on joint recurrence
TRANSCRIPT
C. R. Acad. Sci. Paris, t. 327, SCrie I, p. 837-842, 1998
Systhmes dynamiques/Dynamical Systems
On joint recurrence
Klaus SCHMIDT
Mathematics Institute, University of Vienna, Strndlhofgasse 4: 1090 Vienna, Austria, and Erwin Srhriidinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna, Austria
E-mail: [email protected]
(Rnqu le 7 juillet 1998, accept6 le 11 septemhre 1998)
Abstract. 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 la rfkurrence simultan~e
R&urn& 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
Version francake abrhghe
Nous etudions la recurrence de la marche stationnaire definie par une fonction mesurable f a valeurs dans Rd au-dessus d’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 7 - 3 ‘icl k > 1 alors le resultat principal de cette Note affirme la recurrence de f pourvu que
oti X est la mesure de Lebesgue sur Iw’ (cf. Theorem 1.2).
Note prksentCe par Mikha&l HERMAN.
(1)
0764.4442/98/03270837 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
f’(k .1/k ? I/’ k > 1, est identique B celui des mesures Us f’(k) TIT), k E z, ;-p.p.
(d), k 2 1. La s6paration uniforme des valeurs ex rim&e dans (a)-(b), combin& B une estimation combinatoire asymptotique p
fournit une dCmonstration de (1 j. Les applications les plus inGressantes de (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 gaussienne le 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 measure preserving 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:
f (n,, :z:) =
1
n-l C f(T”2) if 71 > 1,
k=l
0 if 71 = 0, (1.1)
-f(-n,, T”:G) if 7~ < 0.
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)>, 2:) (1 = 0 (1.3)
for p-a.e. 5 E X. If f is not recurrent it is called transient (for terminology and background we refer to [4]).
PROPOSITION 1.1 (see [4]). - Let f : X + R” be a Bore1 map. The following conditions are equivalent..
(1) f is recurrent; (2) ~h({z E X : lim inf 1 n ) -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 CJ~‘“‘(A) I=1
(1.4)
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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,, k a(,‘) = & 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 : 11 11 // < 7)). Zf f is transient, then
and
sup lim sup Ti”‘(B(q))/X(B(7j)) < zc q>O k-33
li!r~lf linngf 7~d)(B(7j))/X(B(7j)) = 0.
(1.5)
The interesting cases are, of course, d = 1 and d = 2. The case d = 1 was discussed in [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 measure preserving 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 measure to a constant, then this constant has to be zero by (1.2). This shows that. if f satisfies the 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.
The proof of the second assertion is analogous. 0 Note that Theorem 1.2 and Corollary 1.3 make no assumptions concerning 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 measure preserving and ergodic automorphism of a standard probability 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 Ok ) rk “I on R” we refer to (1.4).
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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 : 11 f(k::r) )I < E}) = 0 (2.1)
for every k E 7. By decreasing C, if necessary, we may assume that /I,(C) = l/L 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 assume without 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, 3;) = p(76> x)
for every (T”z,s) E RT. Then (1.2) implies that
f(z::c’) + f(x’,x”) = f(x::L.“)!
whenever (x,x’), (:IF:Ic”) E Rr.
(2.3)
(2.4)
We denote by [T] the full group of T, i.e. the group of all measure preserving automorphisms V of (X,S,p) with L’z E {Fx : r~ E Z} for every .r E X. Since T is ergodic we can find, for 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} is this set is nonempty,
0 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 : 71 E 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
9(x) = f(S%,x) = ~(rnC~(S);~) if xEC’.
0 otherwise.
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On joint recurrence
Furthermore, if f’(z) = f(x) + b(Tz) - b(z), and if f’(n,, .) : X + Rd and f’ : RT -+ R” are 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, :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 f and obtain
as in Lemma 3.4 in [4]. that
pI$rl~ (&Q?(rj + rj’)) - a;.(B(rj))) 2 0, yn~ (fqB(rj + rj’)) - @(B(rl))) L 0 (2.6)
for all 77, 77’ > 0. In particular, the inequalities (2.2) will be satisfied if
limsup 7-;(B(7/)) 5 L2”rfe-“! limsup 1 2”7&k(B(2-7L~dq)) 5 dL”2”+1~“~-” (2.7) k-+co ki3il 71=0
for every 71 > 0 and N > 1. The equations (2.1) and (2.5) yield that
C’ n T-“C’ n (3; E X : (1 f’(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:) = 71) ( = 1 {k E z : f’(S”:xr) = II} I = L
for every v E V, and CC E X\N, and that // 71 - 71’ 1) 2 E whenever ‘u, 7:’ E V, and 71 # 7)‘. Hence
1 (0 < I < k : 0 < /I f’(Z, Lx) (I 1. P”r,} 1 5 1 (0 < I 5 k : 0 < (1 f’(Z, ZTr) 1) 5 kl’“rj} (
< (kl’” + E/r,)d Y(X x B(rj))/Y(D) = (AY” + E/?7)dL2d?+-d,
since Sj+“D C X x B(k:l/“n + E), and since the sets S”“D, ‘rn E Z, are all disjoint, By integrating we obtain that
rQ(rj)) = ; 2 c((B(rj)) = ; -& p((2 E x : /I .f’(Z,x) )I 5 Zwj}) I=1 I=1
L 1 ‘r;+-. I’ 1 { 0 5 1 5 k : 0 < 1) f’(Z, x) 1) 5 P’“rj} 1 d/“(Z)
<;+ &“” + &/ rdd . l;&/dE-d
k
and by letting k -+ x, we have proved the first inequality in (2.7).
841
K. Schmidt
Similarly, one sees that
c I{ 0 < 1 5 2”k : 0 < 11 f’(L: x) 11 5 wi2-‘“‘%~} 1 n>o
= L . c [{7~ > 0 : ‘u = f’(Z. II:) for some 1 with 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:,
5L.E c ( 1 {n 2 0 : 1 < 2”k 5 F7/d/jV} I + 1)
.iZl l,Er;;n(E((j+l)E)\B(,E))
k’l” VI’
5 L”2d-ld. c f-1 log (v%d~d) log 2 + 1 < L”2d-1d. 4k@/& i=l
Hence
IV
c n=o
2” &#(2-“““7)) = 2 (4 + 2~‘~~.,,(X(2-~‘~“~~)\{0})) 5 (N ; l) L + Ldd2d+1vd/& 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 -+ R d 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)) I: ~1-0 k-m k
As N > 1 was arbitrary this proves (1.6). The inequality (I .5) is an immediate consequence of the first inequality in (2.2). 0
Acknowledgement. The author would like to thank the referee for pointing out reference [2].
References
[l] Burton R.M., Denker M., On the central limit theorem for dynamical systems, Trans. Amer. Math. Sot. 302 (1987) 715-726.
[2] Conze J.-P., Sur un critbre de rCcurrence en dimension 2 pour les marches stationnaires, Egod. Th. & Dynam. Sys. (to appear).
[3] Dekking F.M., On transience and recurrence of generalized random walks, Z. Wahrsch. Verw. Gebiete 61 (1982) 459465. [4] Schmidt K., On recurrence, 2. Wahrsch. Verw. Gebiete 68 (1984) 75-95.
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