(B2).)This section concludes with a lemma that claries the role of invertibil-ity in our presentation. In particular, it follows from the lemma that ameasure-preserving isometry on Z p must be invertible, that transformationson Z p satisfying the Lipschitz condition in the lemma cannot be totally er-godic when they are invertible (Corollary 4.3), and that any noninvertiblemeasure-preserving transformation on Z p must involve some sort of expan-

sion with respect to the p-adic distance (such as the Bernoulli example).Lemma 4.5 is stated without proof in [3, Theorem 1.1], where the authoralso studies in detail transformations satisfying the Lipschitz condition of the lemma.

Lemma 4.5. Let T : Z p Z p be a transformation satisfying a Lipschitz condition with constant 1: |T (x) T (y)| p |x y| p for all x and y in Z p.Then T is surjective if and only if T preserves Lebesgue measure, in which case T is an invertible isometry.


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Proof. Suppose that T is surjective. Let n 0, let x be a point in Z p, andwrite r = p n . Clearly, T (B r (x))

B r (T (x)) . (This is a direct consequence

of the Lipschitz condition, since if z is in T (B r (x)), then z = T (a) forsome a in B r (x) and |z T (x)| p = |T (a) T (x)| p |a x| p r .) Nowsuppose that T (B r (x)) is properly contained in B r (T (x)), and let z be apoint of B r (T (x)) \T (B r (x)). Since T is surjective, there exists a in Z p suchthat T (a) = z. As T (a) is in B r (T (x)), B r (T (x)) = B r (T (a)). However,because a is not in B r (x), it follows that B r (x) B r (a) = . Now we canwrite Z p as a union pairwise disjoint balls in the form Z p = p

n

i=1 B r (x i)for some points x i in Z p, with x1 = x and x2 = a, say. Then the setT (Z p) =

pni=1 T (B r (x i)) =

pni=1 B r (T (x i )) must be contained in at most

pn 1 balls of radius r (as the terms in the union corresponding to i = 1and i = 2 are equal), contradicting the assumption that T is surjective.Therefore T (B r (x)) = B r (T (x)). Because this is true for each x in X andfor r = p n for each positive integer n, T is an isometry. Injectivity is animmediate consequence of the fact that T is an isometry. It also follows thatT preserves Lebesgue measure.

For the converse suppose that T is not surjective. Then there exists x inZ p \ T (Z p). Since T is measure-preserving,

(Z p \ T (Z p)) = (T 1(Z p \ T (Z p)) = () = 0 .Therefore Z p \ T (Z p) does not contain any ball of positive radius. Thisimplies the existence of a sequence {xn}of points in T (Z p)) converging tox. Select yn in Z p with T (yn ) = xn . Taking a subsequence yn i convergingto some point y in Z

p, we obtain T (y) = x (using the continuity of T ), a

contradiction. Therefore T is surjective.

5. ISOMORPHISMS.

Let X 1 and X 2 be nite unions of balls in Q p. Let T i : X i X i (i = 1 , 2)be a continuous transformation. A map : X 1 X 2 is called a topological isomorphism of (X 1, T 1) and ( X 2 , T 2) if it is a homeomorphism of X 1 onto X 2and the equality (T 1(x)) = T 2((x)) holds for all x in X 1 (i.e., conjugatesthe actions of T 1 and T 2). One can easily verify that minimality is preservedunder topological isomorphism.

If 1 and 2 are probability measures on L(X 1) and L(X 2), respectively,a measurable isomorphism of the measurable dynamical systems ( X 1, T 1, 1)and ( X 2 , T 2 , 2) is a map : X 1 X 2 that is invertible and measure-preserving (i.e., 1(A) = 2((A)) for all sets A in L(X 1)) and has theproperty that (T 1(x)) = T 2((x)) for all x in X 1 outside a subset of X 1 of zero 1-measure. Again, it is not hard to show that ergodicity is invariantunder measurable isomorphism.

Now, we observe that in our case any topological isomorphism is a mea-surable isomorphism, so we henceforth refer to such maps simply as iso-morphisms. In fact, as T 1 and T 2 are minimal isometries (hence, uniquely


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ergodic by Theorem 4.4), if 1 and 2 are their invariant probability mea-sures on

L(X 1) and

L(X 2), respectively, and if : X 1

X 2 is a topological

isomorphism, then the measure 1 1 is an invariant measure for T 2. Itmust therefore coincide with 2. This shows that is measure-preserving,making it a measurable isomorphism.

In our examples, the isomorphism property will follow from a standardlemma whose proof we include for the sake of completeness.Lemma 5.1. Let X 1 and X 2 be nite unions of balls in Q p, let T 1 : X 1 X 1 and T 2 : X 2 X 2 be continuous transformations, and let : X 1 X 2 be a surjective map such that (T 1(x)) = T 2((x)) for all x in X 1. If there is a constant C > 0 such that (6) |(x) (y)| p = C |x y| p for all x and y in X 1, then is an isomorphism of (X 1 , T 1) and (X 2 , T 2).Under these circumstances T 1 is an invertible isometry if and only if T 2 isan invertible isometry, and T 1 is minimal (hence, uniquely ergodic) if and only if T 2 is minimal (hence, uniquely ergodic).

Proof. It is easy to see that (6) implies that is one-to-one and continuousand that (6) holds for 1 (with C 1 replacing C ), so is an isomorphism.Now, if T 1 is an isometry, then

|T 2((x)) T 2((y))| p = |(T 1(x) (T 1(y))| p= C |T 1(x) T 1(y) = C |x y| p= CC 1|(x) (y)| p = |(x) (y)| p,

whence by the surjectivity of the map T 2 is likewise an isometry. Notingthat T n2 = 1 T n1 completes the proof.6. TRANSLATIONS ON Q p.

We start our study of special maps by looking at the translation T b :Q p Q p dened by T b(x) = x + b for xed b in Q p. We rst consider thecase when |b| p 1, in which event T b : Z p Z p. We later show that theother cases can be reduced to this one using isomorphisms. (We note thatT 1 : Z 2 Z 2 is isomorphic to a well-known transformation, the so-calleddyadic odometer.) The reader is referred to Anashin [2], [3] for a study of measure-preserving and ergodic maps on Z p.

Theorem 6.1. If b belongs to Z p and T b : Z p Z p is dened by T b(x) =x + b, then T b is an invertible isometry. Furthermore, T b is minimal and uniquely ergodic on Z p if and only if |b| p = 1 .Proof. It is clear that T b is invertible and is an isometry, for

|T b(x) T b(y)| p = |x + by b| p = |x y| p.Assume that |b| p = 1 and dene : Z p Z p by (x) = bx. Then

(T 1(x)) = (x + 1) = b(x + 1) = bx + b = T b((x)) .


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X . Finally we observe that Corollary 6.3 shows that T b is minimal on B s (a).Since X is T -invariant, this will imply that X is equal to B s (a). Thus it

suffices to analyze T b on balls of radius s = |b| p. We use isomorphisms againto obtain our next result, which together with Corollary 4.3 can be used toobtain the ergodic decomposition illustrated in Figure 1.

Corollary 6.3. If b in Q p has |b| p = p ( Z ) and T b is dened by T b(x) = x + b, then for each a in Q p the map T b : B p (a) B p (a) isan invertible isometry. Furthermore, T b is minimal and uniquely ergodic on B p (a).

Proof. We dene a map : Z p B p (a) by (x) = bx + a. It is easy tosee that this map is invertible, and we have

|(x)

(y)

| p =

|bx + a

by + a

| p =

|b

| p

|x

y

| p.

Moreover,

(T 1(x)) = (x + 1) = b(x + 1) + a= bx + a + b = (x) + b = T b((x)) ,

showing that satises the hypotheses of Lemma 5.1. By Theorem 4.1 T 1is minimal, and uniquely ergodic, on Z p. Lemma 5.1 asserts that T b is aninvertible isometry that is minimal and uniquely ergodic on B p (a).

It is interesting to compare translations on Z p with their real analogs.Translations on the entire real line are plainly not ergodic with respect toLebesgue measure. Translations by rational numbers on, say, [0 , 1) with

endpoints identied (i.e., on R / Z , the reals modulo 1) are periodic and notergodic. On the other hand, translations on R / Z by irrational numbers aretotally ergodic and minimal (see, for example, [20] or [21]). This stands insharp contrast to the situation in Q p, where all translations are ergodic oncertain subsets of Q p, but none are totally ergodic.

7. MULTIPLICATION IN Q p.

While the rst polynomials we have considered have direct analogs in thereals as minimal transformations, the ones we consider for the remainder of this article take advantage of special properties of Q p. In this section weconsider the transformation M a : Q p Q p dened by M a (x) = ax forsome xed member a of Q p, a multiplication (or dilation) by an element of Q p.

The rst study of ergodic properties of transformations on the p-adicsthat we are aware of is in [14], where Oselies and Zieschang considered thecontinuous automorphisms of the group Z p. These automorphisms are of the form M a with a in S 1(0), and in [14] Oselies and Zieschang obtainedtheir ergodic decomposition on Z p. The multiplication transformations M aacting on the group of units S 1(0) were rst studied by Coelho and Parryin [7]. In particular they proved that M a is ergodic on S 1(0) if and only


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if a is a primitive root modulo p and a p 1 1 (mod p2). This is similarto Theorem 7.2, but we note that for = 0 the spheres S p are not mul-tiplicative groups. The motivation of Coelho and Parry was to use thesetransformations to obtain information on the distribution of sequences suchas the Fibonacci sequence, which we describe later in this section.

To make M a a measure-preserving transformation, and in fact an isome-try, it is clear that we require |a| p = 1.One difference between these last two sections and the previous sectionsis that we now demonstrate minimality of our transformations on spheresrather than on balls. However, we briey return to balls in Theorem 8.5.

In any metric space an isometry with a xed point denes a dynamicalsystem on spheres centered at this point, for these spheres are invariant setsfor the isometry. For example, because 0 is a xed point of M a ,

|M a (x) 0| p = |ax | p = |a| p|x| p = |x 0| p,so we see that M a is a well-dened bijective transformation of the spheresS p (0) for each integer . In contrast with the real case, where spheres inR n have measure zero with respect to n-dimensional Lebesgue measure, inthe case of the p-adics spheres have positive measurein fact, a sphere of radius r about a point x has measure greater than or equal to the open ballof radius r centered there. Thus we can use the Lebesgue measure that wedened for the p-adics to analyze dynamical systems on these spheres.

We observe immediately that there are certain choices of a that guaranteethat M a is not minimal on any sphere. For example, when p 3 the ring Z pcontains the ( p1)th roots of unity, so M p 1a is the identity transformationfor each such root a. This means that M a is periodic and thus minimal onlyon nite sets. Also, recall that for p = 3 we may write the sphere S 1(0) as adisjoint union of balls S 1(0) = B1/ 3(1)B1/ 3(2). Consider M a : Z 3 Z 3with a in B1/ 3(1). Then a has the form a = 1 p0 + c1 p + c2 p2 + forsome c0, c1 , . . . . If x is a point in B 1/ 3(1), then it is straightforward to verifythat ax is also in B1/ 3(1). It follows that B1/ 3(1) (and, likewise, B1/ 3(2))is an invariant set under M a , so M a is not minimal on S 1(0). (In this casewe could rene our analysis to treat subsets of spheres, as illustrated inFigure 2, but we will deal here only with transformations that are ergodicon spheres. For a complete classication of the ergodic decomposition of M a , consult [7].)

To see what restrictions we need on a we focus on S 1(0) for the moment.If M a is minimal on this sphere, then every integer that is not divisible by pis a limit point of the orbit of 1, the set {a n}n =0 . Let = p k with k in N ,and let x in Z be such that p does not divide x. Thus there must exist n 0such that |an x| p < p k or, in other words, such that a n x (mod pk)(i.e, an x is in pk Z p). Since this holds for every x in Z not divisible by p, we see that [ a] generates the multiplicative group of units ( Z p/p k Z p)for each k in N. In fact, this is exactly the property that we want a tosatisfy. Furthermore, there is an easy way to characterize the a that enjoy


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Z 3

S 1 (0)

S 3 1 (0)

S 3 2 (0)

Figure 2. Ergodic decomposition of M a on Q 3.

this property using the following well-known lemma from group theory (fora proof of the rst part the reader may refer to [19, Theorem 6.7], and forthe second part to [19, Theorem 5.44]).

Lemma 7.1. Let p be an odd prime, and let a be a member of Z p. Then [a]generates the group of units (Z p/p k Z p) for each k in N if and only if [a]generates the group of units (Z p/p 2Z p) . Furthermore, (Z 2/ 2k Z 2) is not cyclic when k

3.

Notice that Lemma 7.1 implies that M a can never be minimal (hencenever uniquely ergodic) on the sphere S 1(0) of Z 2. Thus we bid farewell tothe even prime and consider only odd primes p for the rest of this paper.We are now ready for our characterization of the minimality of M a . Asmentioned earlier, a theorem similar to Theorem 7.2 (for = 0) was provedin [7].

Theorem 7.2. Let p be an odd prime, let a in Q p have |a| p = 1 , and let M a : S p (0) S p (0) with in Z be dened by M a (x) = ax . Then [a]generates (Z p/p 2Z p) if and only if M a is minimal and uniquely ergodic on S p (0) .

Proof. Assume initially that [ a] generates ( Z p/p k Z p) . Fix x in S p (0).We want to show that, for any y in S p (0) and any > 0, there existsn such that |M na (x) y| p = |an x y| p < . Since is xed, we mayassume that p . Choose k such that p k < . Then k > , so k > 0. Note that |an x y| p = |x| p|an yx 1| p for all n . The fact that|yx 1| p = |y| p/ |x| p = 1, shows that yx 1 lies on S 1(0). Thus [ yx 1] belongsto (Z p/p k Z p) . Since [a] generates ( Z p/p 2Z p) , by Lemma 7.1 [a] mustgenerate ( Z p/p k Z p) , meaning that there exists a positive integer n such


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that an yx 1 (mod pk ) or, stated differently, that |a n yx 1| p < p k .This leads to the conclusion that|M na (x) y| p = |x| p|an yx 1| p < p p k = p k < .

Therefore M a is minimal and, by Theorem 4.4, uniquely ergodic on S p (0).For the converse, suppose that [ a] does not generate ( Z p/p 2Z p) . Then

there exists b in (Z p/p 2Z p) such that an is not congruent to b modulo p2

for any integer n. Now let x = p and y = bp with in Z . Clearly x and yare points of S p (0), as |b| p = 1. Also M na (x) = an p . We have

|M na (x) y| p = |an p bp | p = p |an b| p p 1for each nonnegative integer n, revealing that y is not a limit point of theorbit of x. Since x does not have a dense orbit in S p (0), M a is not minimal

on this space.A theorem equivalent to Theorem 7.2 is used in an intriguing way in [7]

to give the distribution in Z p of the Fibonacci sequence. We outline themain ideas to give a avor of the argument in [7] . We consider only thecase when 5 is an element of Z p (which, by quadratic reciprocity and adirect application of Hensels lemma, which is stated in section 8, occurswhen p 1 or 4 (mod 5); when 5 is not a square modulo p one considers asin [7] an appropriate subgroup of Z p[5]). Now let {f n}be the Fibonaccisequence (i.e., f n +1 = f n 1 + f n , f 0 = 0 , f 1 = 1). The idea is to nd afunction f and a number in S 1(0) so that the sequence f n can be expressedin the form f n = f

M n (1). Once this is achieved it is possible to apply

the ergodic theorem to the function f and the transformation M . In thecase of the Fibonacci sequence we obtain from Binets formula: it is wellknown that f n = (1 / 5)[ n (1/ )n ] for = (1 + 5)/ 2. The requiredfunction f is dened by f (z) = (1 / 5)[z N (z)/z ], where N (z) = 1 if z isa square modulo 5 and N (z) = 1 otherwise. One can verify that indeedf n = f ( n ) = f M n (1). We further assume that M is ergodic (if it isnot ergodic, one must deal with the ergodic decomposition of M , which isdescribed in [7]). Using the fact that M is uniquely ergodic on S 1(0), weconclude that for any open and closed subset A of Z p,

limN

1

N

N

n =0

IA(f n ) = limN

1

N

N

n =0

IA(f

M n (1))

= limN

1N

N

n =0

I f 1 (A) (M n (1))

= I f 1 (A)d = (f 1(A)),which gives us the distribution of the Fibonacci sequence mod pk in termsof the measure f 1. In [7] the authors discuss this measure in detail.


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The next natural transformation we consider is the affine mapping T a,b :Q p

Q p dened by T a,b (x) = ax + b for a and b in Q p with

|a

| p = 1

and b = 0. The rst thing to notice is that this transformation has a xedpoint (assuming a = 1: when a = 1 these transformations are translations,which we covered earlier) at c = b/ (1 a). Thus we analyze the dynamicalproperties of T a,b on the spheres S p (c) centered at c. We start by showingthat affine transformations are in fact isomorphic to the multiplications wehave already discussed and then obtain information on the dynamics of T a,b .

Theorem 7.3. Let p be an odd prime, and let a and b be points of Q psuch that |a| p = 1 but a = 1 . If T a,b and M a are transformations dened by T a,b (x) = ax + b and M a (x) = ax , respectively, and if c = b/ (1 a)is the xed point of T a,b , then (S p (c), T a,b ) is isomorphic to (S p (0), M a ) for each integer . Furthermore, T a,b : S p (c)

S p (c) is minimal and

uniquely ergodic on S p (c) if and only if [a] generates (Z p/p 2Z p) .

Proof. Dene : S p (c) S p (0) by (x) = x c. This map is certainlyinvertible. Furthermore, if x is a point of S p (c), thenM a ((x)) = ax ac = ax ab/ (1 a)

= ax (ab b)/ (1 a) b/ (1 a) = ax + bc= (T a,b (x)) .

Finally, it is clear that |(x) (y)| p = |x y| p. An appeal to Lemma 5.1shows that the two dynamical systems in question are isomorphic. To com-plete the proof we apply Theorem 7.2 .

We note that an interesting property of the p-adic numbers follows fromTheorem 7.3.

Corollary 7.4. Let [a] generate (Z p/p 2Z p) , where a is an element of Q pwith |a| p = 1 but a = 1 , let b be a point of Z p, and let c = b/ (1 a). If xlies on S 1(c), then the set

an x + bn 1

i=0

a i

n =0

has exactly pk 1( p1) distinct elements modulo pk , and it is equal to S 1(c)modulo pk .Proof. The hypotheses ensure that a 1 (mod p). Therefore |1 a| p = 1,c belongs to Z p, and S 1(c) is contained in Z p. We invoke Theorem 7.3 tocomplete the proof.

8. SIMPLE POLYNOMIAL TRANSFORMATIONS IN Q p.

For our last examples we restrict attention to the monomial mappingsM a,n : Q p Q p dened by M a,n (x) = ax n for n in N . (These are simplegeneralizations of the maps n (x) = xn that are studied in [12]). We assume


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in addition that p is an odd prime, a lies in B p 1 (1), and n 1 (mod p).The reasons for imposing these restrictions will soon become apparent.We rst note that, although 0 is a xed point of M a,n , it is immediate

that we do not want to consider the action of M a,n on arbitrary spherescentered at 0. If |x| p < 1, then M ia,n (x) 0 as i , so M a,n cannotbe an isometry on any space containing two such points. If |x| p > 1, then|M ia,n (x)| p as i , so any invariant set containing a ball around xmust have innite measure. If M a,n were to be isometric, this would preventM a,n from being minimal.

However, S 1(0) is an invariant subspace, since for all x in S 1(0)

|M a,n (x)| p = |ax n | p = |a| p|x|n p = 1 .Furthermore, we can show that, modulo certain restrictions on n, M a,n isan isometry on this sphere.

Our approach is to use isomorphisms to reduce M a,n to a map that wehave already considered. In order to do this we rst use Hensels lemma toestablish the existence of a xed point for M a,n . Hensels lemma is one of theearly results in p-adic analysis and can be viewed as a version of Newtonsmethod (for a proof the reader may consult [11] or [18]).

Lemma 8.1 (Hensels Lemma). Let f (x) belong to Z p[x]. Suppose that there exists 0 in Z p such that |f ( 0)| p < |f ( 0)|2 p, where f (x) signiesthe formal derivative of f (x). Then there exists a unique in Z p such that f () = 0 and | 0| p = |f ( 0)| p/ |f ( 0)| p.Lemma 8.2. Let p be an odd prime, let n be a positive integer such that n 1 (mod p), and let a be a point of B p 1 (1) . If M a,n : Q p Q p isdened by M a,n (x) = ax n , then there exists a unique point x0 of B p 1 (1)such that M a,n (x0) = x0.

Proof. Consider the polynomial f (x) = ax n x in Z p[x]. Its formal deriv-ative is given by f (x) = nax n 1 1. We want to apply Lemma 8.1 with 0 = 1. We see that f (1) = a 1, |a 1| p p 1, and f (1) = na 1. Sincena 1 (mod p), |na 1| p = 1. Thus|f (1)| p p 1 < 1 = |f (1)|2 p,

so by Hensels lemma there exists a unique x0 in Z p such that f (x0) = 0(in other words, such that M a,n (x0) = ax n0 = x0) and such that

|x0

1

| p =

|f (1)| p/ |f (1)| p p 1. Therefore x0 is the unique xed point of M a,n inB p 1 (1).We next show that M a,n is isomorphic to T n,a for a value of a to be

specied later. The isomorphism that we use is the p-adic logarithm , whichcan be dened by its power series expansion:

log p x =

i=1

(1)i+1i

(x 1)i ,


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where convergence is with respect to | | p. The region of convergence of theseries is B p 1 (1). (We note that since B p 1 (1) = 1 + pZ p, it follows thatB p 1 (1) is a multiplicative subgroup of Z p = {xZ p : |x| p = 1}.) The p-adic logarithm retains some of the properties of its real analog: log p(xy) =log p(x) + log p(y) and log p(xn ) = n log p x for all x and y in B p 1 (1) and allintegers n. We will also use the fact that log p : B p 1 (1) B p 1 (0) is abijective isometry (see [18, sec. 4.4.2], [12]). We chose a in B p 1 (1) andstudy the action of M a,n on spheres in B p 1 (1).

Lemma 8.3. Let p be an odd prime, let n be a nonnegative integer such that n 1 (mod p), and let a be a point of B p 1 (1). If x0 is the unique xed point of M a,n in B p 1 (1) , then (S p (x0), M a,n ) is isomorphic to (S p (log p x0), T n, logp a ) for = 1 , 2, . . . . In particular, M a,n is an invertible isometry on S p (x0).

Proof. Consider log p : S p (x0) S p (log p x0). This is a bijective isome-

try (i.e.,

| log p(x) log p(y)| p = |x y| p,for x and y on S p (x0)). Now consider T n, logp a : S p (log p x0) S p (log p x0).For x on S p (x0) we compute

log p(M a,n (x)) = log p(axn ) = n log p x + log p a = T n, logp a (log p x).

By Lemma 5.1, log p is an isomorphism.

The following theorem, including the equivalence of minimality, ergodic-ity, and unique ergodicity, was proved by Gundlach, Khrennikov, and Lin-dahl [12] for the case of the transformation M 1,n (i.e., a = 1, x0 = 1). Asalso noted in [12], it is interesting to observe that the spheres S p (x0) arenot multiplicative groups.

Theorem 8.4. Let p be an odd prime, let n be a nonnegative integer such that n 1 (mod p), let a be a point of B p 1 (1), and let x0 be the unique xed point of M a,n in B p 1 (1). Then for = 1 , 2, . . . the transformation M a,n : S p (x0) S p (x0) is minimal and uniquely ergodic if and only if [n] generates (Z p/p 2Z p) .

Proof. Using the isomorphism from Lemma 8.3 in Lemma 5.1, it follows thatM a,n is an invertible isometry on S p 1 (x0). Furthermore, M a,n is minimaland uniquely ergodic if and only if T n, logp a is minimal and uniquely ergodic,

which by Theorem 7.3 occurs if and only if [ n] generates ( Z p/p 2Z p) .While we do not discuss this in detail, the ergodic decomposition of M a,n is

illustrated in Figure 3. In closing, we discuss briey the case when M a,n hasat least one other nonzero xed point, call it x1 . Then x1 is a point of S 1(0),for |x1| p = |ax n1 | p = |a| p|x1|n p = 1. One can use an isomorphism to showthat M a,n is ergodic on the spheres centered at x1 that are contained withinB p 1 (x1). Indeed, an isomorphism : S p (x1) S p 1 (x0) is dened by(x) = xx 0x 11 . It is also possible to use isomorphisms to characterize M a,n


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Z 3

S 1 (1)

S 3 1 (1)

S 3 2 (1)

Figure 3. Ergodic decomposition of M a,n (a = 1) on Z 3.

on spheres around other xed points for the case of values of a outside theball B p 1 (1).

We conclude with a result about the minimality of M a on balls (ratherthan on spheres), remarking that it holds without requiring the condition(as in Theorem 7.2) that [ a] generate ( Z p/p 2Z p) .

Theorem 8.5. Let p be an odd prime number. If a is a point of B p 1 (1),is an integer satisfying

|log

p(a)

| p = p , and c is a point of Q p, then M a is

minimal and uniquely ergodic on B p (c).

Proof. By the proof of Lemma 8.3, M a acting on B p (c) is isomorphic toT logp a acting on B p (log p a). By Corollary 6.3, M a is minimal on B p (c).

ACKNOWLEDGMENTS. The authors would like to acknowledge sup-port from a National Science Foundation REU grant and a Bronfman Sci-ence Center grant from Williams College as part of the SMALL program.We would like to thank Frank Morgan for an early reading of our paperand Joe Auslander for remarking to us that Theorem 4.4 holds in the moregeneral setting of equicontinuous maps. We are indebted to the referees forcomments and suggestions that improved the exposition.

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Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Rutgers University, Piscataway, NJ 08854-8019 USA [email protected]

Department of Mathematics, Williams College, Williamstown, MA 01267,[email protected]

do do trong kh p_adic don gian

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