On joint recurrence

C. R. Acad. Sci. Paris, t. 327, Systhmes dynamiques/Dynamical

On joint

SCrie

I, p. 837-842, Systems

1998

recurrence

SCHMIDT

Klaus

Mathematics Institute, University Srhriidinger Institute for Mathematical E-mail: [email protected]

of Vienna,

(Rnqu

11 septemhre

le 7 juillet

Abstract.

1998,

Version

le

Strndlhofgasse Boltzmanngasse

4: 1090 Vienna, 9, 1090 Vienna,

Austria, Austria

and

Erwin

1998)

Let T be a measure-preserving and ergodic automorphism of a probability space (X:S,IL). By modifying an argument in [4] we obtain a sufficient condition for recurrence of the d-dimensional stationary random walk defined by a Bore1 map f : x H w”. d 2 1, in terms of the asumptotic distributions of the maps (f+fT+...+.~T”-‘)/7~““, I! > 1. Ifd = 2, and if f : X + R” satisfies the central limit theorem wtth respect to ‘I’ (i.e. if the sequence (f + fT + ... + fT”pl)/fi converges in distribution to a Gaussian law on W”), then our condition implies that the two-dimensional random walk defined by f is recurrent. 0 Academic des Sciences/Elsevier, Paris Sur

R&urn&

accept6

Physics,

la rfkurrence

simultan~e

Soit T LUI automorphisme ergodique d’~~n espuce de probabilite’ (X. S. II). En modijiant LUZ argument darts [4] on obtient une condition sujjisante pour la recurrence de la marche aleatoire .stationnaire dtjinie par une,fonction de Bore1 f : X H Ri. d > 1, en termes de la distributinn asymptotique des fonctkms (f + fT + . + f T” -’ )/?I,‘/~, 71,> 1. Si rl = 2 et si ,f : X + iw’ satisjait le the’oreme de la limite centrale relatif a T (c’est-~-diresilasuite (f+ fl’+...+ fT”+‘)/J;- n converge en distribution vers une loi de Gauss sur R’), alors notre condt&m implique que la marche aleatoire a deux dimensions dejinie par f, est rekurrente. 0 Academic des SciencesiElsevier, Paris

francake

abrhghe

Nous etudions la recurrence de la marche stationnaire definie par une fonction mesurablef a valeurs dans Rd au-dessusd’un systeme dynamique invertible et ergodique (X, S, 1-1, T). Si l’on definit ie cocycle f : Z x X H IR” par (1. l), la recurrence de f (ou de la marche stationnaire associee a f) est exprimee sous la forme (1.3). Dans cette Note nous donnons un critere de recurrence au moyen de la distribution asymptotique du cocycle f convenablement normalide. Plus precidment, si o?’ est la distribution de la fonction f(k .)/k ‘icl 7 k > - 13alors le resultat principal de cette Note affirme la recurrence de f pourvu que (1) oti X est la mesure de Lebesgue sur Iw’ (cf. Theorem 1.2). Note prksentCe par Mikha&l 0764.4442/98/03270837

HERMAN.

0 AcadCmie

des Sciences/Elsevier,

Paris

837

K. Schmidt

La d6monstration de cette proposition se fonde sur I’observation suivante : si f n’est pas rkcurrente, alors il existe un ensemble C E S avec p(C) = i, L 2 1, et un E > 0 satisfaisant (2.1) pour tout k E 72.Au moyen de cet ensemble nous construisons une application borelienne f’ : X -+ R” qui est cohomologue B f telle que le cocycle f’ : iz x X -+ R”, dCfini par .f’. posskde les propriMs suivantes p-p.p. : (a) pour tous 1 E Z, [ {k E Z : f’(k, z) = j”‘(I,z)} 1 = L, (b) si /G I E Z et f’(k,:z) # f(l,z), alors )) f’(ll-,z) - f(l,z) )I > E. Puisque f’ est cohomologue B f, le comportement asymptotique des distributions des applications (d), k 2 1. La s6paration uniforme des valeurs f’(k .1/k I/’ ?k > 1, est identique Bcelui des mesuresUs f’(k) TIT),k E z, ;-p.p. exp rim&e dans (a)-(b), combin& B une estimation combinatoire asymptotique fournit une dCmonstration de (1 j. Les applications les plus inGressantesde (1) concernent les cas d = 1 et d = 2. Pour d = 1 nous renvoyons 5 [4]. Pour d = 2, (1) implique le corollaire suivant : ,f est r&u-rent si les distributions de f(lc, .)/,J?/” tendent vers la loi gaussiennele long d’une suite d’entiers k ayant une densit positive (Corollary 1.3). Ceci amCliore un rt%ultat de [2].

1. Recurrence

of d-dimensional

stationary

random

walks

Let T be a measurepreserving and ergodic automorphism of a standard probability space (X, S. CL), d 2 1, and let f = (flj.. . ,fd) : X -+ Rd be a Bore1 map. For every n E Z and 5 E X we set: n-l C f(T”2) if 71 > 1, f

(n,,

:z:)

=

k=l

0

1 -f(-n,,

T”:G)

if

71= 0,

if

7~< 0.

(1.1)

The resulting map f : Z x X -+ Z” satisfies f(m, TTL:r~)+ ,f(n,, 2;) = f(m + n, 2~)

(1.2)

for every m, n E Z and CL-a.e.:I: E X. If 11. )I denotes the maximum norm on Wd, then the map f : X -+ W” is recurrent (or the individual components fi! . . . , fd of f are jointly recurrent) if IimiSf Ij f(7)>, for p-a.e. 5 E X. If refer to [4]).

f

2:)

(1 =

(1.3)

0

is not recurrent it is called transient (for terminology and background we

PROPOSITION 1.1 (see [4]). - Let f : X + R” be a Bore1map. Thefollowing conditions are equivalent.. (1) f is recurrent; (2) ~h({z E X : lim inf 1n ) -rs= ((f(7b, :I:) (1 < zc}) > 0; (3) For every B E S with p(B) > 0 and ever?; E > 0, p(B n T-“B

n {x E X : 1)f(7n.

z)

(I

<

E}

>

0

for some nonzero nl E Z. For every k 2 1 we define probability measures(TF) and 7id’ on Rd by setting: $‘(A)

= c~({z E X : f(kx)/k””

E A}),

$‘(A)

= ; 2 I=1

838

CJ~‘“‘(A)

(1.4)

On

joint

recurrence

for every Bore1 set A c Rd, where 1~ is the indicator function of A. In [3] and [4] it was shown that the recurrence of f can be deduced from certain properties of these probability measures. For example, if d = 1 and limb,, a(,‘) k = & in the vague topology, where s,(A4) =

: i

if

OEil!

otherwise,

then f is recurrent by [3] or [4]. In [4] it was also shown that a map j : X + W is recurrent whenever

The purpose of this paper is to prove the following

extension of this result to higher dimensions.

THEOREM 1.2. - Let T be a measure preserving and ergodic automorphism of a probabipiy space (X,S,p), d 2 1, f : X -+ W1 a Bore1 map, and define the probability measures rk , k 2 1, on W” by (1.4). We denote by X the Lebesgue measure on R” and set, for every q > 0, B(q) = {U E Rd : 1111// < 7)). Zf f is transient, then

sup lim sup Ti”‘(B(q))/X(B(7j)) q>O

and

li!r~lf

< zc

(1.5)

k-33

linngf

7~d)(B(7j))/X(B(7j))

=

0.

The interesting cases are, of course, d = 1 and d = 2. The case d = 1 was discussedin [4]; in order to explain the significance of Theorem 1.2 for d = 2 we say that a Bore1 map f : X -+ R” satisfies the central limit theorem with respect to T if the distributions of the functions f(7~, .)/,/FL, n 2 1, converge to a (possibly degenerate) Gaussian probability measure on R” as n + x (for the existence of such functions, see [1] and [2]). A somewhat weaker form of the following corollary also appears in [2]. COROLLARY I .3. - Let T be a measurepreserving and ergodic automorphism of a probability space (X, S, p), and let f : X -+ R2 be a Bore1 map satisfying the central limit theorem with respect to T. Then f is recurrent. More generally, if there exists an increasing sequence(rll;; k > 1) qf natural numbers with positive density in N such that the distributions of the j.mctions f (71k, .)/fi converge to a (possibly degenerate) Gaussian probability measure on R” as X: -+ x, then f is recurrent.

Proof of Corollary 1.3. - If the sequence (f(n, .)/fi, 71 2 1) converges in measureto a constant, then this constant has to be zero by (1.2). This shows that. if f satisfiesthe central limit theorem with respect to T (with either degenerate or nondegenerate limit), then there exists a positive constant c such that ri’)(B(q)) > c7j2 for all sufficiently large k: and all sufficiently small q > 0. According to (1.6) this means that f is recurrent. 0 The proof of the second assertion is analogous. Note that Theorem 1.2 and Corollary 1.3 make no assumptionsconcerning the integrability of f.

2. The proof of Theorem

1.2

The proof of Theorem 1.2 differs from that of Theorem 3.6 in [4] only by avoiding the use of the total order of R (which is, of course, not available if d > 1). Let T be a measurepreserving and ergodic automorphism of a standardprobability space (X, S! /L), d 2 1, and let f : X + W” be a transient Bore1 map. For the definition of the probability measures (4 on R” we refer to (1.4). Ok ) rk “I

839

K. Schmidt

Proposition 1.1 implies that there exist a Bore1 set C c X with p(C) > 0 and an E > 0 with /*.(C n T-“C

n {x E X : 11f(k::r)

)I < E}) = 0

for every k E 7. By decreasing C, if necessary, we may assumethat /I,(C) = l/L

(2.1) for some L 2 1.

LEMMA 2.1. - For every 7 > 0 and N > 1, limsup ri”‘(B(rl)) _<2”,GKdX(B(r/)). k-+cu Iv limsup C 2” 5-,$!~(B(2-nldrj)) <_2d+‘dLd~-“X(B(q)). L+x n=o

(2.2)

ProoJ -We modify T on a null-set, if necessary,and assumewithout loss in generality that T”x # 5 for every II: E X and 0 # 71,E Z and hence that (I .2) holds for every m, r~,E Z and z E X. Denote by RQ- = { (Tnx:: x) : :LT E

X,

n

E

7)

c

X

X

x

the orbit equivalence relation of T and define a Bore1 map f : RT -+ Rd by setting f(T”z, for every (T”z,s)

3;) = p(76>x)

(2.3)

E RT. Then (1.2) implies that f(z::c’)

+ f(x’,x”)

= f(x::L.“)!

(2.4)

whenever (x,x’), (:IF:Ic”) E Rr. We denote by [T] the full group of T, i.e. the group of all measure preserving automorphisms : r~ E Z} for every .r E X. Since T is ergodic we can find, for V of (X,S,p) with L’z E {Fx any pair of sets B1, Bz E S with p(B1) = ,u(Bs), an element V E [T] with p(VB,A&) = 0. If {C = C’a:Cr , . . ! CL-r} c S is a partition of X with /I, = l/L for %= 0%. . , L - 1, this allows us to find an automorphism W E [T] with ~(WC.~ACi+l) = 0 for i = 0,. . . . L - 2 and IVLx = 3: for every 3; E X. Put mc(.x) =

ruin {,i 2: 1 : Tjs E C} 0

is this set is nonempty, otherwise,

denote by Tcz = “I ‘7rrc(T)the transformation induced by T on C, and set:

There exists a T-invariant p-null set N E S with the following properties: (a> if C’ = C\N, then the sets CL = S”‘C’ are disjoint for k = 0:. . . ! L - 1, and SLC’ = C’, (b) N = X\U;:; Cl, (c) for every 2 E C’, the sets {.j > 1 : Sj.1~E C’} and {:; 1 1 : S-j:r~ E C’} are infinite. Then { S”z : 71E Z} = {T nx : 71 E Z} for every z E X\N. We define a Bore1 map b : X + R” by setting, for every 3: E C’, b(S”:c) = f(SLz, Skx) for k = 1,. . . : L, and by putting b(.x) = 0 for n: E N. The map q(x) = f(S:c,.r) + b(Ss) - 6(x) satisfies that f(S%,x) 9(x) =

840

= ~(rnC~(S);~)

if 0

xEC’.

otherwise.

On

Furthermore, if f’(z) = f(x) + b(Tz) - b(z), and if f’(n,, defined by (1.1) and (2.3) with f’ replacing f, then f’(n,, Lx) = f(n, x) + b(Tn:r)

- b(x).

f’(z,

joint

recurrence

.) : X + Rd and f’ : RT -+ R” are

:I:‘) = f(:c, x’) + b(x) - b(z’)

(2.5)

for every z E X\N, n E Z and z’ E {T’x : k E Z} = {Skx : k E 72). We denote by aft, r; the probability measures defined by (1.4j with f’ replacing as in Lemma 3.4 in [4]. that

f and obtain

pI$rl~(&Q?(rj + rj’))- a;.(B(rj)))2 0, yn~ (fqB(rj + rj’))- @(B(rl))) L 0 for all 77, 77’ > 0. In particular, limsup k-+co

7-;(B(7/))

the inequalities limsup ki3il

5 L2”rfe-“!

(2.6)

(2.2) will be satisfied if 1 2”7&k(B(2-7L~dq)) 71=0

5 dL”2”+1~“~-”

(2.7)

for every 71 > 0 and N > 1. The equations (2.1) and (2.5) yield that

C’ n T-“C’ n (3; E X : (1f’(k, Z) )I < E} = C’ n V-lC’ n {x E X : Vz # 3: and )I f’(Vz,x)

11< E} = 0

whenever k # 0 and V E [T]. We set Y = X x I@, 11 = LL x A, denote by S : Y -+ Y the skew product transformation S(:c, t) = (Ss, t + f’(S, L, 32)) = (SII:, t + g(z)), and obtain that the set D = C’ x H(E/~) is wandering under S, i.e. that S”D n D = 0 whenever 0 # m E Z. For every :I: E X\N we denote by I’? c I@ the discrete set {f’(k,x) : k E Z} = {f’(Sk:c,:c) : k E Z!} and observe that 1 {k

E

z

: f’(k,:r:)

for every v E V, and CCE X\N,

=

71)

and that // 71 - 71’1) 2 E whenever

1(0 < I < k : 0 < /I f’(Z, Lx) (I 1. P”r,} < (kl’”

+ E/r,)d Y(X x B(rj))/Y(D)

since Sj+“D C X x B(k:l/“n we obtain that

( = 1{k E z : f’(S”:xr)

= II} I = L ‘u, 7:’ E V, and 71# 7)‘. Hence

1 5 1(0 < I 5 k : 0 < (1f’(Z, ZTr)1) 5 kl’“rj}

(

= (AY” + E/?7)dL2d?+-d,

+ E), and since the sets S”“D, ‘rn E Z, are all disjoint, By integrating

rQ(rj)) = ; 2 c((B(rj))= ; -& p((2 Ex : /I.f’(Z,x))I 5 Zwj}) I=1

I=1

and by letting k -+ x,

L ‘r;+-.

1

<;+

&“”

I’ 1{ 0 5 1 5 k : 0 < 1)f’(Z, x) 1) 5 P’“rj} 1d/“(Z) +

&/ rdd

. l;&/dE-d

k

we have proved the first inequality in (2.7).

841

K. Schmidt

Similarly, one sees that

c I{

0 < 1 5 2”k : 0 < 11f’(L: x) 11 5 wi2-‘“‘%~}

n>o = L . c

[{7~

1

> 0 : ‘u = f’(Z. II:) for some 1with 0 < I 5 2’“/~ <

I;171d/l(

u II”}

O#uEV,

5 L.

c ( ) (72 2 0 : 1 5 2’“k I k:?+/ll V Iid} / + 1) Ofl~EI:,

c

5L.E .iZl

k’l” 5

( 1{n 2 0 : 1 < 2”k 5 F7/d/jV}

I + 1)

l,Er;;n(E((j+l)E)\B(,E))

L”2d-ld.

c f-1 log(v%d~d) log 2 + 1 < L”2d-1d. VI’

4k@/&

i=l

Hence IV

2” &#(2-“““7))

c

= 2

(4 + 2~‘~~.,,(X(2-~‘~“~~)\{0}))

5 (N ; l) L + Ldd2d+1vd/&

n=O

n=o

and by letting k --) MI, we obtain the second inequality in (2.7). Since (2.7) is equivalent to (2.2) we have proved the lemma. 0 Proof of Theorem 1.2. - Suppose that f : X -+ Rd is transient. Lemma 2.1 yields a constant c > 0 such that N

lhn-s+p C 2”7-$9QB(2-‘“ld77)) 5 cX(B(q)) n=O for every 71 > 0 and N >_ 1. It follows that there exists, for every ~1> 0 and N 2 1, an integer n E {O,.. .,N} with

We conclude that

for some

71

E (0: . . . , N}, and hence that liminf liminf r(“)(B(rl)) k ~1-0

k-m

I:

As N > 1 was arbitrary this proves (1.6). The inequality (I .5) is an immediate consequenceof the 0 first inequality in (2.2). Acknowledgement.

The author would like to thank the refereefor pointing out reference[2].

References [l] Burton R.M., Denker M., On the central limit theorem for dynamical [2] Conze J.-P., Sur un critbre de rCcurrence en dimension 2 pour les appear). [3] Dekking F.M., On transience and recurrence of generalized random [4] Schmidt K., On recurrence, 2. Wahrsch. Verw. Gebiete 68 (1984)

842

systems, marches

Trans. Amer. stationnaires,

walks, Z. Wahrsch. 75-95.

Math. Sot. 302 (1987) 715-726. Egod. Th. & Dynam. Sys. (to

Verw.

Gebiete

61 (1982)

459465.